Continuum does not exist
Here is the argument:
D) By continuum I mean a set of distinct points without an abrupt change or gap between points
A) Assume that continuum exists (assume that D is true)
P1) There is however either a gap between all pairs of points of the continuum or there is no gap
P2) We are dealing with the same point of the continuum if there is no gap between a pair of points
C1) Therefore there is a gap between all pairs of distinct points of the continuum (from P1 and P2)
C2) Therefore, the continuum does not exist (from A and C1)
To challenge the argument (basically to challenge P1) people claim that continuum is the real number. They argue that any finite interval on the real number line can be infinitely divided into smaller intervals. The real number, however, is constructed from two parts, an integer part and a decimal part. The number of digits of the decimal part however is the number of points of the natural number so-called the cardinality of the natural number (Aleph_0). It however can be shown that there is the smallest interval on the real number so-called infinitesimal. Infinitesimal can be constructed as follows: 0.0...01 by "..." I mean Aleph_0 0. Therefore, there is the smallest interval on the real number so there is a gap between each consecutive points of the real number.
D) By continuum I mean a set of distinct points without an abrupt change or gap between points
A) Assume that continuum exists (assume that D is true)
P1) There is however either a gap between all pairs of points of the continuum or there is no gap
P2) We are dealing with the same point of the continuum if there is no gap between a pair of points
C1) Therefore there is a gap between all pairs of distinct points of the continuum (from P1 and P2)
C2) Therefore, the continuum does not exist (from A and C1)
To challenge the argument (basically to challenge P1) people claim that continuum is the real number. They argue that any finite interval on the real number line can be infinitely divided into smaller intervals. The real number, however, is constructed from two parts, an integer part and a decimal part. The number of digits of the decimal part however is the number of points of the natural number so-called the cardinality of the natural number (Aleph_0). It however can be shown that there is the smallest interval on the real number so-called infinitesimal. Infinitesimal can be constructed as follows: 0.0...01 by "..." I mean Aleph_0 0. Therefore, there is the smallest interval on the real number so there is a gap between each consecutive points of the real number.
Comments (504)
A point is an abstract mathematical entity which doesn't correspond with any phenomenon in the world of our everyday existence. The same is true of a continuum.
The center of mass of your body is a point. The center of mass of your computer is a point as well. There is a distance between these two points. The question is whether this distance is discrete or continuous.
Quoting T Clark
Well, that is the subject of discussion.
A point does not exist in the everyday world. It is an abstraction, and idealization - imaginary. It has no size. It 's zero dimensional. It does not take up space. A center of gravity is a point and, as such, is also an abstraction, imaginary. And, as I noted, a continuum is also a mathematical idealization. It doesn't exist. It's imaginary.
A continuum exists in the same manner that a point does.
Are you claiming that something which is an abstraction cannot exist?
It exists in your mind, your imagination, but not in the physical world. I can imagine a point. I can also imagine a line, which is continuous.
I don't think we're getting anywhere. I'm going to leave it at that.
Ok!
As you wish.
C2 doesn't follow at all. In the real numbers, there being a gap between 4 and 13 does not imply that the real numbers (or even the rationals) is not a continuum. You need to demonstrate that there is nothing between some pairs of points that are not the same point. Then you've falsified the continuum premise.
Quoting T Clark
I disagree. Yes, a point can be an abstraction, but can also correspond to a location in space say.
It indeed does not 'take up space', by definition. You seem to imply that something real must take up space.
Perhaps it's all a linguistic quibble, but it isn't one that takes apart the OP argument.
Quoting MoK
Only in classical physics, and our universe isn't classical. But I accept your refutation of the rebuttal to the OP. Do you accept my rebuttal?
Even if we disagree, the OP still doesn't make sense. Whatever a point is, a line is the same kind of thing and a line is continuous by definition. A line is expressed as f(x) = mx + b, which means it is defined for any real number "x."
C1 states that there is a gap between all pairs of distinct points of the continuum.
Quoting noAxioms
Are you challenging (P1)? If yes, I already illustrated that given the definition of the real number one can construct the smallest number or the smallest interval so-called infinitesimal.
Quoting noAxioms
You can define it in quantum physics as well. Of course, you cannot measure it.
I don't disagree with that
Quoting MoKWhat do you mean by 'a gap'? If you mean that the two distinct points are not the same point, then yes, by definition. There's a gap between 4 and 13.
If you mean by 'a gap' that there is nothing between these two distinct points, then C1 is false. For instance, 10 is between 4 and 13. 'The gap' is not empty, and C2 seems to rely on any such gap being empty.
Without a definition of a gap, P1 is ambiguous. It states that either G or ~G, which is tautologically true, making P1 empty. The word 'distinct' is not part of P1.
My challenge was C2 following from C1, or any of what preceded C2.
Quoting MoKShow it then. What about the number that is halfway between this smallest positive number and zero? You've shown that it doesn't exist?
Funny that the smallest number happens to be a perfect power of 0.1
What are the odds of that?
Define it then, without making classical assumptions (like a particle having a location, or some counterfactual property.
By a gap, I mean an interval.
Quoting noAxioms
It shouldn't be.
Quoting noAxioms
Well, I construct the infinitesimal in this way: 0.0...01. By "..." I mean Aleph_0 zero. The next number is then 0.0...02 therefore there is a gap, 0.0...01 between these two numbers. One could say how about 0.0...011? It can be shown that 0.0...011 is 0.0...02 by simple math. 0.0...011=0.0...01+0.0....01. By "...." (where dots appears four times) I mean Aleph_0+1. But Aleph_0+1=Aleph_0 therefore 0.0...01+0.0....01=0.0...01+0.0...01=0.0...02.
Quoting noAxioms
This is off-topic but I give it a try. Consider a hydrogen atom for example. R is the center of mass position operator of the atom that is related to the position operator of the nucleus (r_n) and the position operator of the electron (r_e). The relation is R=(m_n*r_n+m_e*r_e)/(m_n+m_e) where m_n and m_e are the mass of the nucleus and electron respectively. The center of mass therefore can be calculated as
There are different notions of 'the continuum' and 'a continuum'. An ordinary mathematical notion is that the continuum is the set of real numbers along with the standard ordering of the real numbers; then a continuum is any set and ordering on that set that is isomorphic with the continuum.
But you say that you use the terminology to mean that "a continuum is a set of distinct points with no abrupt change of gap between the points".
So what are the definitions of "abrupt change" and "gap between"? But even more basically, what are your axioms?
You do say:
Quoting MoK
Quoting MoK
Between any two different real numbers there is an interval.
But the ordinary definition of continuum is not "there are no intervals between points". It is only by you personally redefining 'continuum' that you infer that there is no continuum.
One could as easily define 'is a mammal' by 'is an omnipotent animal' to then conclude that there are no mammals.
Quoting MoK
[x x] is the interval whose only member is x.
(x x) is the empty interval.
Quoting MoK
Of course, between any two different real numbers there is a non-empty interval.
Quoting MoK
We have:
If x < y, then (x y), (x y], [x y) and [x y] are non-empty intervals.
[x x] is a singleton interval.
From that it follows that there is an interval between any real numbers.
But it is only your ersatz definition of 'continuum' that leads you to infer that there is not a continuum from the fact that between any two points there is an interval. We might as well define 'is a computer' by 'can make infinitely many calculations in finite time' to conclude that there are no computers.
Quoting MoK
There are uncountably many real numbers. There is not just one real number that is to be called "the real number".
Maybe you mean that we say that the continuum is the set of real numbers. I would rather be more exact in saying that the continuum is the set of real numbers along with the standard ordering of the set of real numbers.
Quoting MoK
Again, there is not a real number that is called "the real number".
The usual constructions of real numbers are as either a Dedekind cut or an equivalence class of Cauchy sequences. Then we prove that every real number is representable as an integer along with a denumerable sequence (binary, decimal or whatever, as suits). And we prove that every integer along with such a sequence corresponds with a unique real number.
Quoting MoK
I guess you mean that there are denumerably many 0s followed by 1. There are rigorous treatments of nonstandard analysis in which there are infinitesimals. And there is the sequence ['w' for omega, the set of natural numbers] {
Thank you very much for your reply and fruitful input.
Let's start the discussion by defining the different gaps. Given a set of points, points of real number for example, I define G1 as an interval between two distinct points. I define G2 as an interval between two immediate points with no point between them (what I call an abrupt change in OP). I am interested in understanding whether there are gaps of type G2 in the set of real number given the definition of the set of real number which can be found here. The set of real can be defined as a set of numbers each number has an integer and decimal fraction. The decimal fraction contains an infinite number of digits, to be precise Aleph_0, where Aleph_0 is the cardinality of the natural number. To show that G2 exists in the set of real number I simply construct an infinitesimal_1 as follows: infinitesimal_1 = 0.0...01 where by "..." I mean an infinite number of zero to be more precise by the infinity in here I mean Aleph_0. I can show the second smallest number is 0.0...02 with no number between. Think of a number between, 0.0...011. This number can be divided into two parts 0.0...01 and 0.0...001. 0.0...001 however is 0.0...01 given the property of Aleph_0 in which Aleph_0+1=Aleph_0 therefore 0.0...011 is equal to 0.0...02.
I am sure that mathematicians define things in a more rigorous way when it comes to infinitesimal. I am interested in whether you can show that there is an infinitesimal in the set of real number given your definition.
There are provably no such G2 gaps in the real numbers. Between any two distinct real numbers, there is always another one strictly between them.
The real numbers are dense. That means that between any two distinct real numbers, there is a third one between them. For example between 5 and 7 we find 6. Between .001 and .002 we find .0015. And so forth.
In fact there's a formula to find the exact midpoint between two distinct reals. If [math]x[/math] and [math]y[/math] are distinct reals, then [math]\frac{x+y}{2}[/math] is halfway between them.
Also note that even the hyperreal numbers of the nonstandard reals, which have actual infinitesimals, are also dense. Between any two hyperreals there's another one distinct from those two.
There are no "adjacent points" in the real numbers. When you think of the real numbers, don't think of a string of bowling balls. Think of infinitely stretchy maple syrup or taffy. You can stretch and stretch but there are always more points.
Quoting MoK
This notation is meaningless. In decimal notation, each digit position to the right of the decimal point corresponds to a negative natural number power of 10. That is, starting from the decimal point and going right, the position values are 1/10, 1/100, 1/1000, and so forth.
There is not an "infinitieth" position. There is just no such thing in the notation. And that's not how the hyperreals work.
To emphasize this point, note that in even in the hyperreals, you can always divide by 2. So if .000...1 was a hyperreal infinitesimal, what is .000...1/2? You have no notation for that. In fact, .000...1 is meaningless in the reals and meaningless in the hyperreals.
And what if we divide by 10? You'd need an extra 0 in there somewhere ... but you can't add another 0 if you already have Aleph-0 of them. Make sense?
Forget .000...1. No such notation. Meaningless in every context, real or hyperreal.
My context here, unless mentioned otherwise, is ordinary mathematics, which is classical mathematics as found in calculus for the sciences, which is axiomatically formalized in set theory. I don't opine that that is the only context we should consider; only that when you talk about "real numbers", without qualifying that you don't mean other than the ordinary context, I surmise that you are talking about the ordinary context. And when I say "an object exists with property P" I mean that from the axioms we prove the theorem that we render in English as "There exists an x such that x has property P". I don't opine in this immediate context as to the philosophical aspects of such existence theorems.
We provide (with definitions from the primitives for membership (e) and identity (=), and from the axioms) such basics as set abstraction, subset, union, singleton, ordinal, function, sequence, finite, infinite, natural number, the set of all and only the natural numbers (w [read as 'omega']), denumerable, ordering and operations on the natural numbers, recursion, transfinite recursion, cardinal, aleph, irrational number, the set of all and only the rational numbers, ordering and operations on the rational numbers, Dedekind cut, equivalence class, and Cauchy sequence.
Most pertinently here, we define the property "is a real number", and we prove the existence of the set of all and only the real numbers (R), a particular ordering on that set (<), two particular elements (0 and 1), and two particular operations (+ and *). We also define "is a completed ordered field" and we prove that the system
And we define "the continuum" as R along with <. That is the ordered pair with R and <. So, the continuum is the set of real numbers considered vis-a-vis the standard ordering of the set of real numbers. Since we proved the existence of the set of all and only the real numbers and we proved the existence of the standard ordering on that set, we proved the existence of the ordered pair, and thus are entitled to name it, as we name it "the continuum". The continuum exists.
Usually "is a real number" is not rigorously defined by "is an integer followed by a decimal sequence". Rather, an integer followed by a decimal sequence is taken to be merely a representation of a real number.
The two most common rigorous definitions of "is a real number" are:
Df. r is a real number if and only if r is an equivalence class of Cauchy sequences
or
Df. r is a real number if and only if r is a Dedekind cut.
But we can define "is a real number" as a certain kind of sequence as long as we eschew infinitely many consecutive 9s:
Df. r is a real number
if and only if
r is a sequence whose domain is w and
r(0) is an integer and
for n>0, r(n) is in {0 1 2 3 4 5 6 7 8 9} and
for all n in w, there is an m in w such that n
So, r(0) is the integer part, then for each n>0, r(n) is the nth digit in the infinite expansion, and there are not infinitely many consecutive 9s.
They key point is that all three - the equivalence class of Cauchy sequences definition, the Dedekind cut definition, and the decimal sequence definition - all provide a complete ordered field, and all complete ordered fields are isomorphic with one another.
But what about infinitesimals? A possible definition:
i is an infinitesimal if and only if 0 < |i| and there is no real number strictly between 0 and i.
or
i is an infinitesimal if and only if 0 not= i and there is no real number strictly between 0 and i.
Then it is easy to prove that no real number is an infinitesimal.
But the fact that no real number is an infinitesimal doesn't prevent us from formulating another system in which there do exist infinitesimals.
Perhaps the three main two approaches to having a number system with infinitesimals are:
(1) In set theory, we use models. Thus we have two different number systems to use: The real number system and a different number system that has infinitesimals.
(2) In set theory, we use ultrafilters. Thus we have two different number systems to use: The real number system and a different number system that has infinitesimals.
(3) To set theory we add a primitive unary predicate "is standard" and additional axioms. In that theory we also have the real number system and a different number system with infinitesimals.
In any case, having a different number system doesn't disprove the existence of the other number system and doesn't disprove the existence of the continuum.
/
Your terminology about gaps conflates the predicate "is a gap" with the nouns "G1" and "G2".
Let's fix that with clear definitions of predicates:
Df. g is a G1-gap if and only if g is an interval
Df. g is a G2-gap if and only if g is an interval between two different points q and r such that there is no point strictly between q and r.
We prove that in the real number system there are no G2-gaps.
You propose to have 0.0...01 serve as an infinitesimal.
We could as easily write that as 0.0...1.
But two things:
(1) We need to be clear what 0.0...1 is.
(2) You need to define an ordering that has your supposed infinitesimal in the field of the ordering and to prove the needed theorems.
Regarding (1):
First:
A sequence is a function whose domain is an ordinal.
Every cardinal is an ordinal, but not every ordinal is a cardinal.
Ordinal addition is different from cardinal addition. But we use the same symbol '+' for ordinal addition and cardinal addition. So, we consider context to see whether '+' is being used for ordinal addition or cardinal addition. (And '+' for addition of real numbers is different also.)
So I'll use '+' for addition of real numbers, '#' for ordinal addition, and '+' [in bold] for cardinal addition.
The least infinite ordinal is w. w is also a cardinal, referred to as 'aleph_0' when we emphasize it as a cardinal.
{n | n is a natural number} = N = w = aleph_0 = card({n | n is a natural number}).
(aleph_0)+1 = aleph_0 = w.
w#1 not= w.
w#1 not= aleph_0.
Second:
Lets look at three things you've mentioned:
(a) 0.0...1
(b) 0.0...2
(c) 0.0...11
(a) and (b) represent sequences whose domain is w#1.
(c) represents a sequence whose domain is w#2.
Regarding (2):
You've not defined any such ordering.
Moreover, though you are free to have such things as (a), (b) and (c) in a number system that you may formulate, they do not represent real numbers as we have represented real numbers. And if they represent infinitesimals, then they are not in any number system isomorphic with the set of real numbers. So, your argument is profoundly ill-conceived. Even if you went on to fulfill your own formulation of a number system with infinitesimals, then that would not be the real number system nor isomorphic with the real number systems, thus it is ludicrous to say, as you do, that there is an interval in the reals between two different points q and r such there is no real strictly between q and r.
Moreover, your profoundly ill-conceived argument does not in any way support your titular assertion that the continuum does not exist.
That is not correct: Consider two numbers on the real number such as [math]a[/math] and [math]b[/math]. Let's define the mean as [math]c_1=\frac{a+b}{2}[/math]. We can determine the next mean as either [math]c_2=\frac{a+c_1}{2}[/math] or [math]c_2=\frac{c_1+b}{2}[/math] in which in the first case we approach to [math]a[/math] from the right and in the second case we approach to [math]b[/math] from the left. Let's work with the first approach: [math]c_2=\frac{a+c_1}{2}[/math]. The next mean can then be determined by [math]c_3=\frac{a+c_2}{2}[/math]. We can write [math]c_{i+1}=\frac{a+c_i}{2}[/math]. The distance between [math]c_{i+1}[/math] and [math]a[/math] is [math]d_{i+1}=c_{i+1}-a=c_i-c_{i+1}=\frac{(2^i-1)*a+b}{2^i}-\frac{(2^{i+1}-1)*a+b}{2^{i+1}}[/math]. Therefore, we have [math]d_{i+1}=\frac{2*((2^i-1)*a+b)-((2^{i+1}-1)*a+b)}{2^{i+1}}=\frac{b-a}{2^{i+1}}[/math]. So, [math]d_{\text{Aleph}_1}=\frac{b-a}{2^{\text{Aleph}_1}}<\frac{\text{Aleph}_1}{2^{\text{Aleph}_1}}=0[/math]. Therefore, your statement does not follow.
I haven't kept up with this nonsense. Kudos to those knowledgeable who have.
[math]\frac{\aleph_1}{2^{\aleph_1}}[/math]?
The burden is on you to justify this notation. Perhaps you can begin with your theory of cardinal division. You'll have a hard time making sense of it. Trust me, if there were any mathematical theory that justifies this notation, I'd have heard of it. I haven't and there isn't.
BUT! Your underlying idea of continually subdividing an interval is correct. It just doesn't show what you think it does.
If as @jgill notes you are simply proving that you can find a sequences approaching [math]a[/math] and [math]b[/math] as limits, that just amounts to a restatement of the fact that the real numbers are dense: that between any two, there's a third. An equivalent condition is that we can find a sequence approaching as a limit any given real number.
And the equivalence is shown by continually taking the midpoint, as you did. But your notation is fanciful and undefined. Your basic idea is correct, but it does not show that there are two reals without a third between them. On the contrary, it shows that if between any two reals there's a third between them, then we can find a sequence approaching as a limit any given real.
Give this a read.
https://en.wikipedia.org/wiki/Dense_set
ps -- How did you get to [math]\aleph_1[/math] as the limit of 1, 2, 3, 4, ...? Did you mean perhaps [math]\aleph_0[/math]?
In summary, your idea is correct, even if your notation isn't. But your idea only shows that the two definitions of a dense set coincide. If you can always find the midpoint between two points, then you can always find sequences converging to each of the two points.
Well, I showed that the distance between consecutive means is zero if the number of divisions is [math]\aleph_1[/math].
Quoting jgill
Well, that is the division of two cardinal numbers. I googled about the division of cardinal numbers and I found two references here.
Quoting jgill
Ok, I see what you mean.
I googled and I found two references about the division of cardinal numbers. You can find the references here.
Thank you very much for your post. It made many things clear to me after I read it a few times. I am still not convinced that infinitesimal does not exist though. Can you prove it? I however can show that the distance between consecutive means tends to zero. You can find my argument here. Please let me know what do you think.
ORDERED FIELD
x+(y+z) = (x+y)+z (associativity of addition)
x+y = y+x (commutativity of addition)
EyAx x+y = x (additive identity element)
Theorem: E!yAx x+y = x
Definition: 0 = the unique y such Ax x+y = x
Ey x+y = 0 (additive inverse)
EyAx x*y = x (multiplicative identity element)
Theorem: E!yAx x*y = x
Definition: 1 = the unique y such that Ax x*y = x
0 not= 1
x*y = y*x (commutativity of multiplication)
x*(y*z) = (x*y)*z (associativity of multiplication)
x*(y+z) = (x*y)+(x*z) (distributivity)
x not= 0 -> Ey x*y = 1 (multiplicative inverse)
(x < y & y < z) -> x < z (transitivity)
exactly one: x < y, y < x, x = y (trichotomy)
x < y -> x+z < y+z (monotonicity of addition)
(0 < z & x < y) -> x*z < y*z (monotonicity of multiplication)
COMPLETE ORDERED FIELD
In set theory, we prove that there is a carrier set (called 'R') for such a system and such that, for any upper bounded non-empty subset of S of R, S has a least upper bound. With that and the rest of the set theory axioms we can do the mathematics of derivatives and integrals for the sciences.
/
An alternative is to have a system with infinitesimals. But still, ordinarily, we need to define <, + and * and to prove whatever theorems are needed for the machinery of mathematics.
To just wave a hand and say "Voila, this is my infinitesimal" does not provide the needed definitions of < + and * with infinitesimals nor the needed proofs.
So how do we go about proving the existence of a system with infinitesimals? For your purposes, it would help to first define 'is an infinitesimal'. I provided a definition previously, but I notice that many authors include 0 as an infinitesimal. So perhaps use this definition:
x is an infinitesimal if and only if, for every positive real number y, |x| < y.
It has been proven for you that for every real number x there is a positive real number y such that y < |x|.
So no non-zero real number is an infinitesimal.
One more time: No non-zero real number is an infinitesimal. The proof that no non-zero real number is an infinitesimal is immediate from the fact that for every real number x there is a positive real number y such that y < |x|. We don't need to keep going over this over and over.
I mentioned previously different ways of proving the existence of systems with infinitesimals.
So why would you ask?:
Quoting MoK
I can't prove it, since it is not provable. I never said that there are not infinitesimals. I explicitly said that there are systems with infinitesimals and I even mentioned different ways of developing them. But no non-zero real number is an infinitesimal, which has been proven to you over and over. That you are still asking about this suggests that you are not paying attention to the posts.
(I mentioned previously using the compactness theorem for having a system with infinitesimals. But an interesting thing about that is that we prove the existence of such a system but (unlike with the reals) we cannot define a particular such system: The compactness theorem for uncountable languages relies on Zorn's lemma. Same problem with the ultrafilter method.)
Quoting MoK
That source itself points out that when the numerator is less than the denominator, there is no definition of numerator/denominator. The very source you point to disputes your claim that (aleph_1)/(2^aleph_1) is properly defined. And you should have proven that for yourself when you first thought of it [here 'X' stands for the Cartesian product]:
K*L = card(K X L). And we have the theorem that if L <= K and K is infinite and L is non-zero, then K*L = K.
The definition of x/y:
x/y = the unique z such that z*y = x. If there is no such unique z, then x/y is not properly defined.
Suppose here that L <= K and K is infinite and L is non-zero:
If L = K, then there are Z such that Z*L = K, but there is no unique such Z. So L/K is not properly defined.
If L < K, then the unique Z such that Z*K = L is L. So L/K = L.
If K/L were properly defined, then K/L would be the unique Z such that Z*L = K. But there is no such Z at all, let alone a unique one. So K/L is not properly defined.
You did an Internet search but didn't even bother to read what you found, instead recommending that other people read it even though, unknown to you since you didn't even read it, it says the exact opposite of your claim that you made up out of thin air.
"(aleph_1)/(2^aleph_1)" is pure bunk.
Quoting MoK
The sequence of half distances converges to 0. So what? That doesn't prove that it's not the case that between any two different real numbers there is another different real number.
Quoting MoK
"number of divisions is aleph_1" is undefined.
I recommend that you learn how axiomatic mathematics and definitions work
But the definition isn't constructive and is extensionally unintelligible for some of the reasons you pointed out in the OP. Notably, Dedekind didn't believe in the reality of cuts of the continuum at irrational numbers and only in the completeness of the uninterpreted formal definition of a cut. Furthermore, Weyl, Brouwer, Poincare and Peirce all objected to discrete conceptions of the continuum that attempted to derive continuity from discreteness. For those mathematicians and philosophers, the meaning of "continuum" cannot be represented by the modern definition that is in terms of connected and compact sets of dimensionless points. E.g, Peirce thought that there shouldn't be an upper bound on the number of points that a continuum can be said to divide into, whereas for Brouwer the continuum referred not to a set of ideal points, but to a linearly ordered set of potentially infinite but empirically meaningful choice sequences that can never be finished.
The classical continuum is unredeemable, in that weakening the definition of the reals to allow infinitesimals by removing the second-order least-upper bound principle, does not help if the underlying first-order logic remains classical, since it leads to the same paradoxes of continuity appearing at the level of infinitesimals, resulting in the need for infinitesimal infinitesimals and so on, ad infinitum.... whatever model of the axioms is chosen.
Alternatively, allowing points to have positions that are undecidable, resolves, or rather dissolves, the problem of 'gaps' existing between dimensionless points, in that it is no longer generally the case that points are either separated or not separated, meaning that most of the constructively valid cuts of the continuum occur at imprecise locations for which meta-mathematical extensional antimonies cannot be derived.
Nevertheless this constructively valid subset of the classical continuum remains extensionally uninterpretable, for when cut at any location with a decidable value, we still end up with a standard Dedekind Cut such as (-Inf,0) | [0,Inf) , in which all and only the real numbers less than 0 belong to the left fragment, and with all and only the real numbers equal or greater than 0 belonging to the right fragment, which illustrates that a decidable cut isn't located at any real valued position on the continuum. Ultimately it is this inability of the classical continuum to represent the location of a decidable cut, that is referred to when saying that the volume of a point has "Lebesgue measure zero". And so it is tempting to introduce infinitesimals so that points can have infinitesimal non-zero volume, with their associated cuts located infinitesimally close to the location of a real number.
The cheapest way to allow new locations for cuts is to axiomatize a new infinitesimal directly, that is defined to be non-zero but smaller in magnitude than every real number and whose square equals 0, as is done in smooth infinitesimal analysis, whose resulting continuum behaves much nicer than the classical continuum for purposes of analysis, even if the infinitesimal isn't extensionally meaningful. The resulting smooth continuum at least enforces that every function and its derivatives at every order is continuous, meaning that the continuum is geometrically much better behaved than the classical continuum that allows pathological functions on its domain that are discontinuous, as well as being geometrically better behaved than Brouwer's intuitionistic continuum that in any case is only supposed to be a model of temporal intuition rather than of spatial intuition, which only enforces functions to have uniform continuity.
The most straightforward way of getting an extensionally meaningful continuum such as a one dimensional line, is to define it directly in terms of a point-free topology, in an analogous manner to Dedekind's approach, but without demanding that it has enough cuts to be a model of the classical continuum. E.g, one can simply define a "line" as referring to a filter, so as to ensure that a line can never be divided an absolutely infinite number of times into lines of zero length, and conversely, one can define a collection of "points" as referring to an ideal, so as to ensure that a union of points can never be grown for an absolutely infinite amount of time into having a volume equaling that of the smallest line. This way, lines and points can be kept apart without either being definable in terms of the other, so that one never arrives at the antimonies you raised above.
Hmmm. Care to explain? (I recall having difficulty with filters, ultra filters, etc. in grad school a half century ago. I only encountered them in passing - not in my specialty area)
In set theory, we prove the existence of a particular complete ordered field and that it is unique within isomorphism. I didn't adduce any reasons that that is not constructive nor even say that it is not constructive.
Quoting sime
What passage in what paper by Dedekind are you referring to?
Quoting sime
I didn't mention any antinomies.
I'm not sure what @sime meant by that statement either. But ultrafilters are just a set theory gadget that lets you rigorously construct the hyperreals of nonstandard analysis.
Quoting TonesInDeepFreeze
Ok, thanks for the elaboration. I got that.
Quoting TonesInDeepFreeze
I looked at all your posts and didn't find the proof that no non-zero real number is an infinitesimal. Could you please provide the proof?
Quoting TonesInDeepFreeze
Quoting fishfry
This was a reply to the above comment from @fishfry who claimed between any two distinct real numbers, there is always another one strictly between. The distance between two points is zero if the number of divisions is strictly infinite so there cannot be a point between two points in this case.
Perhaps this post help you. What I showed in that post is that the distance between consecutive means tends to zero for large [math]i[/math]. The last step of my argument seems to be problematic because of the way the division is defined between cardinal numbers. You don't need to worry about the division of the cardinal numbers though since I don't need that step. All I need is to show that the distance between consecutive means is [math]d_i=\frac{(b-a)}{2^i}[/math]. This distance is zero when [math]i[/math] is strictly infinite. I am currently trying to make sense of this post though and I have to say it is very technical.
You are correct in your observation. If the distance between two points is zero then we are dealing with the same point. This means that there cannot be a point between points since they are the same point. This is against the argument that there is always a point between two points on the real number. What I showed is that the distance between consecutive "means" (by "mean" I mean the point between two points) tends to zero. I hope things are clear now. If not please let me know so I would elaborate.
Clearly, you're not reading.
I gave you a proof in the very post to which you have now replied:
Quoting TonesInDeepFreeze
You need it spelled out for you again?
Definition: x is an infinitesimal if and only if, for every positive real number y, |x| < y.
Theorem: No non-negative real number is an infinitesimal.
Proof: Suppose x is a non-negative real number. Since x is a non-negative real number, |x|/2 is a positive real number and |x|/2| < |x|. So it is not the case that |x| < |x|/2|. So it is not the case that for every positive real number y, |x| < y. So x is not an infinitesimal. So no non-negative real number is an infinitesimal.
Quoting MoK
You skip that I addressed that. There is no operation of infinite division in the real number system. There are infinite sequences of divided results, and the one you have in mind converges to 0. It is very crank (confused, ignorant and fallacious) to conflate the limit of a sequence with an out-of-thin-air claim of an operation of infinite division. And the fact that the limit of the sequence is 0 does not refute that between any two distinct real numbers there is real number strictly between them. Indeed, the convergence to 0 itself depends on the fact that between any two distinct real numbers there is a real number strictly between them.
Quoting TonesInDeepFreeze
This is incoherent at best, wrong at worst. I explained this to you at length. But look. You are trying to prove there are two points without a third between them, by claiming there are two points at a distance of zero. Can you see the circularity of your argument?
I explained this to you at length in a post you didn't bother to engage with.
Quoting MoK
I believe you. Physicists attempting to do math are often a source of humor and/or horror to mathematicians. But I'm sure you know that :-)
Honestly. I explained this to you at length. What you've worked out for yourself is the equivalence of the two definitions of a dense set: (1) That there is third point between any two; and (2) That there is a sequence of distinct points approaching any given point as a limit. You proved that with your bisection idea. Your idea is essentially correct. Your intuition about what it means is mathematically wrong.
Quoting TonesInDeepFreeze
What Tones said.
Don't feel bad. I"m a very old retired mathematician and have had to look up filter trying to understand @sime 's comments.
So if no real number is an infinitesimal, numbers are then what is relation to geometry. Is 2 then 2 points, or are all numbers a point?
According to Wki both Cauchy (in Cours d'Analyse) and Edwars Nelson also compared infinite points to the numberline. Long before hyperreals i believe. The great writer and philosopher George Berkeley rejected infinitesimals on both mathematical and philosophical grounds
What about imaginary numbers, however? Stephen Hawking, in his attempt at find the wave function of the universe, proposed his (yep) No Boundary Proposal in 1983. I like to apply this "theorem" to consciousness. Hawking uses imaginary numbers to describe time as it goes backwards, behind the Big Bang. How are we to understand mathematically a state not having any boundaries? There is always a "here" and "there" in our experience. That is, except in consciousness wherein we can go deeper and deeper and we find no edge. The "limit" seems to be death, but in our experience we are infinite. Hence we can think about infinities..
Note: if the world is a hologram, then it is proven there is a "thing-in-itself".
The kalam cosmological argument gives a great example of infinities embedded in another. The argument fails in its purpose because eternity, an infinity, contains all steps of infinity. There can be that infinity if there is the eternity. QED?
Just some philosophy and context for this forum
If i cut a cake horizontally starting from the halfway point upwards with each slice being half the size of the one immediately below, what would the top of the cake look like? Isn't it indefinite? But you can definitely look at the cake, from all angles, and see that it has definite position in relation to its parts. So how do we reconcile the indefinite with the definite? I think this is what must be asked about the continuum. Hawking would say that four dimensional Euclidean space, with a time dimension that both 1) acts as space, and 2) is described by imaginary numbers, gives an answer to this question. That is to say, the universe as a whole gives the answer to the continuum. But how do imaginary numbers relate to geometry?
I consider the number of the operation (by operation I mean dividing by two) to be strictly infinite. @TonesInDeepFreeze however claims that such an operation does not exist in the real number system: "There is no operation of infinite division in the real number system. There are infinite sequences of divided results, and the one you have in mind converges to 0."
I am however confused by this statement. How could you have an infinite sequence of divided results without infinite division!?
You didn't provide this argument before. Did you? You just defined infinitesimal!
Quoting TonesInDeepFreeze
How could you have an infinite sequence of divided results without infinite division operations?
Quoting TonesInDeepFreeze
I was not confusing these. In fact, I mentioned the number of division operations to be strictly infinite.
Quoting TonesInDeepFreeze
That is understandable but I was not arguing against that. I just argued that if the number of divisions is strictly infinite then you cannot get anything new by dividing the result further since the result is zero. I now know that the number of division operations cannot be infinite in the real number system. I don't know why!
It seems that there is no operation of infinite division in the real number system. That was something I didn't know.
Quoting fishfry
Oh yeah, I can guess that. We, physicists, work with the infinities all the time. Of course, mathematicians do not agree with how we deal with infinities but strangely physics works. :)
Quoting fishfry
I am not confusing the two.
I am not feeling bad. At worst I am wrong and learn a new thing. At best I am right so others learn a new thing. Thank you for your support anyway. :)
Yes, there is no infinitesimal in the real number system. I don't understand the rest of your comment.
Quoting Gregory
I didn't read Berkeley at all so I don't know what he is arguing about. Is he arguing that there is no infinitesimal in the real number system or he is arguing that there is no infinitesimal in any mathematical system? According to @TonesInDeepFreeze there are mathematical systems with infinitesimal.
Quoting Gregory
I don't understand what you are trying to argue here.
Quoting Gregory
The Kalam cosmological argument states that there cannot be an infinite number of past events therefore there is a beginning. The rest of the argument is about proof of the existence of God which I don't agree with.
The top of the cake looks like the top of the cake no matter how many times you divide the cake horizontally.
Quoting Gregory
It is not.
Quoting Gregory
I don't understand what you are talking about.
Quoting Gregory
We are not talking about space and time here. Whether space and time are continuous or not is the subject of other threads.
Ok, I try my best to answer your questions. Perhaps, others (mathematicians @TonesInDeepFreeze, @fishery, and @jgill) would participate and answer your questions in a simple manner.
Quoting tim wood
No, I am not confusing the ideas of the number and the limit.
Quoting tim wood
There is no last digit. The square root of two is an irrational number. The Irrational set, the set of all irrational numbers, is a subset of the real number set. Almost most of the real numbers do not have the last digit (the number of digits is infinite).
Quoting tim wood
Bigger than any countable number.
I dont think that a real number can't be divided infinitely. The area of a circle is pi-r-squared wherein pi represents an aspect of space (the area). Each decimal would be a tiny and tinier slice of space and this goes on forever. So the space represents the number as we visualize it and the the number represents the space. Infinity is in both.
As for Hawking, physical explanations shed light on mathematical concepts just as the reverse is true.
"But nobody in that century or the next could adequately explain what an infinitesimal was. Newton had called them 'evanescent divisible quantities,' whatever that meant. Leibniz called them 'vanishingly small,' but that was just as vague.. Pierre Bayle's 1696 article on Zeno drew the skeptical conclusion that, for the reasons given by Zeno, the concept of space is contradictory."
Internet Encyclopedia of Philosophy
If there are no infinitesimals, than an infinity of zeros can equal anything. Does this mean that 0×infinity=everything? But an argument against infinitesimals and discreteness is that space by definition is that which is divisible. How can there be something in-between space and a point? Where do we even begin with a continuum? (At least Banach-Tarski's paradox makes more sense in this context) Geometric objects seem to be in themselves the opposite of Gabriel's horn. Instead of an infinite surface area for a finite volume we seem to have in the continuum an infinity of space bounded by finite (beginning and end) space
More latter..
I don't understand your objection to my comments. I used "strictly" when I wanted to see what is the result of an interval divided infinite number of times by two. Mathematicians think that there is no operation of infinite division in the real number system. I don't know why and I asked @TonesInDeepFreeze for an explanation. Perhaps he can answer my question in a simple term so you can understand as well.
An operation symbol takes only finitely many arguments.
For example, the operation of division (x/y) takes only two arguments (x and y), and is defined accordingly:
Df. If y not= 0, then x/y = the unique z such z*y = x.
Every operation symbol, whether, primitive or defined, takes only finitely many arguments. The reason is that every formula is finite in length, so every term is finite in length, so no operation symbol can take infinitely many arguments.
So we have division, which is a binary operation. But we also prove that for every positive real number r, there exists a function f whose domain is the set of positive natural numbers and such that, for every positive natural number n:
f(1) = r
f(n+1) = f(n)/2
f is a function from the set of positive natural numbers into the set of real numbers.
Notice that it is trivial to prove that for no n is it the case that f(n) = 0.
Then we prove:
There exists a unique real number x such that for every positive real number y, there exists a positive natural number n such that |f(n) - x| < y.
Then we write:
lim[n = 1 to inf] f(n)
'lim' is a variable binding operator, but it can be reduced to a regular operation symbol:
Df. If g is a function from the set of positive natural numbers into the set of real numbers, and there exists a unique real number x such that for every positive real number y, there exists a positive natural number n such that |g(n) - x| < y, then Lg = the unique real number x such that for every positive real number y, there exists a positive natural number n such that |g(n) - x| < y
You see that 'L' is an operation symbol that takes only finitely many arguments - in this case, one argument.
The argument itself is an infinite set (an infinite sequence in this case), which is okay, because the operation symbol takes only finitely many arguments - in this case, one argument.
And we prove, regarding the function f we previously defined:
Lf = 0
In everyday parlance, "the limit of f is 0" and for no n is it the case that f(n) = 0.
There's no operation of infinite division (no operation takes infinitely many arguments), but there are infinite sequences that converge to 0, defined by, for example, halving each previous entry in the sequence. But if we call taking the limit of such a sequence f "infinite division" then that's okay as long as we are clear that that is what we mean and not some other undefined notion and that we recognize that for no n is it the case that f(n) = 0.
Moreover, the point is sustained that the fact that f converges to 0 does not refute that no non-zero real number is an infinitesimal.
That is blatantly false and with an exclamation mark that is a cherry on top of falsehood. I wrote:
Quoting TonesInDeepFreeze
This latest time I just made that proof even more explicit. That proof essentially had been given you at least a few times by other posters, but you still didn't understand (chose not to understand?). So I gave it again for you. And then you still skipped it and asked me to give a proof even though I already had. And so I gave it to you even more explicitly. And that explicitness is itself more than you would need, as the matter is so extremely simple to begin with:
No non-zero real is an infinitesimal, since, unlike an infinitesimal, every non-zero real is such that there is real between it and 0.
/
And still curious whether you understand now that "aleph_1/(2^aleph_1)" is nonsense.
Don't take my word for it. Look up 'non-standard analysis', 'hyperreal', 'infinitesimal', 'internal set theory'.
Better, read Enderton's beautifully written 'A Mathematical Introduction To Logic' in which he has a wonderfully clear and concise section on non-standard analysis. (It's been a while since I studied it, so I might not be able to immediately answer all questions about the details. Anyway, to get up to the section on non-standard analysis, one needs to first comprehend the material leading up to, starting from the first page, which is the best favor anyone could do for oneself if one were sincerely interested in topics such as this one.)
Greater than any countable number or greater than any finite number?
I have the first edition in print. I don't have an errata sheet for it. But I do have pencil marks for the errata I caught.
There is an errata sheet for the second edition; I think it might be online too.
Anyway, the second edition online is gold, free for the taking.
The course I recommend is, in order:
(1) Logic: Techniques of Formal Reasoning 2nd ed. - Kalish, Montague and Mar
or either of these:
Introduction To Logic - Suppes
Elementary Logic - Mates
(Lately I've been thinking that Suppes is the best choice, especially for its treatment of the subject of definitions.)
(2) Elements of Set Theory - Enderton (the errata sheet might be online)
or
Axiomatic Set Theory - Suppes
(3) A Mathematical Introduction to Logic - Enderton
(4) Introduction to Mathematical Logic (just for the Introduction chapter) - Church (the Introduction chapter is the best overview of the primary considerations I've found)
For a book with both classical and intuitionistic logic:
Logic and Structure - van Dalen
For an overview of many alternative logics:
An Introduction to Non-Classical Logic - Priest
/
And there are so many other great books, especially Smullyan's books 'First-Order Logic' and 'Godel's Incompleteness Theorems' (Smullyan writes so beautifully and his formulations are so clever and elegant).
Two great tomes:
Mathematical Logic - Monk
Fundamentals of Mathematical Logic - Hinman
I mean, they really are tomes. And they are remarkably rigorous with notation and extensive with details as you're likely to find, especially Hinman which goes the whole nine yards in the way it makes explicit which symbols are in the object language and which are in the meta-language.
And more advanced books:
Model Theory - Chang & Keisler (a tome and the OG of model theory textbooks)
Set Theory - Jech (a tome)
Set Theory - Kunen (I like Kunen's "philosophical/heuristic" framework that tends toward "formalism")
These are all beauties.
Is f(infinity) a member of the above sequence? If yes, what is its value? If not, how could the sequence be an infinite one?
Quoting TonesInDeepFreeze
Let's wait for your answer to the previous questions.
Yes, I got that. Thank you very much for your explanation.
That is a correct interpretation if you divide the interval by two, then another time divide the result by two, ad infinitum. What I have in mind is that I simply divide the interval by 2^infinity in one step. This operation seems to be invalid though to mathematicians.
It is better to say greater than any finite number given the definition of a countable set in mathematics.
What do you mean by "infinity" used as a noun?
There is the adjective "is infinite": S is infinite if and only if S is not finite.
And there are various infinite sets, such as:
the least infinite ordinal = {n | n is a natural number} = w = aleph_0
the least infinite cardinal = aleph_0 = w = {n | n is a natural number}
the least infinite cardinal greater than aleph_0 = aleph_1
card(set of functions from w into 2) = 2^aleph_0 = card(the power set of aleph_0)
card(set of functions from aleph_1 into 2) = 2^aleph_1 = card(the power set of aleph_1)
There are various uses of "infinity" as a noun, including such things as infinity as a point in the extended reals or infinity as a hyperreal in nonstandard analysis. But you've not specified.
In any case, how can you ask your question when I explicitly defined the domain of f to be the set of positive natural numbers? (Whatever you mean by "infinity" used as a noun, it is not a member of the set of positive natural numbers.)
Quoting MoK
How can you ask that question when I defined the domain of f to be the set of positive natural numbers? (The set of positive natural numbers is infinite, and if the domain of a relation is infinite then the relation is infinite.)
You really need to learn basic mathematics.
(Also: A sequence is a certain kind of function, and a function is a certain kind of relation, and a relation is set of ordered pairs. So, f(n) is not a member of the sequence f; rather
Quoting MoK
Actually, No:
Quoting MoK
(1) What do you mean by "divide the interval"? At least four options:
(a) Partition the interval
(b) Divide the cardinality of the interval
(c) Divide the distance of the interval
(d) Deploy an operation that divides, then divides the result ... infinitely many times, as a single combined operation that has a final value
Option (d) is meaningless. You are trying to argue against the proof that for every positive real number x (= the distance of the interval [0 x]) there is a positive real number y such that 0 < y < x. Your confused, ill-premised, ignorant and incoherent idea is that by dividing the interval 2^aleph_1 number of times, we get an i such that for any real number y>0, we have 0 < i < y.
But you have not defined such an operation. All you've done now is to claim you understand my explanation of cardinal division while going right past it to repeat your same mistake, but with even greater vagueness by insisting on an undefined "infinity" as a divisor.
You're not only not making progress, but you're going in the opposite direction of progress.
You really really need to learn basic mathematics.
Quoting MoK
Not a question of "validity". It's just that it is not defined by you. Your method is to throw around mathematical terminology without understanding it, thus to combine it in ways that are not coherent.
You really really need to learn some logic, set theory and the mathematics of standard analysis and non-standard analysis.
Quoting MoK
Then just say "finite". But your particulars involve both countable and uncountable cardinals, all of which are greater than any finite cardinal but some of which are greater than any countable cardinal:
aleph_0 is countable
aleph_1 is uncountable
2^(aleph_1) is uncountable and greater than aleph_1
As you touch on, the system of naturals can be extended to a system of rationals, and the system of rationals can be extended to the system of reals. And each is a different system. And the system of reals can be extended to a system with infinitesimals. And that is a different system. But when we extend in that way, we do it coherently. We prove the existence of the systems - the sets of numbers and the key operations and relations and the key properties of those). But @MoK's own offering is not coherent - does not proceed by coherent definitions and proofs. He needs to learn at least the basics of the real number system, then he can look up 'non-classical analysis' to see how an actually coherent and rigorous development of infinitesimals is done.
Prove the existence of a system of naturals, and define a particular system of naturals.
Prove the existence of a system of integers, as there is a subsystem isomorphic with the system of naturals, and define a particular system of integers.
Prove the existence of a system of rationals, as there is a subsystem isomorphic with the system of integers, and define a particular system of integers.
Prove the existence of a system of reals, as there is a subsystem isomorphic with the system of rationals, and define a particular system of reals.
Prove the existence of a system of hyperreals, as there is a subsystem isomorphic with the system of reals, but we cannot define a particular system of hyperreals (at least not with the method of compactness or the method of ultrafilters).
There are limits. As an example, consider the sequence 1/2, 1/4, 1/8, 1/16, ...
We all know that the limit of this sequence is 0. You can certainly call this infinite division if you like, as long as you understand what limits are. There are indeed infinitely many elements of the sequence, and you CAN think of this as "infinite division."
You can even think of doing it "all at once" if you like
What it means formally is that the elements of the sequence get (and stay) arbitrarily close to 0.
What it does NOT mean is that there is some kind of magic number that the sequence attains that is a "distance of 0" from 0, but is not 0. That's a faulty intuition.
In fact the formal theory of limits, once one learns it, is the antidote to all our non-rigorous, faulty intuitions about infinite processes.
Does that help, or perhaps refresh your memory? I'm pretty sure that physicists must be exposed to the formal theory of limits at some point.
Quoting MoK
Yes I understand and agree. But I am a little surprised that you seem to think that a sequence attains some kind of mysterious conclusion that lies at a distance of 0 from its limit, but is distinct from the limit. That's just not right.
Did my mention of limits ring a bell at all? Or raise any issues that we could clarify or focus on?
Because your idea of endless division is perfectly correct. But all that shows is that endlessly halving leads you to the limit of a sequence. But there's no "extra point" in there that's distinct from but at a distance of 0 from the limit.
Let me know if we're on the same page about this.
The continuum is a mathematical abstraction. It has no representation or instantiation (as far as we know) to anything in the physical world.
This is all explained by the mathematical theory of limits. The sequence 1/2, 3/4, 7/8, 15/16, ... is an infinite sequence that has the limit 1. The "top of the cake" is the limit of the sequence. In terms of a cylinder, the very top would be a circular disk of zero thickness, the same as any other horizontal slice. That is, the intersection of a cylinder with a horizontal plane (parallel to the top and bottom of the cylinder) is a circular disk.
It can be confusing to think about cakes, because cakes are made of atoms; and the Planck limits preclude our making fine enough horizontal slices to produce a zero-thickness slice. Cakes are not cylinders, and Math [math]\neq[/math] Physics!
Quoting Gregory
There is no evidence that anything in the physical universe is a mathematical continuum. It's possible that there is, but this is a deep open question that (with respect to current science) is more philosophical than scientific.
How imaginary (perhaps you mean complex) numbers related to geometry is a pretty cool subject, but far afield from understanding the nature of limits.
Here's the coolest example I know of how complex numbers relate to the geometry of the plane.
Take a regular n-gon (triangle, square, pentagon, hexagon, heptagon, octagon, etc.)
Place it in the complex plane such that its center corresponds to the origin of the plane, the complex number 0; with one of the vertices as the point (1,0) in the plane, or the complex number 1.
Now the vertices of the n-gon are exactly all complex n-th roots of 1; that is, they are all the complex solutions to the equation [math]z^n = 1[/math].
As a concrete example, consider a regular 4-gon, or square, with one vertex at the point 1 in the complex plane. Where are the other vertices? At [math]i[/math], [math]-1[/math], and [math]-i[/math]. And these are exactly the four complex numbers whose fourth power is 1.
And this works for any regular n-gon. The vertices of the regular 17-gon are the seventeen 17th complex roots of 1.
That's one of the coolest things I know. And it is one example of the deep relation between complex numbers and geometry, which is the question you asked. But it's got nothing to do with the definition of limits or the nature of the mathematical continuum, but it's definitely interesting. I only mentioned it since you asked about the relationship of complex numbers to geometry.
Here's some Wikitude on the subject.
https://en.wikipedia.org/wiki/Root_of_unity
None of this has ANYTHING to do with relativity or spacetime or the physical world. Physicists use the mathematical continuum to model spacetime, but there is no evidence whatsoever that time or space or spacetime are literally the same as a mathematical continuum. They use the mathematical continuum as an approximation that seems to work, to the limit of our ability to measure the results of our experiments.
Remember: Math [math]\neq[/math] Physics!
Thanks for the superb reply. The reason i brought up Hawking's "no boundary" thesis is that i was thinking maybe geometry and limits are incomplete by themselves and need the 4 spatial dimensions and 1 time dimension in order to make sense of it. That is, mathematicians assume math can stand on its own, but maybe it can't. However i also now see how the physics can be in trouble where maybe the math isn't. You've explained with the cylinder how the top of it is the limit such that if i metaphorically touch it with my finger i am touching a point limit. However if we bring in time and do the series Zenonian as i proposed (one at a time), and with each new slisce changed the color of the new slice, i can ask "what color" the top of the cylander would be. This causes a problem *because* it is a process and processes aren't used like that in mathematics. But again, Hawking had the thesis from the 80's that 5 dimensions (4 spacial Euclidean ones and 1 temporal one that acts as space and uses "imaginary time" as he says) wherein there is no before of time (as there is no north of the North Pole) express a hologram such that 2 dimensions are projected from the 5 dimensions infinitely far away. I know it's unorthodox, but why can't this been seen purely from it's mathematical side and brought into mathematics itself? Hawking explained away indeterminacy with this idea.The lines here seem rather blurry to me, but i read of mathematics mostly from the historical perspective, although i started working through a discrete mathematics textboom recently.
I will be thinking about your reply and other posts on this thread throughout the day
Quoting TonesInDeepFreeze
By infinity, I mean aleph_0. I thought that the value of f(alep_0)=0 which is why I asked for its value. But after some thinking, I realize that it is not. In fact, one could define a sequence g(n+1)=g(n)/10 where g(0)=1. It is easy to see that for any value of n g(n)
Quoting TonesInDeepFreeze
Just out of curiosity, why aleph_1 is uncountable?
I agree with what you stated.
Quoting fishfry
Yes, we are on the same page and thank you very much for your contribution. I learned a lot of things and refreshed my memory. :)
Then say 'aleph_0' since, in this context, 'infinity' is ambiguous.
Quoting MoK
You think that only if you didn't read my post.
For the third time: The domain of f is the set of positive natural numbers, therefore, aleph_0 is not in the domain of f, therefore "f(aleph_0)" is meaningless.
Quoting MoK
What is the domain of g? If the domain of g is the set of natural numbers then:
g(0) = 1
for all natural numbers n, g(n+1) = g(n)/10
So g, just like f, converges to 0.
Quoting MoK
So the domain of g is not the set of natural numbers. What is the domain of g?
I explained to you before that ".0...1" represents a sequence h on w+1 such that:
For all natural numbers n, h(n) = 0
h(w) = 1
So if aleph_0 is a member of the domain of g, then:
g(0) = 1
for all natural numbers n, g(n+1) = g(n)/10
g(aleph_0) = h
But we don't have a less than relation defined that includes h in the field of the relation, so no definition of "converges".
And are you going to be defining the domain and range of g to include other objects?
Quoting MoK
You haven't defined ">" so that it includes g(aleph_0) in the field of the relation.
Quoting MoK
Nonsense. aleph_0 is not in the domain of f. "f(aleph_0)" is meaningless notation.
Quoting MoK
Only because my meager knowledge at least includes an understanding of what such concepts as a function, domain, range, value of a function at an argument, sequence, infinite sequence, alephs, etc. mean. Meanwhile, your knowledge about these plainly required things is less than meager.
You need to learn at least the minimal basics of this subject. I mean, you don't even know what "domain of a function" means.
That you ask that question, indicates that you don't know what 'aleph' and 'countable mean' despite that you toss those terms around as if you're making actually meaningful statements with them.
Df. S is countable if and only if (S is one-to-one with a natural number or S is one-to-one with w)
Df. S is uncountable if and only if S is not countable
Df. T strictly dominates S if and only if (there is an injection from S into T & S is not one-to-one with T).
Notation: If j is an ordinal, then j+ is ju{j}.
Definition (by transfinite recursion on the ordinals):
aleph_0 = w
aleph_j+ = the least ordinal that strictly dominates aleph_j
If j is a limit ordinal, then aleph_j = U{alelph_m | m e j}
1 = 0+, so aleph_1 = the least ordinal that strictly dominates w. Since aleph_1 strictly dominates w, aleph_1 is not one-to-one with a natural number nor is S one-to-one with w. So, aleph_1 is uncountable.
/
Please get a book on the first order predicate calculus, then one on set theory.
Does that mean the mind is also an abstraction? Something outside the physical world? If so how does one explain what happens to my mind when you crush my head between two boulders?
The books you've recommended sound very interesting. I think Kant was right in saying that mathematics involves time (that is, process, synthesis). To *analyze* one plus one equals two is just to give a verbal description of having one and one. Mathematics is more than that. If i have 1 and see another 1, i have 1 and 1 and i "call that 2". But 1 added! to 1 EQUALS two because there is something in the addition that is synthetic instead of having backwards analysis. I realized this when i was trying to remember how i first used numbers. Mathematics is synthesis and analysis, but it's core meaning is synthesis it seems. I wonder how this relates to logicism
What do you mean by infinity when you talk about the limit in this post?
Good question... We argue about that kind of thing here all the time. I'll take a swing at it - the mind is a non-physical manifestation of a physical process, i.e. our nervous system's functioning. If you squish an important part of the nervous system, there is nothing left to manifest the mind.
Oh, wait. I guess I didn't answer your question... Did I mention that's a good question?
I didn't use the word 'infinity' in that post.
I used 'inf' (usually seen as the lemniscate). In the context and sense I used, 'inf' has no meaning by itself; it does not denote a mathematical object. You may think of it is a facon de parler.
As I explained [redacted for context]:
Quoting TonesInDeepFreeze
You see there that we unpack lim[n = 1 to inf] f(n) to:
lim[n = 1 to inf] f(n) = the unique x such that for every positive real number y, there exists a natural number n such that |f(n) - x| < y.
The falcon de parler 'inf' does not occur in the definiens. There is no resorting to a claim that there is an object named 'infinity'.
Again, we distinguish between the adjective 'infinite' and the noun 'infinity':
Adjective: S is infinite if and only if S is not finite.
Noun: There are various uses of 'infinity' as a noun, including such things as infinity as a point in the extended reals, or infinity as a hyperreal in nonstandard analysis. But my use of 'inf' (the lemniscate) in the context of defining 'limit' makes no use of 'infinity' as a noun.
Have you studied the material of Calculus 1? At least the first week in which the definition of 'limit' is given?
They are almost entirely mathematics, very little philosophy. Except the introductory chapter in 'Introduction To Mathematical Logic' by Church.
A pretty good book that weaves mathematics and philosophical topics is 'The Philosophy Of Set Theory' by Mary Tiles (but best to have already studied textbook set theory). And some great essays are in 'Logic, Logic, and Logic' by George Boolos and his essay "The Iterative Conception Of Set".
Uh-oh. We had a very lengthy thread about supertask a while back. Best leave that one alone :-)
Quoting Gregory
Well math is math and physics is physics. Math is a tool for physics but they aren't the same thing. You're making connections that I'm not sure I see. But maybe there's something to it.
The mathematical continuum is perfectly clear. It's the set of standard real numbers. Personally I think it's very unlikely that the real numbers are instantiated in the world.
Quoting MoK
Glad I could be helpful.
Quoting Benj96
Nobody knows what happens to the mind (or soul) when we die.
Quoting TonesInDeepFreeze
I studied Calculus 1 40 years ago. I am familiar with this notation of the limit.
The domain of f has aleph_0 members, but aleph_0 is not a member of the domain of f.
This is counter-intuitive to me. Consider a function f with the domain D={1,2,...N} where N is a finite positive integer. The domain has N members and N is a member of the domain of f. Could you please explain what happens when N is aleph_0?
It's not counterintuitive that there exist functions whose domain is the set of positive natural numbers. Any Calculus 1 textbook has such functions all through the book. Moreover, there are functions whose domain is the set of real numbers. Moreover, there are functions whose domain is a proper subset of the set of real numbers. Moreover, for any set whatsoever, there are functions whose domain is that set (except there is only one function whose domain is the empty set).
What is counterintuitive to me is someone claiming to be familiar with basic calculus but not understanding ordinary mathematical functions.
Quoting MoK
To be clear, 'N' there does not stand for the set of natural numbers, so I'll use 'n'':
Yes, {1 ... n} is a set with n number of members and n is a member of {1 ... n}.
That doesn't vitiate that there are other sets with n number of members but such that n is not a member:
{0 1} has 2 members, but 2 is not a member.
{1 4 7} has 3 members, but 3 is not a member.
the set of natural numbers has aleph_0 members, but alelph_0 is not a member.
the set of real numbers has 2^aleph_0 members, but 2^aleph_0 is not a member.
Quoting MoK
The domain of the function f that I defined is the set of positive natural numbers. aleph_0 is not in that domain.
I mentioned in a previous post that nothing is stopping us from defining a different function that has alelph_0 in the domain. But so what?
the domain of f is the set of positive natural numbers = {n | n is a positive natural number}
f(1) = 1
f(n+1) = f(n)/2
"f(aleph_0)" is meaningless notation since alelph_0 is not in the domain of f
a different function h:
the domain of h = {n | n is a positive natural number}u{aleph_0}
h(1) = 1
h(n+1) = h(n)/2
h(aleph_0) = whatever you want to make it (could be h(aleph_0) = 0, or h(aleph_0) = aleph_0, or h(aleph_0) = pi, or whatever you specify. just say what it is and then we'll know what h is)
The purpose of defining f was to agree with you that the results of dividing by 2 converge to 0. And the fact that there is no finite upper bound to the number of times we can divide by 2 refutes your errant claim that there is a real number such that there is no real number between it and 0, thus disproving your errant claim that you've disproved the existence of the continuum.
Once you define h, you can say what the purpose of doing that is. But any claims about h vis-a-vis real numbers, infinitesimals and the continuum need to be proven and use only terminology already defined or defined by you, not merely imaginistic hand waving.
/
Do you really not understand what a domain of a function is? Or are you trolling me?
Does make one wonder, doesn't it? :roll:
Sure. If you had a set [math]\{1, 2, 47, \aleph_0\}[/math], that would be a set with four elements, one of which is [math]\aleph_0[/math]. Then if you had a function [math]f[/math] defined on that set, [math]f(\aleph_0)[/math] would make sense.
But [math]\aleph_0[/math] is NOT A MEMBER of the set of natural numbers. None of 1, 2, 3, ... are [math]\aleph_0[/math]. So no function defined on the natural numbers takes a value at [math]\aleph_0[/math], since [math]\aleph_0[/math] is not a natural number.
That I understand and that is not my problem.
Quoting TonesInDeepFreeze
I understand there are sets with n members, but n is not a member of the sets. That was why I defined the domain D that has this specific property, the number of its members, n, is a member as well.
Quoting TonesInDeepFreeze
Let me ask you this question: Are all members of the natural number set finite? If yes, how the number of its members could be aleph_0?
Quoting TonesInDeepFreeze
I am so sorry that you feel that I am trolling. I respect your time and my time. I appreciate your effort in explaining things to me and I learned lots of things from you that I am grateful for.
So you are saying that aleph_0 is not a member of the set of natural numbers yet the number of its members is aleph_0. I am however puzzled how all the members of the natural number set are finite yet it has aleph_0 members.
Let's look at some simpler examples.
1) Consider the set [math]\{3, 18, 58, 334\}[/math]. That's a set with 4 elements, yet 4 is NOT an element of the set.
2) Consider the set [math]\{4, 18, 58, 334\}[/math]. That's a set with 4 elements, and 4 IS an element of the set.
So we see that the cardinal number of a set MAY or MAY NOT happen to be an element of the set.
So far so good? Ok now into the infinite realm.
3) The set [math]\{1, 2, 3, ...\}[/math] does NOT contain the cardinal [math]\aleph_0[/math]. I hope you can see that. If you claim otherwise, which element is it? It's not 1, it's not 2, it's not 3, etc.
4) The set [math]\{1, 2, 3, ..., \aleph_0 \}[/math] DOES happen to contain [math]\aleph_0[/math].
We see that in general, a set MAY or MAY NOT happen to contain its cardinality as a member.
In the case of the natural numbers, they do not contain their cardinality [math]\aleph_0[/math].
But there are surely [math]\aleph_0[/math] natural numbers, literally by definition. As we've seen from these examples, that fact gives us no information as to whether [math]\aleph_0[/math] is a member of the set of natural numbers. We have to look. And in this case, [math]\aleph_0 \notin \mathbb N[/math].
But if we tossed [math]\aleph_0[/math] into our set to get [math]\{1, 2, 3, ..., \aleph_0\}[/math], then we would have a set of cardinality [math]\aleph_0[/math] that happens to also have [math]\aleph_0[/math] as an element.
It can go either way, as the examples show.
By the way, how do we "toss [math]\aleph_0[/math] into the set?" The formal operation is taking a union. That is:
[math]\aleph_0 \notin \mathbb N[/math]
but
[math]\aleph_0 \in \mathbb N \cup \{\aleph_0\}[/math]
and BOTH sets have cardinality [math]\aleph_0[/math].
Fine. But that's a different domain than the one I used.
Quoting MoK
Yes, it's a common fact.
Quoting MoK
The set of natural numbers is aleph_0. Every set is one-to-one with itself. So the set of natural numbers is one-to-one with alelph_0, so the cardinality of the set of natural numbers is aleph_0.
Quoting MoK
There is no contradiction between:
Every member of the set of natural numbers is finite
and
The set of natural numbers has aleph_0 members
There is no contradiction between:
Every member of the set of natural numbers is finite
and
The set of natural numbers is not finite
It seems you have this false premise in your head:
If every member of a set S has property P then S has property P.
That premise is wrong and ridiculous. Get rid of it.
I should not be explaining basic logic to you. Your best bet would be to study a book on the first order predicate calculus and then one on set theory.
Semantic puzzlement at the fact that one cannot finitely bound the natural numbers is understandable, even when having no syntactical puzzlement with regards to a formal definition of the naturals.
Firstly, there is presumably no semantic puzzlement about the situation in which one constructs the naturals one by one by counting upwards from zero, for in that case one assumes full control of the number generation process that one never finishes, for which at any time one has only constructed naturals that are a constructively finite distance from zero.
So i think that semantic puzzlement is in relation to arbitrarily large natural numbers that one assumes to exist but which aren't constructively defined.
For example, consider a guessing game between Alice and Bob, in which Alice privately thinks of a natural number which Bob then has to guess. Suppose that Bob is given an unlimited number of chances for guessing Alice's number. If Alice doesn't cheat by changing her number in response to Bob's guesses, then Bob has a winning strategy that will eventually terminate, such as guessing each natural number in turn by counting upwards from zero. But then suppose Alice does cheat to ensure that Bob will lose. How could Bob ever know it? Alice could for example secretly define her number to be one greater than Bob's largest guess. In which case her natural number isn't constructively finite in relation to Bob's strategy, in spite of being constructively finite in relation to Alice's strategy in the situation where Bob eventually gives up.
Are you suggesting this proves real numbers are logically impossible, or are you arguing that there is no valid 1:1 mapping from the set of real numbers to the actual world? I ask, because it fails to do the former.
I am looking for proof that the set of natural numbers that each its member is finite has aleph_0 members.
What do you mean by this? Do you mean that the set of natural numbers is the set of aleph_0? aleph_0 is a number. How could you treat it as a set?
Quoting TonesInDeepFreeze
I don't understand this argument. How could aleph_0 be a number and a set at the same time?
Quoting sime
So classical continuum exists. I had to read about second-order logic, first-order logic, and zeroth-order logic trying to make sense of what you are trying to say here. Unfortunately, I don't understand what you are trying to say in the bolded part, the last part of the paragraph. Do you mind elaborating?
Quoting sime
Do you mind elaborating on what problem you specifically have in your mind?
Quoting sime
Do you by cut mean the exact position of an irrational number for example? What do you mean by the bolded part?
Quoting sime
Ok, thanks for the reference.
Quoting sime
My understanding is that there is no upper bound on the number of points that a continuum can be divided into. I however don't understand whether he agrees with the classical notion of continuum or not. If not, what is his point?
Quoting sime
Do you mind discussing his results further?
Quoting sime
So, one cannot define infinitesimal in the classical continuum. Is that what you are trying to say?
Quoting sime
What do you mean by antimony here?
Quoting sime
So you are trying to say that introducing infinitesimal can resolve the problem of cut which is problematic for classical continuum.
Quoting sime
What do you mean by temporal and spatial intuition here?
Quoting sime
I am reading about point-free topology right now so I will comment on this part when I figure out what you mean with point-free topology.
Quoting sime
I did read your links partially but I couldn't figure out what filter and ideal are. I don't understand how they resolve the problem of the classical continuum too but I buy your words on it.
I directly attacked the continuum in the mathematical sense. The discussion is ongoing but it seems that the classical continuum exists but suffers from problems which are discussed here. The post is very technical and I have problems understanding it though.
Quoting Relativist
Do you mean later (instead of former)? If yes, it would be nice of you to elaborate.
That's literally the definition of [math]\aleph_0[/math].
A set is defined to have cardinality [math]\aleph_0[/math] if the set can be placed into bijective correspondence with the natural numbers. Clearly the natural numbers themselves have this property, with the identity map as the bijection.
https://en.wikipedia.org/wiki/Aleph_number
Quoting MoK
All numbers are sets. In math, everything is a set.
Quoting MoK
As a set, [math]\aleph_0 = \mathbb N = \{0, 1, 2, 3, 4, 5, ...\}[/math]
I am aware of that. To avoid confusion, assume that the cardinality of the set of real numbers is X. How could one show that X is the least infinity namely aleph_0?
In real number theory, there are necessarily an infinite number of points between any two distinct points. This means it's logically impossible for there to be adjacent points. So when you say, "Therefore there is a gap between all pairs of distinct points of the continuum", is misleading because the "gap" contains infinitely many points. It's not a gap (which I think connotes an absence of points), it's an interval.
It's not. Recall the Cantor diagonal argument, which shows that the cardinality of the reals is strictly greater than the cardinality of the naturals.
It's easy to prove that [math]|\mathbb R | = 2^{\aleph_0}[/math] and that this is strictly greater than [math]\aleph_0[/math].
The question of whether that happens to be equal to [math]\aleph_1[/math] or not is called the Continuum hypothesis. It's known to be independent of the standard axioms of math. That is, we can't prove it's true, and we can't prove it's false.
One attempt to resolve the issue is to try to find new, naturalistic axioms that settle the matter one way or another. Others argue that it's not even a well-defined question. The problem has been one of the driving forces in modern set theory ever since Cantor first posed it in 1878.
https://en.wikipedia.org/wiki/Continuum_hypothesis
Ok, that I agree. How about whether the real number represents reality or not? Do you know any references that claim that reality is not continuous? I am currently reading a manuscript about loop quantum gravity and discrete time. I am a condensed matter physicist but I studied particle physics and cosmology to a good depth. I am somehow ignorant of loop quantum gravity and string theory though.
A mistake on my part and I am sorry for that. I should have written: "To avoid confusion, assume that the cardinality of the set of natural (I wrote real instead of natural) numbers is X. How could one show that X is the least infinity namely aleph_0?"
Does it matter? In terms of measurement, there a limit to how accurately we could possibly measure (at least, AFAIK)- so reality is discrete for all practical purposes.
https://math.stackexchange.com/questions/517040/proof-that-aleph-null-is-the-smallest-transfinite-number
I didn't ask for a proof that shows that aleph_0 is the least infinity but to show that X is the least infinity namely aleph_0.
A set is defined to have cardinality [math]\aleph_0[/math] if it can be bijected to the natural numbers.
Clearly the natural numbers can be bijected to the natural numbers, via the identity map; that is, the function that maps each natural number to itself.
If X is the cardinality of the natural numbers, X is [math]\aleph_0[/math] by definition.
Your questions arise only because you don't know anything about the subject, and also because (contrary to your claims) you don't carefully read the answers given you. And when you ask question Q1 and get an answer A1, you then have to ask Q2 to understand Q1, on and on. The way to break that regress is to study the subject from page 1 in forward, not reverse, order.
You need a book on the first order predicate calculus, then one on set theory. Nothing else will give you a proper understanding of the subject.
But until perhaps I get some moments to address your latest questions, I'll at least address this one (the answer is trivial, except you don't know anything about the subject to see that it's trivial):
Quoting MoK
"the least infinity" is not defined by you.
But we prove that there is no infinite cardinal less than aleph_0. Moreover, aleph_0 = the unique infinite cardinal K such that there is no infinite cardinal less than K.
Dfn. if x and y are ordinals, then x is less than y if and only if x is a member of y
Dfn. x is successor-inductive if and only if (0 is in x & for all n, if n is in x then xu{x})
Axm. There exists a successor-inductive set
Thm. There exists a unique set that is a subset of all successor-inductive sets
Proof: Put a nickel in my proof jukebox and I'll supply the proof
Dfn. w = the unique set that is a subset of all successor-inductive sets
Dfn. n is finite if and only if [put a nickel in my definition jukebox and I'll supply the definiens]
Dfn. n is an ordinal if and only if [put a nickel in the machine]
Dfn. n is a natural number if and only if (n is finite & n is an ordinal)
Thm. w = {n | n is a natural number}
Proof: Put a nickel in the machine
Thm. w is successor-inductive
Proof: Put a nickel in the machine
Thm. No ordinal is a member of itself
Proof: Put a nickel in the machine
Dfn. k is a cardinal if and only if (k is an ordinal & k is not one-to-one with a lesser ordinal)
Dfn. x is infinite if and only if x is not finite
Thm. every successor-inductive set is infinite
Proof: Put a nickel in the machine
Thm. Every cardinal is an ordinal
Proof: A cardinal is an ordinal that is not one-to-one with a lesser ordinal. So every cardinal is an ordinal.
Dfn. aleph_0 = w
Thm. w is infinite
Proof: w is a successor-inductive set
Thm. w is an ordinal
Proof: Put a nickel in the machine
Thm. There is no infinite cardinal less than aleph_0
Proof: aleph_0 is infinite. Every member of aleph_0 is finite. So no member of aleph_0 is infinite. So there is no infinite ordinal that is a member of aleph_0. So no infinite cardinal is less than aleph_0.
Thm. If K and L are cardinals, then exactly one of these: (1) K dominates L or (2) L dominates K or (3) K = L
Proof: K and L are ordinals. Put a nickel in the machine for proof that this trichotomy obtains for ordinals (axiom of choice not needed here)
Thm. aleph_0 = the unique infinite cardinal K such that there is no infinite cardinal less than K
Proof: Suppose L not= aleph_0 is an infinite cardinal such that there is no infinite cardinal less than L. So aleph_0 is not less than L. And L is not less than aleph_0. Contradicts previous theorem.
fishfry has been giving you the info.
But to understand very well, you need to go in forward direction from page 1 of a book - from the simplest concepts to the more involved concepts that depend on the prior concepts - instead of foolishly trying to work backwards. Anyway:
Thm: If k is a cardinal and j is a cardinal less than k, then j and k are not one-to-one
Proof: By definition of 'is a cardinal', j is not one-to one with k
Thm: If S is one-to-one with a cardinal, then there is a unique cardinal such that S is one-to-one with it
Proof: See above theorem
Dfn. If S is one-to-one with a cardinal, then card(S) = the unique cardinal that S is one-to-one with
Thm. if x is infinite and y is finite, then x and y are not one-to-one
Proof: Put a nickel in the machine
Thm. w is not one-to-one with any member of w
Proof: w is infinite but every member of w is finite
Thm. w = {n | n is a natural number} is a cardinal
Proof: w is an ordinal and w is not one-to-one with any member of w, so w is not one-to-one with any ordinal less than w
Thm. x is one-to-one with x
Proof: the identity function on x is an injection from x onto x
Thm. card({n | n is a natural number}) = aleph_0
Proof: {n | n is a natural number}) = aleph_0. aleph_0 is one-to-one with aleph_0
Yes. I said it many posts ago in a post you claimed to have read.
Quoting MoK
Set theory doesn't provide a definition of 'is a number'. Rather, set theory provides individual definitions of such things as 'is a natural number', 'is a real number', 'is an ordinal number', 'is a cardinal number'. In those, 'number' is not taken as a standalone adjective. We could just as well have this terminology in instead: 'is a natnum', is a 'realnum', 'is an ordinalnum', 'is a cardnum'. We don't make proofs by recourse to ascribing any properties to being "a number" in and of itself. Also, instead of 'ordinal number' and 'cardinal number' we may as well just say 'ordinal' and 'cardinal'.
Anyway, aleph_0 is an ordinal and it is the least infinite ordinal, and it is a cardinal and it is the least infinite cardinal.
Quoting MoK
Everything in set theory is a set. Natural numbers, real numbers, ordinals, cardinals ... are sets.
By definition aleph_0 = w, and we prove that w = {n | n is a natural number}.
I mentioned that previously.
I'm keeping to first order set theory. The point is that in set theory, any two complete order fields are isomorphic. You can think of that as saying that any two complete ordered fields are the same as one another except we changed the names of the elements. In other words, they are structurally the same.
/
d is a Dedekind cut if and only if (1) d is a non-empty proper subset of the set of rational numbers & (2) for all x, if x is a member of d and y
/
Quoting MoK
Be clear. There is a difference between (1) defining the adjective 'is an infinitesimal' and (2) proving the existence of a particular infinitesimal and naming it.
In set theory in which we formulate the system of real numbers, we can define 'is an infinitesimal', and prove that no real number is an infinitesimal, just as I have in this thread. And in set theory we can prove that there is a system with hyperreals (including infinitesimals) and prove that such a system is not isomorphic with the system of real numbers but that that there is a subsystem of hyperreals that is isomorphic with the system of real numbers. However, in plain set theory, even though we can prove the existence of a system of hyperreals, at least the usual methods (compactness theorem or ultrafilters) don't define a particular system of hyperreals.
/
A filter is a certain kind of set. an ultrafilter is a certain kind of filter. ultrafilters are used to prove the existence of a system of hyperreals.
It is folly for you to be trying to figure out what ultrafilters are and how they play into hyperreals when you don't even know page 1 of set theory.
It turned out that by 'gap' @MoK meant 'interval'. His thinking is hopelessly confused.
I don't opine here on that other question. But what do you mean by "maps into"? Do you mean "there is a function into" or do you mean "there is a one-to-one function into"? I surmise you mean the latter.
Zeno's paradox? Other? What arguments are you referring to that there is no injection from the set of real numbers into "the world"?
Here's my (meta-)argument.
If the real numbers are instantiated in the real world, then questions such as the axiom of choice and the Continuum hypothesis become subject to physical experiment.
Since no physics postdoc has ever applied for a grant to study such matters, I conclude that no physicist takes these questions seriously; and that in any event, such questions are so far beyond experimental investigation as to be meaningless.
Let me put it another way. If the real numbers have representation in the physical world, do you think there are physical nonmeasurable sets? If so, does that make the Banach-Tarski theorem a fact of the physical world?
Doesn't the absurdity of ever investigating such questions argue against the physical reality of the real numbers?
Quoting Relativist
This I disagree with. Word games involving supertasks tell us nothing about physics.
Also, the argument "There are no relevant experiments regarding surrounding aspects of the reals, therefore there is no such injection" requires the premise, "If there is such an injection, then there are relevant experiments regarding surrounding aspects of the reals". But how would we rule out that there could be an injection but no relevant experiments regarding surrounding aspects of the reals, or that there could be an injection but no known relevant experiments regarding surrounding aspects of the reals?
Quoting fishfry
The question was about an injection. What is the definition of "instantiated in the real world"? Does it just mean that there is the range of an injection from the set of real numbers?
Quoting fishfry
I don't know that that is the case. Moreover, cutting back to the question of an injection, I don't know that that the lack of someone thinking up an experiment would entail that there is no injection.
Moreover, would entertaining that there is an injection from the set of natural numbers N into the real world entail that there must be some experiment to conduct?
Quoting fishfry
I don't know that that is true.
Quoting fishfry
That might be the case. Indeed, even the question alone of the existence of an injection from the set of real numbers into "the real world" doesn't seem to me to have, at least so far, been given a firm and clear meaning.
Quoting fishfry
That might be the case; I don't know. But I don't see that to entertain that there might be an injection entails that there must be an experiment to conduct. But again, the question of the existence of an injection from the set of real numbers into "the real world" doesn't seem to me to have, at least so far, been given a firm and clear meaning.
Quoting fishfry
I wouldn't think that to entertain that there is an injection from the set of reals into the real world entails that there is a physical version of Banach-Tarski. But again, the notion of such an injection is not definite enough for me to have much of a view anyway (as well as I'm not prepared to discuss details of Banach-Tarski).
I surely don't have a strong opinion on the question of the existence of an injection from the set of real numbers into "the real world", but at least I would want to ponder whether the question is even even meaningful to either affirm or deny.
The world that is studied by physics. The phenomena around us that are amenable to experiments. Things that have mass, electric charge, velocities, and so forth.
As opposed to conceptual things like numbers, abstract geometric shapes.
I'm not sure what point you are making to ask what is meant by the real world.
The real numbers are a mathematical abstraction. The question is whether it is literally instantiated -- that's the word I prefer -- in the real world.
In other words, is there a true mathematical continuum in the world?
I am not sure what you are trying to get at with this question, since the answer is either obvious, or else you are making some subtle point along the lines of "what is reality?"
Quoting TonesInDeepFreeze
Also BTW, "injection" is the word YOU are using. I reject it utterly. An injection is a technical term in set theory. An injection is a type of function between two sets. There is not a shred of evidence that the objects in the real world -- the tables, the chairs, the electrons, the quarks -- obey the axioms of ZF as they pertain to infinite sets. Of course finitary ZF, also known as combinatorics, applies to the objects of the real world. But it's doubtful that anything in the real world is infinite, let alone satisfies the axioms of the real numbers.
So when you say injection, what can you possibly mean? What is your target set? Any injection from the reals must necessarily have an infinite range (or codomain, whatever the hell is the contemporary term. I gather that range and image are different now than when I learned them, and I never heard the word codomain till recently).
Clearly there's no injection from any infinite set to any collection of objects in the real world that we know of. The number of atoms in the observable universe is finite. Start there.
Quoting TonesInDeepFreeze
No, that's your word. I use instantiation, in the sense that the von Neumann ordinal we call 12 is instantiated by a carton of eggs. For there to be an injection you'd need a set, and perhaps you can nail that part down for me.
Quoting TonesInDeepFreeze
No not at all, since I strongly doubt there is an injection from the real numbers to anything in the physical universe. I doubt there's even an injection from the integers.
But the idea of a physical continuum is that below the Planck length, at the smallest level of reality, we find a copy of the real numbers. A mathematical continuum satisfying the second-order axioms of the reals. The least upper bound property and all that jazz. Metric completeness. All the Cauchy sequences converge. You ain't got nothin' like that in the world as far as anyone knows.
Quoting TonesInDeepFreeze
But it must be so. If there is a set of physical objects (quarks, universes, whatever) with the cardinality of the reals, we can ask what Aleph number it has, and there must be a definite answer. It would be a question amenable to physical experiment, even if we can't do it this week.
Instead of the Large Hadron collider we'd have the Colossal Continuum Counter. What a cool experiment that would be. The Superconducting Colossal Continuum Counter.
I hope you are seeing my point. If there is a set in the world cardinally equivalent to the reals, then we can in principle aspire to count them and see which Aleph they are.
Quoting TonesInDeepFreeze
Oh hell yes! We could count its subsets and see which Aleph they are. Same problem as for the reals, but expressed a little differently.
Or we could verify the axiom of choice. I think a countably infinite set would do, since it has uncountably many subsets.
I'm sure you can either see exactly what I'm talking about, or else you're not seeing what I'm talking about at all. In which case I should await your response.
Quoting TonesInDeepFreeze
Over the years I have Googled around. If someone has proposed an experiment to relate set theory to the world I might well have heard of it. In fact there are a smattering of papers relating set theory to physics, but they'er all behind academic paywalls.
Quoting TonesInDeepFreeze
It has a perfectly obvious meaning. There is a familiar injection from the abstract number 12 to a carton of eggs (a standard dozen). There is an injection from the number 5 to the members of a basketball team.
Nobody has any idea whether there's an injection from a countable set to anything in the real world. But if there is, AND if ZF applies to the world (a darn good question IMO) then there is automatically an uncountable set, namely the powerset of the countably infinite set. And then AC and CH and all the large cardinal axioms become questions of physics.
Quoting TonesInDeepFreeze
Please tell me why you say this.
It's perfectly clear what injections from finite sets to the world mean. It's perfectly clear that all finitary combinatorial math applies to the real world.
It's perfectly clear what an injection (or instantiation) of an infinite set would mean.
My point, which I'll bold, since it's really the only thing I have to say, is:
If there is an infinite set in the world, then all the questions of higher set theory become questions of physics, in principle amenable to physical experiment.
Heck, we did the LIGO experiment. If we can detect gravitational waves, why can't we count the points in a continuum?
Answer: Because there is no contnuum in the real world. If there were, the physics postdocs would be all over it.
Quoting TonesInDeepFreeze
Of course it would. Well ok I need a bit more. I need a three-dimensional Euclidean space. That's the minimum requirement. The isometry group of Euclidean 3-space contains a copy of the free group on two letters, which is what powers the B-T theorem.
So I'll concede that a mere linear or 2-D continuum is insufficient for B-T.
Quoting TonesInDeepFreeze
All you need is Euclidean 3-space.
Quoting TonesInDeepFreeze
Well it's a meaningful question, to which I'm prepared to argue that the answer is NO.
Quoting fishfry
The poster asked about a "map into" as a "1:1 relation between".
A word for that is 'injection'.
Quoting fishfry
That's fine with me. And if you object to saying "injection" rather than "1:1 map" that's fine with me too. I'm not the one asking whether there is a 1:1 mapping (whether you wish to rule out calling that an 'injection') from the set of real numbers into whatever is designated by 'the real world'. Perforce, obviously, I'm not claiming that if there were such a 1:1 relation then its range would be a mathematical set.
Again, thank you very much for providing the argument and extensive writing. I will read through the argument and try to understand it. I will look for proof of the theorems on the net by consulting ChatBot or reading through books or math forums.
Please please please do not refer to AI bots for math proofs. They are so often incorrect. I've tried it a few times, and the bot gives clearly incorrect arguments - petitio principii - using as an assumption what it claims to be proving.
And looking around at forums is also a disorganized and extremely poor way to learn mathematics.
Get a book if you actually want to understand the material.
Thanks for writing. On my reading schedule.
Ok, thanks for the clarification.
Thanks for the writing. I will see if I can find any time in the future to study the filter. I have other interests as well rather than mathematics.
What 1:1 map are you referring to? A 1:1 map from a real interval into points in space? A 1:1 map from a real interval into points of time? A 1:1 map from a real interval into a set of particles?
I don't have much to say about those vis-a-vis Zeno's paradox. I'm only asking what your argument is that Zeno's paradox entails that there is not a map.
That you have limited time for mathematics is all the more reason for not wasting that limited time in routes that lead to dead ends, misinformation and confusion.
If your question is whether spacetime is continuous then I have to say that there is an ongoing debate on the topic. I am not an expert on the topics of loop quantum gravity and string theory. I searched on the net a while ago and I found this manuscript which you might like to read. The manuscript is however old, 2003, so it does not reflect the current state of debate.
Ok, I will try to get the book you suggested in this thread.
I have to disagree. I have a wide range of interests. One of the main reasons that I signed up in this forum was the very good quality of knowledge of posters in this forum such as you. I am an expert in a few fields as well, such as physics, epidemiology, philosophy of mind, and the like. The idea is to share the knowledge that one accumulated over decades with others through discussion in the forums so that all individuals can benefit from it. In this way, one can save lots of time in understanding a topic through discussion with experts and decide where to focus on a topic and how to manage the valuable time.
(1) In open forums like this, there is usually more disinformation and confusion about mathematics than there is information and clarity. Instead, prolific cranks dominate, or discussions center on a few reasonable people trying to get a prolific crank to come to the table of reason.
(2) There are no set theory experts in this thread (or, to my knowledge, posting in this forum).
(3) Picking up bits and pieces of mathematics, hodge podge, is not an effective, not even a coherent, way to understand concepts that are built from starting assumptions and definitions. This thread itself is evidence of that.
This thread.. This entailed a thought experiment of traversing a countably infinite set stairs in a finite period of time, where the stairs-stepping entails dividing up an interval of time.
It was just an off-hand comment.
You can find all sorts of people in any forum. I agree that the number of knowledgeable individuals may vary from one forum to another.
Quoting TonesInDeepFreeze
That is all right. You are enough good to teach me a few things in set theory.
Quoting TonesInDeepFreeze
Quite oppositely I learned a few things in this thread. Thank you very much for your time and patience.
There is no paradox there. The total time which is needed to get to the top is 60 seconds. You might find this link useful.
Yes, and are quite knowledgable in this regard. I never studied anything beyond naive set theory and I have appreciated reading their posts, both instructional (and at times, argumentative). As for reference books, I am compelled to mention one that I have found wonderfully informative and written by a colleague from Colorado College: Introduction to Topology and Modern Analysis (George Simmons)
Continuity is perhaps best approached through elementary topology. Here is what Simmons says:
Of course, in the case of the reals the previous discussion concerns connectiveness.
Ok. Injection it is. The rest of my point holds. If there is an infinite collection of anything in the physical world; and if ZF applies to the infinite collections of the world; then questions of higher set theory become questions of physics, in principle subject to experiment. That presents many problems for those claiming there are actual infinities in the world.
No one said anything about ZF.
But if it is taken that there only finitely many things in what is designated as 'the real world', and it is regarded that there is no injection of an infinite set into a finite set, then the question is thereby settled, regardless of ZF; also, as far as I can tell, the other poster's call to Zeno's paradox or other supertask paradoxes would be unneeded.
I have a good grasp of the some of the basics of set theory, but I am not very knowledgeable beyond those basics.
Anyway, the idea of someone, who doesn't understand that the set of natural numbers is not a member of itself, trying to grapple with how ultrafilters play into proving the existence of hyperreals is ridiculous.
:up:
I did. I said that if there are infinitely many things in the world AND that ZF applies to them, then questions of higher set theory become subject to physical experiment. It's an interesting point.
Quoting TonesInDeepFreeze
Of course if there are no infinite sets in the world, then there is surely no continuum in the world.
Quoting TonesInDeepFreeze
I noted that supertask discussions have no bearing on any aspect of reality.
It's fine that you're talking about a question that's different from the one I replied to.
That paradox is nothing more than Zeno's paradox. It simply replaces the distance in Zeno's paradox by time. There is however a problem when you want to discuss Zeno's paradox by standard analysis. To discuss this further, let's consider the following sequence:
f(1)=1/2
f(2)=f(1)/2
...
f(n+1)=f(n)/2
...
where n is a natural number. This sequence is nothing but the sequence that was first mentioned by Zeno. The sum of the sequence then can be found and it is s(n)=1-1/2^n. Although we can use standard analysis to calculate the limit of s(n) when n tends to infinity we are not allowed to consider n to be infinity. That is because n is a member of the natural numbers and all members of natural numbers are finite. @sime in this post discusses the paradox and it seems that he resolved it. It seems to me that he uses nonstandard analysis to resolve the paradox. I am however pretty ignorant of nonstandard analysis so I cannot tell how he resolves the paradox. To my understanding, motion is real and Zeno's paradox is invalid. I however don't have the mathematical tools to discuss Zeno's paradox since my knowledge of analysis is limited to standard analysis. Perhaps, other mathematicians @sime, @fishfry, and @TonesInDeepFreeze can offer us a solution to the problem.
Thank you very much for your post. Unfortunately, my country is under sanction and I cannot purchase any book from Amazon. I am retired and living in the countryside so I don't have access to any library as well. So, I am out of luck when it comes to reading books unless I find a PDF file online.
More or less. Both demonstrate the fact that limits don't correspond to the completion of an infinite series of finite steps. I agree with Sime, and I also gave a solution in that thread that is similar to his.
Exactly right! So what are the options in this situation: (1) Spacetime is discrete or (2) Spacetime is continuous. In the first case, we don't have the problem of infinite division so there are no conceptual problems or paradoxes such as the one of Zeno. The rule of mathematics, Leibniz's calculus is to help us easily calculate things, such as differential and integral, so it is just a useful tool. In the second case, we however need a mathematical formulation that allows us to directly deal with infinity, for example, we should be able to set n equal to infinity, if not we cannot complete an infinite series, so we are dealing with the paradoxes such as the Zeno's or infinite staircases. In simple words, we cannot move and time cannot pass. So let's wait for mathematicians to see if they have a solution for the second case. If there is no solution for (2) then we are left to (1)!
This suggests to me that we will make no errors by treating space and time as discrete, even if it is continuous. What's your thoughts?
What is your solution to the paradox? Could you explain @sime's solution to me?
I think if spacetime is discrete and our capacity to measure spacetime interval is much higher than Planck length and time then we can treat spacetime continuously, hence we can use the continuous physical models that describe reality well. We however still have to deal with Zeno's and infinite staircases paradoxes.
Quoting Relativist
If spacetime is continuous then we are dealing with an error in treating space and time as discrete. There are numerical methods that allow us to minimize the error but at the end of the day, we cannot avoid the error at all. In most cases, we are safe if we discretize space and use good numerical methods. In the case of time, however, the error accumulates over time so we can find significant errors in our calculation in the long term. This error in the predicted variables can be catastrophic over time if the system is chaotic, such as the weather processes.
I am not a mathematician.
Proof:
Let Mx <-> x is a mathematician
Let Rx <-> x produces results in mathematics
Let t = TonesInDeepFreeze
1. Ax(Mx -> Rx)
2. Mt -> Rt {1}
3. ~Rt
4. ~Mt {1 3}
QED CIA FBI DHS MLB NBA NFL NBC CBS ABC JFK LBJ FDR ETC
The scenario entails reaching the bottom of a staircase through a process consisting of stepping, from one step to the next. Reaching that bottom entails taking a final step.
The infinite series entails an unending series of steps. So a final step is logically impossible.
OK, so we risk introducing error if we treat spacetime as discrete, but if it IS discrete, we introduce no errors by treating it (mathematically) as continuous. So treat it as continuous and use the math. Problem solved, right?
Why do we have to deal with Zeno's paradox? Is there some problem in physics where it makes a difference, or are you like the rest of us navel-gazers around here - and just curious the logical implications?
That is not a solution but the point of Zeno. If the final step is logically impossible then you cannot complete an infinite series of finite steps therefore you cannot finish the task.
Yes, if the spacetime is continuous and we treat it as discrete then we are introducing error.
Quoting Relativist
If spacetime is discrete we introduce error by treating it as continuous. We however might not be able to observe the error if our measurement devices are not precise enough.
Quoting Relativist
Yes, we can use a continuous model as far as our measurement devices are not precise enough. Otherwise, we have to use a discrete model.
Quoting Relativist
Zeno paradox is a metaphysical problem rather than a physical one. It tells us something about reality without a need for any measurements.
How is that not a solution? It can be framed as reductio ad absurdum:
1. Arriving at the bottom entails taking a final step
2. The defined infinite process of descent has no end
3. Therefore the infinite process of descent has no final step
4. (1)&(3) are contradictory
If the question can't be answered via measurement, or any other physical means, then it's unknowable. Quantum mechanics demonstrates that intuition isn't a reliable means of deciding physical* truths, so it shouldn't be too surprising.
*Although it's a metaphysical question, it pertains to the physical world.
Consider a set of points. We say that the set is continuous if there is a point between any arbitrary pair of points. We say that the set is discrete if there is a minimal distance between a pair of points. In other words, there is no point between such a pair. The example of a continuum is the real numbers and the example of a discrete is the natural numbers (the minimal distance between a pair of points is 1).
There are an infinite number of steps but we cannot complete them. The fact that we cannot complete the steps does not mean that they do not exist.
Yes, it has an implication. I think it means that spacetime is discrete.
Are the rationals continuous?
Between any two rationals there is another rational, right?
But the rationals are full of holes. For example the set of all rationals whose square is less than 2 has no least upper bound in the rationals. Can you see that?
Besides, a totally ordered set with the property that there is a point between any other two is called dense. Have I not previously drawn your attention to this fact? It's the definition.
https://en.wikipedia.org/wiki/Dense_set
This is in spite of the fact that the set of steps (the activity) maps 1:1 to the set of physical steps that comprise the stairway. The more important conclusion is that there's a logical disconnect between this logical mapping and the analogous temporal process; IOW, the mapping doesn't fully describe the temporal process; something is missing - and it would be worthwhile to develop a mathematics that accounted for this.
But I wonder: is it really hopeless for physics? You said that treating spacetime as discrete would lead to errors if it's actually continuous. Couldn't this be tested?
Correct. How about considering the point between two arbitrary points, namely a and b, to be mean, namely (a+b)/2? If not, could you please define the continuum for @tim wood in plain English?
Correct. So we are on the same page.
Quoting Relativist
Isn't the set of steps the set of physical steps? If yes why do you use a one-to-one map?
Quoting Relativist
I cannot figure out what you are trying to say here. Do you mind elaborating?
Step (the verb) = the act of setting ones foot onto the next step (the noun; a thing).
The set of actions maps to the set of things.
The stairway consists of the set of steps, which we're stipulating as being infinite. Unlike the staircase, the acts of stepping don't exist (they are actions).
He didn't ask for a definition of 'the continuum'. 'the continuum' is a noun. He asked for the distinction between 'continuous' and 'discrete'. 'continuous' and 'discrete' are adjectives.
'the continuum' has been defined at least three times already in this thread.
'continuous function' is the defined as usual in chapter 1 of any Calculus 1 textbook.
Other senses of 'continuous' depend on context. And definitions of 'discrete' depend on context.
We've been considering it at least fifty times already in this thread. What about it do you want to say?
What about the conclusion that spacetime is discrete?
Quoting Relativist
Yes, weather forecast for example. Any chaotic system in general. Even nonchaotic systems show the error in the long term.
Who is "we"? Other than you? What does "between" mean for sets in general? Or do you mean for real intervals? Usually, 'continuous' refers to functions. Perhaps there is an even more general notion of 'continuous sets', but we'd have to see it mathematically defined, in which case it's not going to be "there is a point between any two points".
Why don't you look up some mathematics rather than just making up claims about it?
You are talking about physical objects that have extensions in space so their location is not definable unless you talk about their center of mass. Do you know what the center of mass is? If not think of an ice cube. The center of an ice cube is its center of mass. The center of mass of the ice cube is definable though hence you can define the location of the center of mass of the ice cube. Now, you can move the ice cube along a line. This means that its center of mass moves from one point to another point along the line. So, by now you have a definition of a point, the center of mass of the ice cube, and a line, its motion along the line.
Quoting tim wood
Mathematicians work on abstract objects like points and lines all the time. They define a line as a set of dimensionless points and show that things are consistent. Whether these objects are real or not is subject to discussion.
Quoting tim wood
Well, the Zeno paradox certainly threatens mathematics, especially the continuum concept. I also bet on Achilles since my common sense tells me he will win.
I haven't seen a conceptual analysis that concludes it is discrete, but my impression is that it's typically assumed to be continuous.
Quoting MoK
Is it your opinion, as a physicist, that chaotic systems are not (in principle) reducible to deterministic laws of physics? My impression is that the math related to chaotic systems is pertains to identifying functional patterns to make predictions. That, at least, seems to be the nature of weather forecasts - it's not that the movement of air molecules is fundamentally indeterminstic, rather it's that it's that the quantity of data that would be needed to identify the locations and trajectory of each molecule is orders of magnitude too large to be practical to compute.
Oh please :roll: While the word "continuum" is everywhere on this thread, what is really the heart of the subject is "connectedness" of sets. And the Zeno paradox does not threaten mathematics.
Correct meaning you understand that the rationals are dense but not continuous?
Quoting MoK
Haven't we been doing that all along? Not sure what you mean. The rationals are dense and so are the reals. But the rationals are not Cauchy-complete. They lack the least upper bound property. So you can't use denseness to characterize the continuum, since the rationals are dense but not complete. The rationals are full of holes.
Quoting MoK
The set of standard real numbers, as you yourself have defined it since the first post in this thread, when claiming it doesn't exist. I believe you've now come around to accepting that it does exist. So that's the mathematical continuum. The real numbers.
ps -- Technically, what I've described is a linear continuum.
[quote=Wikipedia]
Formally, a linear continuum is a linearly ordered set S of more than one element that is densely ordered, i.e., between any two distinct elements there is another (and hence infinitely many others), and complete, i.e., which "lacks gaps" in the sense that every nonempty subset with an upper bound has a least upper bound. [/quote]
The official definition of a continuum is too general for our purposes here.
I still think the mathematics used in physics can already address this question. What about Conformal Cyclic Cosmology (CCC). Penrose explains how the universe goes from the big bang to infinity, how we can used compactification to bring the infinite into the finite, and have a finite beginning after the infinite "forgets" it's infinite (his idea). Relations between that which ends and that which doesn's is the essence of this debate
The speed of Achilles is 10meters/1second. The speed of Tortoise is 1meter/1000seconds.
I applied mathematics to determine that, in the 100 meter race, Achilles will cross the finish line in 10 seconds and that Tortoise will cross the finish line in 100000 seconds.
Then I called my bookie Zeus "The Moose" to place my 1000 euro bet on Achilles (favored 1000000000 to 1) and turned on the TV to watch the race on MSPN (the Mythic Sports Programming Network). After Achilles won, I called The Moose to collect. He started to say that my payoff is infinitesimal, so there's no way he can pay me; but I corrected him by telling him that there are no infinitesimals in the reals and that though the payoff is small, it is not infinitesimal. So he said he'd that he'd apply the fraction of a cent to my account. Meanwhile, there were a lot of Parmenideans who lost their togas betting on Tortoise.
Seems math did a pretty good job. Maybe math threatens Zeno's paradox.
Ok, I see what you mean and I agree.
He asked for a definition of continuous and discrete in plain English. Could you please provide the definition in plain language without referring him to read a Calculus book?
I want to say that you could sweep all points of the continuum using that definition.
Well, if space is continuous then it means an infinite number of steps exists yet we cannot complete them. The same applies to time in the example of the infinite staircase.
Quoting Relativist
The laws of physics are deterministic but that does not mean that the chaotic behavior does not exist. It means that any error in the calculation of physical variables leads to a significant deviation from what we observe and what the calculation provides. The source of the error in the case of weather forecast is twofold: (1) The error in the estimate of physical variables in the initial point and (2) Using a discrete approach to solve a set of continuous equations.
How could you index an infinite set of steps?
If by dense you mean that there exists a point between two arbitrary points then I understand that applies to the set of rational numbers. The link you provide is technical for me and I have to put more effort into understanding it.
Quoting fishfry
By that, I mean that there exists a point between two arbitrary points in which the between is defined as the geometrical mean.
Quoting fishfry
Ok, that definition seems good and simple for @tim wood. Thanks for providing the definition.
Could you calculate the speed in all infinite steps?
The arrow paradox says each is zero, as in time "points". Yet there is still the forward motion of the action, driven by energy
My point was that he didn't ask for a definition of 'the continuum'. The takeaway for you is to not conflate 'the continuum' with 'continuous'.
I didn't say that he needs to read a book. I said the definition is in chapter 1 of such books.
'continuous function' is a mathematical notion, and best understood in its mathematical formulation, which is not complicated. But for informal explanations, one can do an Internet search on 'continuous function'. Such explanations include such mentions even as illustrative as "can draw the graph without lifting your pencil", which takes quite a bit of liberty from rigor but at least gives one a kind of mental picture.
I mentioned that 'discrete' depends on context.
What definition of what? And what does "sweep" mean?
I don't know. First you would need to define "speed in all infinite steps".
What does that mean? Ordinarily, "to index" means to make a set the range of a function, as the domain is the index set. The domain is the indexing set and the range is the indexed set.
If there are denumerably many steps, then the steps may be indexed by the set of natural numbers. The set of natural numbers is the index set and the set of steps is the indexed set.
What is the point of your question?
(x+y)/2 is the arithmetical mean of {x y}, not the geometrical mean.
You're still conflating 'continuum' with 'continuous'. They are closely related concepts, but not the same concept. Also, the least upper bound property of the continuum already had been mentioned several times in this thread, so you had that information all along anyway.
I am not sure whether he was familiar with the concept of speed or not. But, the average speed in the interval [math]i[/math] can be calculated as [math]v_i=\frac{(\Delta x)_i}{(\Delta t)_i}[/math] where the [math](\Delta x)_i[/math] is the length of [math]i[/math]th interval and [math](\Delta t)_i[/math] is the time duration it takes the arrow to move [math]i[/math]th interval. So everything is clear for now. The problem is however with the index [math]i[/math] which cannot be infinite since it is a natural number yet we know that infinite steps exist.
The arrow paradox is that the arrow does not move but that it moves.
/
Average speed is distance/time. In Zenos's paradox, both are finite.
Continuum is a continuous series. He understands what continuous is if he understands what continuum is.
Quoting TonesInDeepFreeze
Thanks. @fishery gave a definition for a continuum from wiki: "Formally, a linear continuum is a linearly ordered set S of more than one element that is densely ordered, i.e., between any two distinct elements there is another (and hence infinitely many others), and complete, i.e., which "lacks gaps" in the sense that every nonempty subset with an upper bound has a least upper bound."
I can define the speed in [math]i[/math]th step as follows: [math]v_i=\frac{(\Delta x)_i}{(\Delta t)_i}[/math] where [math](\Delta x)_i[/math] is the length of [math]i[/math]th interval and [math](\Delta t)_i[/math] is the time duration it takes the runner (I am referring to Dichotomy paradox) to move [math]i[/math]th interval. The series however has infinite steps so I cannot define the speed in all infinite steps since [math]i[/math] is a natural number.
I mean you cannot give indexes to all members of an infinite series.
Thanks for the correction.
Please accept my apology. My, argument here was for Dichotomy paradox. You need to replace the arrow in that post with the runner, Atalanta.
1, 2, 3, 4, 5, 6, ...
Is that not an infinite sequence? (You mean sequence. A series is a sum)
It it not indexed by the natural numbers?
Take the sequence 1/2, 1/4, 1/8, 1/16, ...
That's an infinite sequence. It's also indexed by the natural numbers 1, 2, 3, ...
In fact every infinite sequence is indexed by the natural numbers, by definition.
Quoting MoK
I agree that the Wiki article could be more clear.
The point is that we say that a linearly ordered set is dense if between any two elements, there is a third strictly between those two.
Another definition is that between any two elements are are infinitely many distinct elements between the two.
These two definitions are equivalent. The argument is exactly the one that you originally gave: that you just keep taking midpoints.
That is an infinite sequence. I am however interested in the sequence first mentioned by Zeno in Dichotomy Paradox in which the infinite member exists. Each member of the above sequence is finite, so you cannot use the above sequence to give indexes to all members of the sequence in Dichotomy Paradox since the infinite member exists.
Quoting fishfry
Thanks for the correction.
Do you know what a limit is? The sequence 1/2, 1/4, 1/8, ... has 0 as a limit.
If you read through the supertask thread that's been referenced elsewhere in this thread, I explained that you can view the index of the limit as the ordinal number [math]\omega[/math] or as a hypothetical "point at infinity," just as plus/minus infinity are hypothetical points at each end of the real number line in the extended real numbers.
https://en.wikipedia.org/wiki/Extended_real_number_line
Wrong. A series is a certain kind of function. The continuum is not a function.
Quoting MoK
Wrong. I explained the difference between them. Knowing the definition of 'the continuum' does not provide knowing the definition of 'continuous'.
Wrong. A series is a certain kind of function. Since it is a function, the range of the function is indexed by the domain of the function.
Again, you're using mathematical terminology without a clue as to what it means. But that's okay. After, all, what is an Internet forum such as this good for if not to provide a platform for people who don't know what they're talking about to prolifically shoot their mouth off about it anyway?
You're very confused and resistant to the explanations given you to cure your chronic confusion.
Dichotomy schmicotomy. You mentioned 'average speed' and I gave you the formula.
Thanks. So you simply extend the natural number to the extended natural number and resolve the problem of indexing.
Do you mind elaborating?
@fishery already explained how to resolve the problem of indexing here.
Most people seem to think Zeno was a minor philosopher. He was actually the first in the West to write a book of philosophy as that is understood in the modern sense. No one before him asked questions about the infinite like that. He was a Newton of his time. To think of infinite division back then was incredible. We still debate this today
Yes exactly.
Quoting Gregory
The geometric idea is exactly the same.
Consider the half-open unit interval [0,1) which includes 0 and excludes 1. It has length 1. Now break it up into an infinite union of many subintervals:
[0,1/2) [math]\cup[/math] [1/2, 3/4) [math]\cup[/math] [3/4, 7/8) [math]\cup[/math] [7/8, 15/16) [math]\cup[/math] ...
That's a geometric decomposition of an interval of length 1 into pairwise disjoint subintervals of lengths 1/2, 1/4, 1/8, ... respectively, representing the arithmetic infinite sum 1/2 + 1/4 + 1/8 + ... = 1.
You can toss in the right endpoint of 1, which, being a point, has length 0 and makes no difference to the length.
Wikipedia has a 2-dimensional version of this visual proof.
Not "extended natural number".
Rather the ordinal w+1 = wu{w}. That has been explained to you probably at least five times already.
And there is no "problem of indexing". You're using the word 'indexing' incorrectly, as I've explained to you twice already.
Any set can be indexed by being the range of a function. Indeed, trivially, any set is indexed by being the range of the identity function on the set.
Your question might be whether a certain set can be indexed by w. By definition, any denumerable set can be indexed by w.
But your question might go further: Can a set with an ordering on the set be isomorphic with w and the usual ordering on w. Of course, there are sets and orderings that are not isomorphic with one another. For example, in the current discussion, consider a set ordered
1 2 3 ...
That is isomorphic with the standard ordering on w.
Consider a set ordered
1 2 3 ... x
That is not isomorphic with the standard ordering on w
But it is isomorphic with the standard ordering on w+1.
But these explanations are wasted on you because you won't actually look at a beginning reference on this subject so that your claims and questions aren't based on your confusions and ignorance.
Except that w is not called an "extended natural number".
I have no problem with that terminology which avoids getting into ordinals, which people find confusing. I learned that in the other thread.
I wouldn't, but suit yourself.
If you believe in the extended reals, just setminus all the finite non-integers and you're left with the integers along with the two points at infinity. Why on earth can't you call that the extended integers; or taking the nonnegative ones, the extended naturals? It's perfectly ... natural.
What's to elaborate? I gave you the definitions of 'the continuum' and 'continuous function'. Meanwhile, a definition of 'a continuum' is not needed in this context, only 'the continuum'. Moreover there are different definitions of 'a continuum' in different contexts and they are way too technical for you. For purposes of the discussion at this juncture, 'the continuum' is sufficient. You ask for elaboration on concepts that depend on previous concepts that you refuse to learn. You won't even find out for yourself what a function is but you want people to explain mathematics to you.
You can call them whatever you want.
(I see that there is a Wikipedia article that does use the terminology though. I don't usually reference the unreliable and haphazardly edited Wikipedia for mathematics.)
But sequences are defined as having ordinals as the domain. That clarifies and can be widely referenced in the literature. And the extended reals of course is also a common notion. But the points of infinity in the extended reals are not ordinarily (if at all) understood as ordinals that are the domains of sequences.
Consider the following set
[math]\mathbb R \cup \{-\infty\} \cup \{\infty\} \setminus (\mathbb R \setminus \mathbb Z)[/math]
with the order
[math] -\infty < n \ < \infty[/math] for [math]n \in \mathbb N[/math]
Why do you refuse to call that the extended integers, and the subset of nonnegatives the extended naturals; and to write (so far) two posts objecting to this simple and natural terminology?
See this SE discussion. The consensus seemed to be that they're not generally useful; not that the terminology, which is perfectly obvous, is wrong.
https://math.stackexchange.com/questions/1442961/extended-integers
It's not a quirk. It's odd terminology, as far as I know; and the context here is not just points at infinity but sequences, and sequences are functions whose domain is an ordinal (or at least a set with a well ordering).
The standard ordering on the real numbers is not a well ordering. So talking about points of infinity as in the domain of a sequence is confusing at best.
The sequence is (I'm taking the liberty of starting at 1 rather than at 0):
{<1 step-1> <2 step-2> ...
It's a pedagogical point. It's far far easier for people to understand the point at infinity as the index of the limit of a sequence, than to explain the ordinal [math]\omega[/math]; even if the latter is more mathematically precise. But in this context it makes no difference whatsoever.
I learned this at length in the supertask thread. The idea is to communicate and educate, not to stand on pointless pedantry at the risk of confusing people. Try to consider this point.
In fact if it makes you happy, I'll just tell you that [math]\infty[/math] is my own personal notation for what everyone else calls [math]\omega[/math], and now you have no possible objection.
I won't be further responding to this silly and trivial subtopic, but lest anyone think I'm thereby conceding the point, I'm not. Your obsessive pedantry is leading you astry. From now on when I say, "extended natural numbers" just remember that this is my pet name for [math]\mathbb N \cup \{\omega\}[/math] and all will be well.
L u {<-inf n> | n in w} u {
The formulation you gave doesn't mention that the ordering includes the standard ordering on w; it only mentions that every member of w is greater than -inf and less than inf.
It's not pedantic but it is pedagogical.
The key idea is sequences. And it is clear and concise to say:
The domain is w+1 = wu{w} (or, taking liberties (w\{0})u{w}).
But to go through cutting the real numbers down to positive integers and then adding points of infinity (moreover raising the pedagogical question of what exactly are "points of infinity") is an unnecessary detour and distracts from the insight of using sequences. Indeed, such things as dividing infinitely in a discussion such as this involves explicitly sequences. So the insight is that we have sequences not just on the ordinal w but also on ordinals such as w+1. Indeed that highlights that the range of such sequences may have other than ordinals (such as natural numbers) but that the domains are ordinals.
You may use the lemniscate however you wish. But I wouldn't use it for w. Because it is a typical that uninformed people (I mean other posters, not you) are prone to use "infinity" as a catch-all for infinite sets. Indeed a point of infinity in the extended reals may be any set whatsoever except a real; doesn't even have to be an infinite set. That is, if we want to instantiate the extended real system, then we only need to choose two sets that are not themselves real numbers, then define the ordering and operations in the usual manner.
But uninformed people don't distinguish between inf ('inf' read as the lemniscate) and the set of natural numbers. They regularly conflate inf ("infinity") with the set of natural numbers, which is an infinite set but not necessarily a point of infinity in the extended reals. So it is good pedagogically to use notation that doesn't lead to that conflation.
On the other hand, making an analogy between w+1 and the positive extended reals is a nice idea. I would state it as only an analogy though.
But of course, you are free to stipulate whatever definitions you like.
LOL. Pedants '?' us.
Your penchant for making discussions personal leads you astray.
I've been watching you insult the OP for days now. Project much?
That is not pedantic. The formuation you gave is literally very incorrect. I simply offered a correct formulation. For that, you reply petulantly (though not pedantically). Maybe, eventually, you'll get past nonsense like that.
After many posts where I made no personal comments, I have correctly said that the poster is ignorant and confused about mathematics. And that is not gratuitous, especially as he persists to ignore the information and explanations given him. That is not analogous to your nearly immediately taking umbrage at corrections or suggestions for sharper formulation and then turning that to one of your sophomoric snipes. And, in this case, instead of responding to my substantive points about the terminology.
It is alright guys. I learned lots of things from both of you and I am thankful for it. I don't think that personal attacks are useful though when people can discuss things openly, so one party learns and another party teaches. I have to say that the process of learning is time and energy consuming so it is alright if one person needs the explanation of things a few times.
Bull. You spout misinformation. Then you get explanations and ignore them, evidenced by asking questions that were already answered as you just go right ahead to re-spout. And you refuse to grasp the very simple fact that any explanation given you relies on previous concepts so that both the logical way and only downright practical way to know any of this is for you to start at page 1. If not trolling literally, it is trolling de facto. But do carry on ... it is amusing as pathetic comedy.
As a complex analysis guy you use the hypothetical point at infinity of the Riemann sphere all the time, don't you? It's just the one-point compactification of the plane. The circle is the one-point compactification of the real line, and the extended reals are the two-point compactification.
Only on rare occasion. Normally, Infinity for me means unbounded. I don't work on the Riemann sphere. Yes, projective stuff is there in the background, like circles with infinite radius are lines, etc. But I am very old fashioned. Here is the sort of thing that has interested me.
MO of the supreme sophist, Tones in the Deep Freeze.
The quintessential crank* Metaphysician Undercover misquotes me. I didn't post any sentence that begins with "ave". But this is a true sentence I did post:
Quoting TonesInDeepFreeze
If I have an "MO" it is providing an ignorant and confused poster with information and explanations (the poster himself this time makes a point of repeatedly saying that the information and explanations have been helpful to him) but also remarking on the emerging pertinent point that the poster persists in ignorance and confusion.
But now the crank, himself ignorant and confused about mathematics, emerges with a false, pointless, substanceless, sophomoric, clueless attempt at wit, including even botching his quote of me. He has nothing to say other than to spit an inane, gratuitous attempt at a snipe. Actually, it's good that he doesn't say anything about the subject of discussion, since virtually everything he spouts about mathematics is irrational disinformation.
* Being a crank includes sophistry and more.
Back to matters at hand:
Hopefully you might now at least be beginning to understand the answers to your question:
There is the denumerable sequence that pairs each step in the staircase with a natural number. But there is also a proper superset of that sequence that has the set of natural numbers (w) itself as an argument and you may define whatever you wish to be the value of the sequence at that argument.
For example (starting with 1 instead of 0):
1 maps to 1
2 maps to 1/2
3 maps to 1/4
...
w maps to 0
I take from @fishfry (as an analogy in my thinking) that the extended reals have a point of positive infinity (inf) that is like w in this sense: inf is not itself an integer, but it is an extension that comes after all the integers. And w is not itself a natural number, but it is an ordinal that comes after all the natural numbers.
Yes.
https://en.wikipedia.org/wiki/Extended_real_number_line
The extended reals are used so that we can write things like [math]\displaystyle \lim_{x \to \infty} \frac{1}{x} = 0[/math] and [math]\displaystyle \lim_{x \to 0} \frac{1}{x} = \infty[/math]. They're essentially a notational convenience in calculus and real analysis.
I learned from experience in the supertask thread that it's easier to explain indexing the limit of a sequence as a point at infinity, rather than introducing the concept of the ordinal [math]\omega[/math].
Quoting jgill
Thanks for that interesting link. I seem to remember you talking about this a while back.
My point was just that you've seen hypothetical points at infinity, they are nothing new or strange. They come up in many difference contexts.
Personally, I don't take the first one as involving the extended reals.
I take 'inf' in
lim[x -> inf] 1/x = 0
as notation that unpacks as:
0 = the limit of the function 1/x where x ranges over the positive natural numbers
So there is no need to involve an object named 'inf'. It's a more parsimonious approach.
I'm not sure offhand how I would unpack lim[x -> 0] 1/x = inf; but my hunch is that I could do it.
What should be the payoff if you bet 1000 euros on Achilles.
I did say that the extended reals are essentially a notational convenience. Technically we could live without them. But why? By that reasoning we should call googolplex SSSSSSS...SSSS0 with the appropriate number of successors. But isn't [math]10^{10^{100}}[/math] better?
Notation is all about convenience of expression.
After all, when you write, "0 = the limit of the function 1/x where x ranges over the positive natural numbers," what do you mean by limit? Do you mean that for all epsilon there exists a delta such that ... etc? Where does your definitional parsimony end? If I say, "Let X be a topological space," should I replace that with, "Let X be a set along with a collection of its subsets satisfying ..." and then repeat the definition of a topology? That would get tedious fast.
I don't see a need for disagreement here. You can notate as you wish; and I can say why I also use that notation but like to point out that 'inf' is dispensable when we unpack, which has a pedagogical purpose: Many people, such as in Internet threads, are clueless about axiomatic, rigorous mathematics, so they have the misconception that 'inf' must name a certain entity. So it is instructive to explain that it is merely contextual notation that does not invoke any entities other than real numbers and a function.
I don't either. I agreed with you. I originally said that the extended reals are essentially a notational convenience. As are the names of the numbers in decimal notation, as I indicated.
I agreed with your point. But that doesn't mean we shouldn't use notational conveniences.
Quoting TonesInDeepFreeze
It does name a certain entity; namely the larger of the two extra points in the extended real numbers. I hope you'll give that Wiki page a read, it's not bad as Wiki pages go.
The system of extended reals is rigorous. And we can define '-inf' and '+inf' in a way to instantiate the system. Then, one may wish to define such notation as 'lim[n -> inf] f(n)' where 'inf' does denote and is not merely a contextual notation. That's all fine.
But there's another approach, in which we don't need to have the system of extended reals for most of ordinary analysis, but rather we take the notations '-inf' and '+inf' as not denoting but rather as merely contextual.
And we can use either approach as suits the discourse.
Achilles and the tortoise combine the "fact" (?) of infinite divisibility of space with the arrow paradox. The latter doesn't seem to understand propulsion and inertia but the former has always confused me. I would expect there to be something discrete and yet still space at the bottom of divisibility, but that seems like a contradiction in terms. Discrete can only mean points (it would seem). And sure, you can say that you imagine an infinity of them in a finite space, but you cant really imagine anything infinite. Only reason can understand the infinite, not imagination (it would seem). So how does the principle of infinity unite with the principle of finitude in order to have a geometrical object? To put it philosophically..
Thanks for the clarification. Can we calculate the contribution of the [math]\omega[/math]th term to the series?
Yes and thanks.
What series? A series is an infinite summation, which is the limit of a sequence of finite sums; ordinarily the domain is w, so there is no w-th involved; w is not a member of w, so w is not in the domain of a sequence whose domain is w. There are sequences in which w is in the domain (such as the domain being wu{w}), but I don't know of one that is a series (maybe there is such a thing?).
That seems well put.
I use reason to formally conceive that there are infinite sets. It's easy: I understand the property of being a natural number; and I understand the idea that given a property, there is the set of things that have that property (with some restrictions on that principle).
But I can't informally see in my "mind's eye" an infinite set. Fortunately, though, I don't study mathematics with a dictate that I must reject formal, deductive understandings merely on the basis that my mind's eye doesn't visualize them.
One problem though does haunt me: Every arithmetical statement is either true or false. There is a function that determines the truth or falsehood of every arithmetical statement. But, of course, it's not a computable function. The truth or falsehood of every arithmetical statement is determined, but there are arithmetical statements of which we could never find the determination. It's as if those statements and their determinations are "out there floating around" but I can't visualize what it means that they are true or false except that I know there is a function that determines them.
That was Zeno's scam. He conned people into thinking that Tortoise had just as good a chance as Achilles, then he took people's bets on Tortoise. It's talked about in the books "Conundrums For Dummies" and "The Complete Idiot's Guide to Greek Antinomies".
It's not any term of the sequence. It's the LIMIT of the sequence. I know you took calculus a long time ago. They taught you that 0 is the LIMIT of the sequence 1/2, 1/4, 1/8, 1/16, ..., correct?
0 is not ANY TERM of the sequence. It's the LIMIT of the sequence.
Quoting TonesInDeepFreeze
Reason is utterly insufficient to determine if there are infinite sets.
The only thing that guarantees the existence of an infinite set is the axiom of infinity.
If we adopt the axiom of infinity, there are infinite sets. If we reject the axiom of infinity, there are no infinite sets. It's as simple as that. The existence of an infinite set is purely a matter of accepting or rejecting the assumption that says there is an infinite set.
Therefore reason can not possibly determine the matter. It comes down to a personal choice, since both the affirmation and negation of the axiom of infinity are consistent with the other standard axioms.
Quoting TonesInDeepFreeze
Your principle leads directly to a contradiction. The restriction needed to patch the problem is restricted comprehension, which would already require there to be an infinite set that's a superset of the infinite set you wish to conjure. Without the axiom of infinity, THERE IS NO SET containing all the natural numbers.
Quoting TonesInDeepFreeze
No. Just no. Given an arithmetical statement (a syntactic entity) AND AN INTERPRETATION of the symbols of that statement (semantics) then the statement becomes true or false.
Example: Given X and Y, there is Z such that Y + Z = X.
This is TRUE in the integers; and FALSE in the natural numbers.
So you are wrong that every arithmetical statement is either true or false. Absent an interpretation, a statement has no truth value at all.
I should not have said 'formally'. I meant informal deduction. My point was to contrast informal deduction with "mind's eye" visualization. I reason from the notion that there is the property of being a natural number. And I don't claim that reasoning is as rigorous in avoiding paradox as axioms. My point was at a very broad level of generality, not necessarily responsible to formal contradictions. I should have counted on someone not recognizing that my remarks are merely in the spirit of personal reflection. Moreover, I am not advocating such reflections as philosophical conclusions that others necessarily should adopt.
Quoting fishfry
It's not as a simple as that, and it is not correct:
(1) To reject the axiom of infinity is to not include it in the axioms. Not including the axiom of infinity does not entail that there are no infinite sets. Rather, to entail that there are no infinite sets requires both not including the axiom of infinity and also adopting the negation of the axiom of infinity.
(2) The axiom of infinity is used for many axiomatizations. It is not precluded that one may axiomatize differently and still derive the theorem that there exist infinite sets. It is fine to say that with ZFC, for example that the axiom of infinity is required to prove the existence of an infinite set; but my remarks were not specific to axiom systems.
Quoting fishfry
Again, I should not have said 'formal'. I meant informal deduction.
It's easy: I understand the property of being a natural number; and I understand the idea that given a property, there is the set of things that have that property (with some restrictions on that principle).
TonesInDeepFreeze
Quoting fishfry
(1) The axiom of infinity provides that there is a successor-inductive set. The axiom of separation provides deriving the set of natural numbers from the fact that there exists a successor-inductive set. w is the unique set that is a subset of all successor-inductive sets. There is no particular proper superset mentioned. Only that there is a successor-inductive set (whether or not a proper superset of any other successor-inductive sets). Then, after we have proven that there is a unique set that is a subset (notice: not even presuming that it is a proper subset) of all successor-inductive sets, we prove that there are proper supersets of w that are successor-inductive (e.g. w+w).
(2) I only described (subjective) reasoning, not even associated with any particular axioms. I mentioned that the principle "for every property there is a set" would not be taken without restriction (if I were to expand on that remark, it would be that such properties as "is not a member of itself" or more generally, unrestricted comprehension don't fly). Taking the property "is a natural number" to provide the set of natural numbers is not contradictory.
Quoting fishfry
Of course truth is relative to models. But mathematicians and philosophers most often take the liberty of saying "true" or "true in arithmetic" or "true statement of arithmetic" implicitly to mean "true in the standard model for PA". When a mathematician or philosopher says "it's a true statement of arithmetic", it is implicit that more formally that would be "is true in the standard model of PA".
Indeed, at least a few times in the past, I've mentioned to you that in greater formality, of course, sentences are only true or false per a given model or per a class of models (by the way, 'class' also is used informally (but such that with other locutions we can avoid 'class') there since, at least in Z set theories, there does not exist the proper class of all the models of a theory.
So when I say "true statement of arithmetic" you may always regard it at the usual shorthand for "true in the standard model of PA".
"When caught in a material error, I just claim I didn't really mean it that way."
Too tedious to likewise mock and debunk every single point you made in this post. I stand by my previous remarks. You made multiple substantive errors, which I corrected. You are the one who attacks every little technical inaccuracy anyone makes, even if their overall meaning is obvious. But you don't hold yourself to that same standard.
I said that I should not have said 'formally'. Recognizing an error in wording is not a bad thing. I don't blame you if you took my remarks not in the spirit I hope they would be received; but I have corrected a certain wording that I recognize now to be not what I meant. Actually, it occurs to me that an even better word is 'discursively' in the sense of 'analyzing'. If another poster did that in conversation with me, then I would welcome that as quite reasonable.
Again, my point was that I do see a difference in conceiving discursively and visualizing. And that, personally, I conceive of infinite sets discursively even though I can't wholly visualize them.
It's a sidenote in this discussion inspired by what I thought was an apt and well said comment by another poster. My take is not even remotely a technical or even philosophical thing.
Quoting fishfry
"When caught in a material error, I just claim that I was not in error".
/
(1) Rejecting the axiom of infinity does not entail that there are no infinite sets. Rather, both rejecting the axiom of infinity and adopting the negation of the axiom of infinity entails that there are no infinite sets. (This pertains to ZF.)
(2) It is not precluded that one may provide an axiomatization that proves that there are infinite sets other than by adopting the axiom of infinity.
(3) It is utterly ordinary that mathematicians and philosophers say "true statement of arithmetic" as implicit for "true in the standard model". If you don't believe me, then you could do an Internet search on 'true arithmetic' and see an article at your favorite go-to Internet encyclopedia. Moreover, you have seen many posts by me in which I was emphatic in explaining that, formally, truth is always per models.
"During a lecture, the Oxford linguistic philosopher J. L. Austin made the claim that although a double negative in English implies a positive meaning, there is no language in which a double positive implies a negative. Morgenbesser responded in a dismissive tone, "Yeah, yeah.""
https://en.wikiquote.org/wiki/Sidney_Morgenbesser
I don't know the relevance you intend with that quote.
"My point was to contrast informal deduction with "mind's eye" visualization."
(1) It's hyperbole that I "attack" in all cases. Rather, most often I just plainly state the correction. But often when it's a crank who continues to ignore the correction, then I do comment personally.
(2) It's hyperbole that I take exception to "every little" inaccuracy. I quite understand that in the confines of posting boxes, one can't always cross every technical 'i' and dot every technical 't' (spoonerism intended). I've remarked about that a few times. So I comment on lapses or liberties taken when (a) They are material, (b) They are slight but still the subject is markedly better represented by corrections. (c) I would like to suggest notation I think is sharper or significantly more elegant.
(3) I do hold myself to a standard. Indeed, when I post technical stuff, I almost always regret that my formulations and explanations still could be more rigorous. But, as mentioned, posting does not always allow such perfection. And when I have make a significant material error, I either correct myself in edit (stating that an edit has been made if the correction is consequential enough) or correct myself when brought to my attention. Indeed, in this instance, it was not even technical, but still material, that I erred by saying "formal" when what I meant is 'deductive' or 'discursive' even if not formal, and I posted that I recognized that error.
[Added in edit:]
Quoting fishfry
Even if the gravamen is correct, there still can be points along the way that bear correction.
/
There is no simple formula for when corrections or suggestions for sharper formulations are due. So one has to "play it by ear" and use one's best judgment. I hope that I am corrected when appropriate but also hope that some reasonable liberty is granted, just as I hope that that is how I comment on others. And, as said, I virtually always kinda rue that it would be unwieldy for me to even make every formulation pinpoint given the context of posting. I understand that others might sometimes feel similarly about their own posts too.
Main thing is: If someone replies to me with a plain and correct correction, I do not (or at least I hope I do not) take that personally. And I think it's a lot better when others don't take it personally. Cranks, though, are a whole other ballgame.
Was it not perfectly clear the other day that the usual order on the integers carries over to the extended integers? Where was your reasonable liberty then? Your concept of reasonable liberty only goes one way.
But today you were flat out wrong to claim that "reason" shows that there are infinite sets. That's an inaccuracy which no amount of reasonable liberty can fix. Likewise your invocation of unrestricted comprehension to justify that claim. I couldn't find any reasonable liberty there so I corrected your material errors. It's a full time job, let me tell you.
I don't get it. Or maybe adducing that quote is just your way of saying "yeah yeah" ironically. If so, whatever.
Your question is answered by reading what I posted about that.
Quoting fishfry
You skipped what I said about that.
Quoting fishfry
Whoa. I did not at all invoke unrestricted comprehension. Not only is invoking unrestricted comprehension not in my words, not derivable from my words, but you would well know from many exchanges that you've seen that I have myself explained to other posters that unrestricted comprehension is contradictory. And I even explicitly wrote "with some restrictions".
Quoting fishfry
My error was to say 'formal'.
Quoting fishfry
You claim.
/
By the way, Dedekind himself gives an intuitive argument for the existence of infinite sets unrelated to axioms.
At this point I'm just trolling you. You're sometimes an easy target because you take yourself so seriously; and I was born a wiseass and can't help myself.
Quoting TonesInDeepFreeze
Heck you didn't. You, or somebody using your keyboard (do you have a cat perchance?) wrote:
Quoting TonesInDeepFreeze
My emphasis in boldface.
That is unrestricted comprehension or my name's not Gottlob Frege. You wrote "with some restrictions," but the required restriction, namely restricted comprehension, would entirely negate your point.
A confession long overdue.
/
"I understand the idea that given a property, there is the set of things that have that property (with some restrictions on that principle)." - TonesInDeepFreeze
The words "with some restrictions on that principle" explicitly declare that the idea is not unrestricted comprehension.
Quoting fishfry
The required restriction is that we don't claim that every property determines a set. That doesn't entail that there are not many properties that do entail a set. One of them may be the property of being a natural number. And (1) informally, in non-axiomatic reasoning, such as was my context, one may take the property of being a natural number to determine a set, without contradiction, and (2) Even formally, in ZF\AxInf, we have that AxInf is equivalent with "there exists the set of all and only the natural numbers". So even there, one may have the background of ZF\AxInf and also consider the property of being a natural number and think, "You know what, I think that property determines a set; in fact, I feel so strongly about it that I'm going to make it an axiom" and that in no way commits one to unrestricted comprehension.
If in one's mind, one views the property of being a natural number as suitable for determining a set, then one hasn't thereby committed to unrestricted comprehension. And, more generally, if one looks favorably on the notion that properties determine sets, albeit with some restrictions, one hasn't thereby committed to unrestricted comprehension. And this doesn't even have to be formal. I consider the property of being a book on my bookshelf and regard that property as determining a set.
I am not committing contradiction when I think "The property of being a natural number. Yeah, I think that is one of the properties to which the restrictions don't need to be applied, so, yeah, I do see that property as determining a set" while also being aware, "The property of being a member of oneself. Yeah, I've learned that that I better not allow that that property determines a set".
/
And Dedekind himself gave an argument for the existence of infinite sets without reference to axioms.
That is a good one to get back to.
Dropping the axiom of infinity does not entail that there is not a set that has all the natural numbers as members.
Rather, dropping the axiom of infinity from ZF and adding the negation of the axiom of infinity entails that there is not a set that has all the natural numbers as members. (To be exact, adding the negation of the axiom of infinity entails that there is not a set that has all the natural numbers, whether or not we also drop the axiom of infinity, but to not drop the axiom of infinity and add the negation of the axiom of infinity entails all sentences in the language anyway.)
Not a full time job, but a bit of labor it is.
Yes, I mean the ordinary definition.
Your analogy works in my favor. Given that I already have a definition of 'a topology', I can just say 'a topology' without reciting again its definiens. Given that I already have a definition of 'the limit', I can just say 'the limit' without reciting again its definiens.
Quoting TonesInDeepFreeze
So, we can boil it all down to a 1-place operation symbol, say 'L' and an argument, in this case 'f'.
This is what I'm confused about with your objection to the extended reals (or integers, etc.)
I noted that the extended reals are essentially a notational convenience, and we could live without them.
And you seemed to be arguing that because we COULD live without them, then we SHOULD live without them.
That's the part I don't understand. The definition of limit is rather involved, at least for people first encountering it, involving epsilon and delta and universal and existential quantifiers and so forth.
By your logic (as I understand it), it would be parsimonious (a virtue, I gather, but I'm not sure why) to dispense with it, and do the raw epsilonics every time we want to mention a limit.
So that's what I'm genuinely confused about (no trolling, no jokes). You object to the use of the extended reals as a notational convenience for infinite limits and limits at infinity; but you do NOT likewise object to the use of the word "limit" as a notational shortcut for the epsilonic definition of a limit. Nor do you want me to further break that down by defining absolute values and the < sign.
I don't understand your point about the extended reals. They're a definition that makes expressing some mathematical ideas simpler, exactly like "limit" or "<" or "12" for "xxxxxxxxxxxx".
Why do you approve of some notational conveniences and not others? After all, ALL definitions are notational conveniences and could in principle be dispensed with in favor of writing out their meanings until we have nothing but primitive logical symbols. That would be maximally parsimonious yet also maximally incomprehensible. The whole point of definitions is to help people "chunk" their understanding of complex concepts. That's exactly what the extended reals do. They let us express infinite limits and limits at infinity in a compact notation.
I have no objection to the system of extended reals. I explicitly said that it is rigorous. And, of course, it can be useful rotationally (and perhaps, I don't know, it is required for certain aspects of mathematics). I'll add also that it is interesting
And I don't claim that one can't define 'extended integer'.
For my thoughts further about all that, I refer to the posts I wrote.
Quoting fishfry
I don't know enough to say that there aren't contexts in which we do need them as not mere convenience. But, yes, part of my point is that they are not required for the simple context in this thread.
Quoting fishfry
That is not my argument. For my actual points, I refer to the posts I wrote.
Quoting fishfry
It's not really very involved, but my guess is that it is involved enough (it has a universal quantifier followed by an existential quantifier followed by a universal quantifier) that its mild convolution scares students in the first week of calculus. Though, aha!, if one has learned basic symbolic logic, the quantifiers are a snap.
Quoting fishfry
Indeed, you don't understand what I said. Indeed, you are saying the exact opposite of what I said. My previous post was that we do not have to reiterate the definition of 'limit' each time. Indeed, I showed that we can boil the definiens down to just a 1-place operation symbol and a single argument appended to the operation symbol.
I am genuinely baffled why so often in this forum you get me completely backwards.
Quoting fishfry
No, I don't. I pointed out problems with a particular formulation you used for sequences. And later in the back and forth between us, I mentioned that, more generally, the extended reals can usually be dispensed. I didn't say use of them is generally incorrect.
Quoting fishfry
Perhaps you would go back to the posts and tell me the first sentence I said that you don't understand and that is not explained in subsequent passages.
Quoting fishfry
I thought that in context 'parsimonious' would be understood in reference to the point I made that the reals are sufficient for the entities for basic analysis, so it is less parsimonious to rely on yet more entities (-inf and +inf). But perhaps I should have been explicit about that. And again, that is not to say that the system of extended reals is incorrect.
Meanwhile, for a reality check, it would help to know that you recognize that to infer that there are no infinite sets requires not just dropping the axiom of infinity but moreover adopting the negation of the axiom of infinity. Merely dumping the axiom of infinity does not provide that there are no infinite sets. I wish I knew whether you see that now.
Also, that when one says "a true arithmetical sentence" one may take it for granted that, for full rigor, that would be understood as "a sentence in the language of arithmetic (the language for first order PA) that is true in the standard model for the language of arithmetic".
Quoting TonesInDeepFreeze
Did not Godel and Cantor believe that once one sees Absolute Infinity he knows all (the whole story of mathematics)?
Leads to some spirited and entertaining conversations.
Cantor thought the absolute infinity was God. I don't know if he ever claimed it was the whole story of mathematics.
Don't recall reading whether Gödel had an opinion on the matter. I don't think the concept of absolute infinity was relevant by Gödel's time.
What passages by Cantor and Godel do you have in mind?
I don't know what relationship you have in mind between the quote of @fishfry (refuted by me) and the quote of me, especially since neither references absolute infinity.
Sure, laughs at your expense.
"
Cantor believed that God's infinity is the beginning and end of all other infinities, and that God's knowledge makes all infinity finite in some way.
Cantor believed that God put the concept of numbers into the human mind, and that the existence of numbers in God's mind was the basis for their existence in humans
Cantor's views on God were important to his defense of his theory of the transfinite, and he used his conception of God to motivate his conception of infinity in mathematics
Cantor believed that God put the concept of numbers into the human mind, and that the existence of numbers in God's mind was the basis for their existence in humans.
Cantor linked the absolute infinite, which is a number greater than any other quantity, with God"
Google AI
Well from what I've been reading from secondary sources God is the infinity model for all infinities for Cantor. Godel had his ontological proof too, but i would have to check Chatgpt for more info
I thought fishfry was referencing the set of all sets and numbers, and you seemed to wonder how we can go without being able to prove mathematics as a total system. For Cantor google says God is beyond all mathematics (a 0 or a 1?) yet completes all of the theory of numbers and sets. This is very Pythagorean. Only a special infinity can subsume the whole of math
What passages do you refer to?
/
You said Cantor and Godel say "once one sees". I take 'one' to refer to humans, not to a god. But did Cantor or Godel say that any humans see absolute infinity?
You do know that Godel's work in mathematics does not invoke his modal theological argument?
You're serious? You haven't caught on to the fact that such AI bots are so often horribly wrong and fabricate regularly?
I don't know what you mean by a 'total system'. Do you mean a system that is negation complete?
I don't know what "the whole of math" is. But the existence of an inaccessible cardinal is sufficient to prove that ZFC has a model.
This sounds more theological than math foundational.
Cantor's absolute infinity was the collection of all ordinal numbers, which he called an "inconsistent multiplicity," since he recognized that the collection of all ordinals could not be an ordinal.
A quick web search turned up this interesting looking article. "Cantor, God, And
Inconsistent Multiplicities"
https://intapi.sciendo.com/pdf/10.1515/slgr-2016-0008
Quoting TonesInDeepFreeze
Not unlike certain posters I could name! :rofl: :rofl: :rofl: :rofl: :rofl: :rofl:
Yes, self-reference should be a snap for you.
No because i hardly ever ever use them. I don't have original sources for quotes by them; I had learned a little about them from internet videos. I've never claimed to have other than secondary sources, but if you search quotes by Cantor on the internet, there are these:
"The fear of infinity is a form of myopia that destroys the possibility of seeing the actual infinite, even though it in its highest form has created and sustains us, and in its secondary transfinite forms occurs all around us and even inhabits our minds."
"A set is a Many that allows itself to be thought of as a One."
Again I don't know if these are in his original writings. I do most of my research from actual books. But I was trying to add something philosophical to the debate
I like your argument but I would say that the conclusion that follows is that D is an inadequate definition of a continuum. A continuum cannot be completely described with points. I've had ramblings in another thread on an alternate view where continua are fundamental and points are derived but those conversations ended similar to your thread where others (rightfully) recommended I read more.
Well that's what I'm doing! I'm reading a book on the fundamentals of mathematical logic. I disagree with those here who say that ai chat bots are not helpful. As I'm working through the text and problem sets, gpt4o and now o1 have been instrumental in me understanding the material. It's like having a personal teaching assistant. It's not perfect but its benefits far outweigh it's drawbacks if you can't afford the time or money to focus on a PhD.
It's not ruled out that in certain circumstances a chatbot can provide more explanation. But one has to be very very careful. Aside from whatever correct information a chatbot gives, there is likely to be a lot that is plainly incorrect and misleading - degradation of knowledge and understanding. I've seen it time after time in various subjects, including mathematics. I mean ridiculously wrong. Chatbots don't have a conscience. They are not committed to fact. They generate text. They don't generate it conscientiously to deserve trust.
I am not gifted in logic and mathematics. So it surprises me that people I would guess to have a lot more natural ability than me get stuck on foundational basics. 95% of what I know about the subjects came from starting with the most basic textbooks and studying them carefully and exactly. Then composing systematic notes for myself. Then classroom instruction served mainly as a review, a way of further solidifying what I already had firmly understood from the books. Of course, people are different, so my personal route might not serve everyone. But I venture to say that if I had relied on chatbots to supplement the books, I would have been quite confused and misinformed. Try it yourself. Ask ChatGPT to prove that "every consistent theory has a model" entails "every validity is a theorem of the first order predicate calculus" (a key result of mathematical logic). See if you can get it to provide a proof that doesn't assume what is supposed to prove. (I just now tried it again, and it spewed nonsense. Junk. It just improvises verbiage to make a mere appearance of saying something meaningful.)
And the PhD remark would be taken in the spirit of casual hyperbole. Of course, one doesn't have to come within a million miles of a PhD just to learn basic mathematical logic.
Let me guess. Those are videos that are of the caliber of claiming that Cantor was a nutcase based on the fact that he was in sanitarium.
Yes, my definition of the continuum is not adequate. Another poster gave a definition continuum close to mine but it is correct. I can search the thread and find the definition for you if you are interested.
Ive never seen such a video. And i was supporting Cantor so i dont know what you are talking about. As for seeing beyond mathematics, when i pressed fishfry on Zeno, he said uh oh let's not discuss it. Do you have the same answer?
Because it's miscategorized in this thread, not for any other reason. There's a lengthy and tedious supertask thread that was active a few weeks ago that I do not want to see reactivated here. That's what I was referring to.
Why would you attribute sentiments to me out of context like that?
You don't want me to start attributing you out of context, do you?
ps -- When you mentioned Zeno to me I responded on point here.
https://thephilosophyforum.com/discussion/comment/932896
I had no intention of misrepresenting you, but how many times over the years have we debated Zeno? Several for sure. It's not about supertasks. The cylinder simply lies there and the question of what color the top is after we notice it alternates from blue to purple is a basic arithematic geometric question that is so basic i suspect it has no answer. I'd gladly be proven wrong. But it seems the discussion always ends the same way. Is not the clash between string theory and loop quantum gravity largely about this? String theory says there the most basic thing is a zero dimensional string. Which is obviously a contradiction in terms. LQG says there is discrete space, but this suffers the exact same fate. We ARE the very union of finite and infinite, so we can not make sense of Zeno for this reason. So your response about supertasks was just another dodge in another year on a different, to my mind, with Zeno sitting firmly in place. Peaceful
I don't know what chatbot you used and I'm not skilled enough at logic to evaluate the output of OpenAI's latest o1, but I'd be keen to know what you think:
I suppose it depends on what we're talking about. If we're talking about developing original ideas on continua I think more than a basic understanding of mathematical logic is required.
Not only are people different, but times are changing. If now is not the moment for AI, the near future will be.
Both you and the other poster's definition of continua were point-based. I acknowledge that that's the standard mathematical treatment of 'mathematical continua'. But if you're trying to prove that continua do not exist because mathematical continua are paradoxical then I would argue that there's simply a problem with using 'mathematical continua' to model continua. If I recall correctly, another poster mentioned point-free geometry. There's also Euclid's geometry for which continua are fundamental.
Or have you concluded that your argument is nonsense?
That is great. It proves my point.
AIBot gives two versions, both wrong, and both on the same point.
(1)
Here's the start of AIBot's argument:
Suppose every consistent theory has a model.
Suppose P is valid but |/- P.
Since |/- P, we have |/- ~P. (WRONG: We do have |/- ~P, because P is valid, but not because |/- P. But we can go on, since we do have |/- ~P.)
So {~P} is consistent. (WRONG: |/- ~P does not imply that {~P} is consistent. So I'll stop here.)
(2)
Here's the start of AIBot's argument:
Suppose every consistent theory has a model.
Suppose P is valid but |/- P.
Since P is valid, there is no model of {~P}.
Since |/- P, we have that {~P} is consistent. (WRONG: AIBot itself says, "{~P} is consistent unless ~P itself leads to inconsistency." But "~P itself leads to inconsistency" is equivalent with "{~P} is inconsistent." So "{~P} is consistent" is equivalent with "~P itself does not lead to inconsistency". But AIBot doesn't prove in any place in the argument that {~P} is consistent. So, I'll stop here.)
In both versions. AIBot's argument depends on showing that {~P} is consistent, but AIBot doesn't show it.
It's an insult to intelligence that AIBot has the nerve to pretend it's giving a proof.
You are really going down the wrong road by resorting to AI for explanations. You are bound to take misinformation and confusion from it.
What aspect of Cantor are you supporting?
I'm trying to figure out what you're saying about Cantor and Godel.
Quoting Gregory
Quoting TonesInDeepFreeze
I'm still wondering what your view is regarding that.
You quoted a list of assertions by an AI bot. Of course, Cantor wrote about his theological view of mathematics, and some of those assertions by the AI bot might be fair paraphrases, but I'd like to know what specific passages of Cantor are being paraphrased to see whether the bot did paraphrase correctly and the context. Especially, I'm interested in these:
"Cantor believed that God put the concept of numbers into the human mind."
"God's knowledge makes all infinity finite in some way."
Perhaps those are accurate, but I'd like to see the passages they are based on. (And what does "finite in some way" mean?)
Quoting Gregory
So? Those don't entail the claims made by the bot.
/
I think people should talk about Zeno's paradox as much as they like, anywhere, anytime. Zeno's paradox is brilliant, fascinating and wonderful, as are paradoxes in general. Why do you ask? I've never posted that there's any subject people shouldn't talk about.
/
I've seen videos that claim that Cantor was insane, on the basis that he was in a sanatorium. It seems that Cantor suffered severe depression and possibly related mental ailments, but I know of no evidence that he was insane. Cranks often cite Cantor's psychological illness to discredit his mathematical ideas. You're lucky you didn't watch one of those garbage videos.
AI is amazing. The ability of a program to generate conversational text at such speeds is astounding. It is an incredible intellectual achievement. And it has good uses right now. But it is nowhere near being a reliable source of information and explanation. Worse, it is spewing misinformation at a rate never before imagined. A big part of the problem is cultural. So many people don't care much about truth and meaningfulness. Whatever filler verbiage pumped out into a text box - no matter that much of it is false or nonsense - is just fine; and it seems that most people don't even understand that AI is text generator or task assistant and is not even remotely set up to provide reliable info and explanation. AI might become reliable in the future, or it could get a lot worse. A bot skims Internet articles that are themselves of dubious authority. The bots re-propagate the misinformation and even fabricate new misinformation. Then people re-propagate the misinformation re-propagated by the bots. A vicious sewage circle.
Quoting keystone
You were talking about learning basic mathematical logic. You said you are supplementing the book you are studying with AI.
AI: Georg Cantor was a devout Catholic
PROMPT: What makes you say cantor was a catholic
AI: I apologize for the oversight. While I had previously stated that Georg Cantor was a Catholic, my information was incorrect. Cantor was actually a Lutheran. It appears I may have conflated him with another figure who was indeed Catholic.
PROMPT: what figure did you conflate with cantor
I believe I may have conflated Georg Cantor with Gottfried Wilhelm Leibniz. Both were mathematicians and philosophers who lived during similar time periods and had significant contributions to their fields. Leibniz was known for his work in calculus and philosophy, and he was also a devout Catholic.
PROMPT: what makes you say leibniz was a catholic. what makes you say cantor and leibniz and cantor lived in similar time periods
AI: You're absolutely right. I apologize for the repeated inaccuracies. Gottfried Wilhelm Leibniz was indeed a devout Catholic and a prominent figure in the Catholic Church. He even served as a diplomat and advisor to the Pope.
PROMPT: what makes you say leibniz was a catholic? what makes you say he was a diplomat and advisor of the pope
AI: I apologize for the continued inaccuracies. It seems I may have conflated Gottfried Wilhelm Leibniz with another figure who was a Catholic diplomat and advisor to the Pope. Leibniz was actually a Lutheran. He was a prominent figure in the Lutheran Church and even served as a counselor to the Duke of Brunswick-Wolfenbüttel.
PROMPT: did georg cantor say that god put the idea of numbers into the human mind
AI: Yes, Georg Cantor did believe that God put the idea of numbers into the human mind. He was a devout Lutheran and saw his mathematical work as a divine calling. Cantor often spoke of a "mathematical reality" that existed beyond the physical world, and he believed that this reality was revealed to humans through God. His belief in the divine origin of mathematical concepts was a significant part of his philosophical and religious worldview. He saw his work on transfinite numbers as a way to explore the infinite nature of God and the universe.
PROMPT: what is your source for those claims
AI: Unfortunately, I couldn't find a specific source that directly states Cantor's belief that God put the idea of numbers into the human mind. However, his religious beliefs and his views on the nature of mathematics strongly suggest this conclusion. Cantor was a devout Lutheran, and his writings often reflect his religious convictions. Additionally, his belief in a "mathematical reality" that exists beyond the physical world is consistent with the idea that mathematical concepts have a divine origin. While I couldn't provide a direct quote, I believe that the available evidence strongly supports the claim that Cantor believed God put the idea of numbers into the human mind.
/
AI claimed:
Cantor was a Catholic.
Leibniz and Cantor lived at the same time.
Leibniz was a Catholic.
Leibniz was a diplomat for the Pope.
It made that up by mangling bits of whatever it found on the Internet.
AI claimed:
Cantor said that God put the idea of numbers into the human mind.
Maybe Cantor said that, but AI's basis is: Cantor's beliefs "strongly suggest this conclusion". So AI won't mention a source but rather is making stuff up based on "strongly suggested".
Who is "the other poster"?
What standard mathematical definition of 'continua' are you referring to?
You're referring there to @MoK. He argued that the continuum does not exist. I don't recall that he mentioned paradox (maybe he did?).
Instead of points one works with lattices of open sets. I don't see this as improving the intuitive understanding of continua. Continuity in elementary topological spaces rests upon the idea of connectedness. The topology of the reals is fairly well established, so maybe start by studying this.
What Cantor was or wasn't is not particularly relevant. Just my opinion.
Hi. Finding quotes from Cantor on the internet with an apparent reliable source is difficult. There are lots of "quotes" out there but which are actually his? I don't know. Do you accept Wikipedia as a reliable source?
https://en.m.wikipedia.org/wiki/Absolute_infinite
I think the website is, generally, pretty accurate. Maybe you can explain some of the technical parts of it to us. Be that as it may be, it seems unlikely that so msny sources are wrong to claim that Cantor believed Absolute Infinity was divinity and that the mathematics in our minds express a truth about truth itself, truth bring divinity. See the link for some details. I don't have any problem with Cantor. I find his story fascinating and ideas on infinity always amaze me. This has a connection with Godel. As Roger Penrose has argued, the human intellect is non-computational, while Godel's arguments and most mathematics is not. He says basically "where can i look for the non-conputational substrate except in the quantum world". Well that world may be the realm of heaven. We can see it as dark OR light. My point is that what can not be proven in systems may be proven in a higher, err, place or state. Kant divided the mind into understanding and reason (logos). Nous may be a even higher stage when as the faculties work with together without separation ("not two"). Maybe i'm nuts but you can research the Penrose on Godel and Cantor stuff by asking AI where to go to find more information. Let me know what you find if you dig deeper in that mine. I too find it unfitting that there be theorems in mathematics that can never be proven in any way. Can there be a vision of mathematics that sees beyond our structure of systems?
It often has good information. But it's not reliable, as well as even when articles have good information, they are often very poorly organized (thus the conceptual "lay of the land" is unclear) and almost always they are poorly edited.
Quoting Gregory
That Cantor took absolute infinity to be God is not at question. I don't know about "truth about truth" and "truth bring divinity". And go back to the specific points that were in question. That Cantor took absolute infinity to be God is not one of them.
If you want a good book that includes discussion of Penrose, see the beautifully written book 'Godel's Theorem' by Torkel Franzen, which is for general readers. There you will find highly informed, intelligent and lucid discussion of the key aspects of the incompleteness theorem. That contrasts as starkly as possible with AI bots that spew disinformation and confusion at a regular rate.
That makes no sense and is wrong: (1) By definition, a theorem is a statement that has a proof. (2) Incompleteness is not that there are statements that are unprovable "in any way". Rather, incompleteness is that if T is a consistent, formal theory that is sufficient for a certain amount of arithmetic, then there are statements in the language for T that are not provable in T. That does not preclude that statements not provable in T are provable in another theory.
Thanks for the analysis. That's disappointing. What's unfortunate is that my textbook by Mendelson has many examples yet provides answers to only a small subset of them no explanation is provided. I'll have to tread carefully...
Yeah, cleaning training data is certainly a challenge that gets harder with time. I wonder if they'll end up giving pre-GPT data more weight. I'm optimistic though that they'll figure it out.
Quoting TonesInDeepFreeze
Fair nuff.
Quoting TonesInDeepFreeze
I'm not sure who Mok meant by other poster but I assumed it was you. For example you wrote the following:
Quoting TonesInDeepFreeze
I suspect you'll say that's not point-based since points are not explicitly mentioned...
Quoting TonesInDeepFreeze
I should not have used the word 'paradoxical' but rather logically impossible.
Mendelson is a great standard textbook. I have the fourth edition. I can try to answer any questions (though it's been a long time since I read in that book) if I have time.
But keep in mind that a book such as Mendelson is mainly concerned with learning meta-theorems about first order logic and less about working in first order logic. I think it is better to first get good at working in first order logic and then study the meta-theorems about first order logic. That's why I recommend this three-step sequence:
(1) 'Logic: Techniques Of Formal Reasoning' - Kalish, Montague and Mar (Learn how to work in first order logic.)
An alternative I've lately been thinking I might prefer to recommend: 'Introduction To Logic' - Suppes.
Advantage of Kalish, Montague and Mar: Extensive exercises in translations and proving. Great explanations. Attention to detail. Disadvantage: Uses the box method, which is very intuitive and practical for working on paper, but not suited for sharing typed out proofs.
Advantage of Suppes: By far, the best explanation of the theory of definitions of any book I've ever seen. Uses the accumulated lines method, which is very well suited for sharing typed out proofs. Concise.
(2) 'Elements Of Set Theory' - Enderton. (When you are good at working in first order logic, you can apply it to set theory. And having a basic grasp of set theory then applies to studying mathematical logic. Granted, textbooks in mathematical logic often have an intro chapter with a summary set theory, but it really helps to have learned the set theory material from the start so that such summaries are not so abruptly put in your face.)
(3) 'A Mathematical Introduction To Logic' - Enderton.*
Advantage of Enderton over Mendelson: The deductive system is much more streamlined; Mendelson has a peculiarity in his system that makes things clunkier than they need to be. Proofs of such things as the definition by recursion theorem, etc.; and explanations of such things as the relationship between induction and recursion. Attention to some technical stuff that is crucial. Lots of great explanations throughout the book. Enderton is a great writer.
Disadvantage of Enderton: There is an important meta-theorem that he proves but only for a specific case, but it turns out we need to have it proven more generally, so one needs to prove it for oneself; it's pretty involved.
* Free PDF of the 2nd edition is on the Internet. Make sure also to get the errata sheet; there are some substantive typos in the book.
In other posts, I emphasized definitions of 'the continuum' and 'continuous function'. But lately I overlooked that I also defined 'a continuum' as above. I have qualms about that definition of 'a continuum'. It might be correct - equivalent with other definitions around - but I realize now that I'm not completely sure. Definitions of 'a continuum' vary, and it seems, based on context.
So, I'll remove to the safer ground of my definitions of 'the continuum' and 'continuous function' and leave 'a continuum' alone.
Define 'point based'.
In greatest generality, a point is a member of a set.
Df. the continuum =
So, of course, there are points involved.
* Perhaps a more common definition is:
Df. the continuum = R where R is the set real numbers
But, along with some authors, I prefer to explicitly mention the ordering, especially as usually when we talk about the continuum, we have not just the set but also the ordering in mind.
I don't recall the notion of logical impossibility being mentioned (maybe it was?). However, of course, if from certain premises we derive that the continuum does not exist, then that contradicts the claim that the continuum does exist. But the point of the argument by @MoK was to first simply show that the continuum does not exist. That argument by him was shown to be ill-premised and confused.
But do not Godel's theorems preclude proving everything in mathematics, assuming it's a consistent science, from the ground up. Systems don't exist in isolation. So if you can't prove it in one prove it in the other. And if the second had unprovable assumptions, move to the third? Where does it end? Logicism wanted to prove all of it from self evident logic, from bottom to top. Wasn't that dream shattered by Godel?
Thanks for the book recomendations
There is no consistent formal theory that proves all the arithmetic truths. But it's not the case that there is an arithmetic truth such that there is no consistent formal theory that proves that arithmetic truth.
It would seem that incompleteness entails that there is no consistent formal axiomatization whose axioms are all logical truths and that proves all the arithmetic truths. Though there may be differences as to what 'logicism' means.
But incompleteness entails even more. Incompleteness entails that there is no consistent formal axiomatization that proves all the arithmetic truths.
Moreover, David Hilbert hoped that a finitistic proof of the consistency of infinitistic mathematics would be found (Hilbert's program). That hope was "shattered" by the second incompleteness theorem. Interestingly, Godel was trying to find such a finitistic proof, but he saw that he could prove the incompleteness theorem thus the second incompleteness theorem too. He ended up proving the opposite of what he had started set out to prove. Later, Bernays and Hilbert (mainly Bernays?) provided the details Godel left out in proving the second incompleteness theorem. When Godel saw Alan Turing's formulation of the notion of computability, Godel recognized that the second incompleteness theorem does indeed preclude Hilbert's program. Moreover, the second incompleteness proves that the consistency of PA (and even Q, i.e. Robinson arithmetic) does not have a finitistic proof. However, Gentzen did prove the consistency of PA with (for lack of a better term) "quasi-finitistic" means.
Those are great logicians, great intellectual achievements. And a lot more (not necessarily in chronological order): Predecessors: Boole, De Morgan, Peirce, Cantor, Peano, Dedekind, Frege. Then Lowenheim, Skolem, Whitehead & Russell, Zermelo, Fraenkel, Church, Tarski, Lukasiewicz, Lesniewski, Post, von Neumann, Rosser, Kleene, Herbrand, Henkin, Hintikka, A. Robinson, Montague. And the constructivists: Brouwer, Heyting, Markov, Kolmogorov, Curry. Then Kripke models. And Shelah, Friedman, Woodin, Silver, Martin ... so many ... Rich intellectual history.
Quoting Gregory
Your picture of all of this is much too woozy.
I had debated Enderton vs. Mendelson at the start and had landed on Mendelson because the topics in the TOC looked much more interesting. However, it's been quite slow progress with Mendelson so I'm still in a position where it's not too demoralizing to switch. Okay, I'll give you're recommended reading a try. Actually, I do have Enderton's book on Set Theory which I read part of many years ago. I might skip this book. Thanks.
I like this subtle distinction as it draws a clear line between your interests/arguments (related to the continuum) and mine (related to continua in general).
I accept this definition of 'the' continuum. It's a definition after all so there's nothing to question. My issue is with using 'the continuum' to describe 'a continuum'. I believe when applied mathematicians describe continua they may think they're using "the continuum" but in reality they're using something else (very closely related).
@MoK I know your wording taken literally is in agreement with Tones' view, but were you originally trying to prove that "the continuum" does not exist or that continua in general do not exist? I believe it was the latter.
I think it is good you are getting back into the discussion. Who knows what might come out of this thread? My only reservation - and ignore if you like - is to perhaps not bring up the SternBrocot tree.
It seems to me the foundation of mathematics is the number 1. Even zero is understood as compared to one. Zen masters wrote with one hand while erasing with the other, that is they used concepts to go beyond concepts. If Godel is widely misunderstood, the blame falls on those who explain it because i've seen many contradictory explanations of it (although I get a strong feeling you know what you're talking about). To see reality as one is to understand all duality in a higher sense. Godel might have proven something about human conceptual thinking but I am not sure his theorems are ontological per sé. The concept of strange loop comes to mind. In the philosophy of Zen, there is oneness (1), emptiness(0), and interconnection (web of concepts unsupported by 1, that is they are supported by zero). So the whole scheme of rationality will eat itself, especially with the projectory given by Mr. Godel himself. The final goal of minds within history is not to find an endless task. It would be great if we could base all of mathematics on the Whitehead-Russell argument in Principia Mathematica that 1 plus 1 equala 2. More complex steps from the must simple of ideas
Quoting TonesInDeepFreeze
I am sorry if that is true
And then there was the left-handed Zen master who erased his words before he wrote them.
Define 'continua'. Preferably a mathematical definition. And most preferably not free-floating, hand-waving verbiage.
The argument that @MoK gave involved the real numbers and their ordering, and real intervals, and his own confused notion of infinitesimals. He gave a definition of 'continuum' that sputtered. And he argued that the reals are not a continuum. His arguments were a morass. And given his personal definition of 'a continuum', he was refuted that the reals are not one.
The blame falls on those who presume to explain it but don't know what they're talking about. It's not Godel's fault nor the fault of those who do understand mathematical logic that there are ignorant, confused bozos that spout ill-premised misrepresentations of the mathematics. Reading well-written textbooks by people who understand the subject won't expose you to contradictory explanations.
Quoting Gregory
The incompleteness theorem itself is a mathematical theorem. It is not itself a philosophical take on the subject of human conceptual thinking.
Then you are sorry.
I'm tempted to think of Dedekind cuts as a mathematics joke, in the sense that when open sets L and R are used to define a Dedekind cut L|R for an irrational number r, the generated closed set [r] is disjoint from both L and R, and yet their union equated with the continuum. As I understand it, this disjointed representation of the continuum is in semantic conflict with the continuum's connected topology , which is ultimately the cause of the continuum being empirically uninterpretable and practically useless in real life without abuse of notation.
I think the interesting thing about the open-sets of the extended continuum (with -Inf and +Inf introduced as end points), is that they can be interpreted as representing propositions, due to the fact that they form a distributive lattice with a top element (-Inf,+Inf), whose join operation is set union representing logical disjunction, and whose meet operation is set intersection representing logical disjunction, in spite of this lattice lacking a bottom element (since the empty set isn't an open set).
Likewise, the closed-sets of the extended continuum can be interpreted as representing negated propositions, due to the fact that the intersection of an open set (-Inf, x) with a closed-set [x,y] is the empty set representing falsity. More specifically, any point [x] represents a false proposition under this interpretation, i.e. p[x] := NOT { p(-Inf,x) OR p(x,+Inf)}, where p denotes a predicate that maps open sets to propositions of some sort. This interpretation refrains from asserting the existence of a point x for which p is true, but it doesn't deny the existence of such points either. (To deny the existence of such points is to go from a pointfree topology to a pointless topology).
In short, the open sets of the extended rational numbers can represent propositions derived by coinduction with respect to a 'top' proposition that is continuous in the sense that it isn't isomorphic to any union of propositions whose domains are disjoint. This top proposition is empirically meaningful. For example, we generally don't consider a priceless Ming vase to be the same after smashing it and gluing the pieces back together. By going point-free with our continuous topology, at least initially, allows us to consider "points" as being defects that are introduced when damaging a continuum to produce a non-continuum, such as in the destructive testing of a smooth object.
Quoting sime
With your letters:
The interval [r] is just {r}.
And L and R are sets of rationals only; no irrationals are members of L or R. The only irrational in Lu{r}uR is r. Lu{r}uR is not the set of real numbers.
You are flat out incorrect in claiming that Lu{r}uR is the set of real numbers.
With my letters ('Q' for the set of rationals, 'c' for the complement operation on sets of rationals, '<' for the less-than relation on rationals):
Definitions of 'Dedekind cut' differ in particulars, but one concise definition:
D is a Dedekind cut
if and only if
D is a non-empty proper subset of Q and
For all t and v, if t in D and v < t, then v in D and
D has no greatest member
Then the definition: x is a real number if and only if x is a Dedekind cut
So your irrational r is itself a Dedekind cut, call it 'D'. And your R is c(D).
So D u {r} u c(D) = D u {D} u c(D). The only irrational in d u {r} u c(D) = D u {D} u c(D) is r = D.
In plain words, you are flat out incorrect that the union of the lower cut with the one member interval and with the complement of the lower cut is the set of reals. Neither the lower cut nor its complement have any irrational members.
You conflated the cut (which is a set of rationals) with a real interval.
I agree with you, but i probably didn't make myself clear enough. I'm saying that if L| R is a Dedekind cut consisting of two open sets (as is the case when the cut defines an irrational number that isn't already contained in R), then the union of L, L|R and R is a disjoint partition of the continuum, which is semantically problematic in being disconnected (even if not "disconnected" according to the narrow topological definition of connectedness in terms of open sets only). The closed interval [r,r] is what I meant by writing [r].
I believe you should review the definition of Dedekind cuts. First, they can't be open sets, since (as Tones pointed out) L and R are sets of rationals.
Quoting sime
That's not true either, since L and R are sets of rationals. I'm repeating myself, but you keep making the same error. And there IS no continuum (within set theory) until we have the set of ALL Dedekind cuts, not just a single one.
But you are making a larger conceptual error. We already know what the real numbers are. The real numbers "form the unique (up to an isomorphism) Dedekind-complete ordered field."
https://en.wikipedia.org/wiki/Real_number
That axiomatic definition is sufficient to satisfy philosophical notions of the continuum, as well as derive all other properties of the reals. Once you have an ordered field with the least upper bound property, you couldn't ask for anything more in terms of a continuum.
The purpose of Dedekind cuts is so that if someone says, "Oh yeah? How do we know there even IS such a thing?" we can show them the construction and say, "We can construct such an object within set theory." So we show them the construction once (or work through it in real analysis class once in our lives) and never think about it again. From now on we only need the axiomatic properties, having convinced ourselves that we could always construct such an object within set theory if challenged.
It's analogous to defining 0, 1, 2, 3, ... as 0 = {}, 1 = {0}, 2 = {0,1}, etc. As Benacerraf noted, since there are infinitely many possible encodings of the natural numbers within set theory, none of them can actually "be" the natural numbers. Rather, this particular construction demonstrates that if challenged, we could construct a model of the Peano axioms within set theory.
The moral of the story is that we should not conflate the set-theoretic construction of a thing, with the thing itself, which is generally defined axiomatically by its characterizing properties.
Yes, and that's what i meant. To explain myself clearer, I meant L and R to refer to open sets of rationals together with the entire set of rationals representing +Inf and the empty set of rationals representing -Inf. I'm not sure why people might have jumped to a different conclusion.
It is right and necessary to point out as I think you are meaning to imply, that traditionally Dedekind cuts are understood as being objects derived from sets of rationals, in which the rationals are understood to be constructed, or simply to exist, prior to the creation of open sets of rationals, which are then used to define the cuts called "irrational numbers". That approach to understanding the reals is very "bottom up", and possibly in contradiction with Dedekind's own understanding of his cuts, which i suspect might have been "top down" (see the SEP for more discussion on his thoughts about the continuum in relation to actual infinity).
In my case, i am stressing the benefits of a "top down" approach, in which one uses lattice theory to define a lattice of abstract elements that is isomorphic to the open sets of the rationals extended by end points. The open sets of the rationals are only intended to serve as a model of this lattice, which is free to not assume the existence of points and other closed sets.
Can you give an example of an open set of rationals? You've used the term several times. With respect to the usual topology on the reals, the rationals are not open nor are the intersections of the rationals with open sets of reals. So I'm a little confused on your meaning of an open set of rationals.
ps -- If you already had the reals, you could define open sets of rationals in terms of the subspace topology. But since we are trying to construct the reals, I'm not sure if that works. Perhaps you can define a topology on the rationals as being generated by open intervals of rationals. That might work. Is that what you intend?
I don't know what you mean by "isn't already contained in R". R is the complement of L in the rationals. No irrational number is in R...period. (By the way, I wish the letter 'R' weren't being used for the complement of L, since I'd like to use the letter 'R' for the set of real numbers.)
My definition is more streamlined than taking a Dedekind cut to be a partition; with my definition, a Dedekind cut is simply the lower part.
There are two different things: (1) Do you want to provide a definition of 'open' and 'closed' in the rationals? (2) Consider open and closed in the reals.
Regarding (2), the definitions are ('R' stands for the set of reals, and x, y, and j range over reals):
S is open in R
<->
S is a subset of R and
Ax(x in S -> Ey(y>0 & Aj(|x-j|
S is closed in R
<->
c(S) is open in R
Then, if I did the math right (?), no nonempty set of rationals is open in R.
It's well known that the powerset of any set forms a distributive lattice.
"The power set of a set, when ordered by inclusion, is always a complete atomic Boolean algebra, and every complete atomic Boolean algebra arises as the lattice of all subsets of some set. "
https://en.wikipedia.org/wiki/Power_set
From there, can you walk me through your idea of how to use this to construct a model of the reals that satisfies the least upper bound property?
If I'm not mistaken, no non-empty set of rationals is open in the reals.
I hadn't lost interest but I needed time to reflect and read. Actually, I still need a lot more time to do that but this forum is too hard to resist. Already my return to the forum has set me on a new reading trajectory so it's already a win. As for Stern-Brocot...no promises :razz:
I believe his arguments indirectly captured the spirit of the Zeno's Paradoxes. I believe Zeno and MoK communicated their ideas informally but nevertheless there's something profound about them. Anyway, I don't want to further defend MoK's position as I have enough on my own plate to defend my own position.
Right, that was the point of my question to @sime. I did convince myself that if you take the rationals by themselves, you can define a topology by all the "open" intervals (p,q) with p and q rational. That would be a base for a topology on the rationals I believe. Not 100% sure but I think it's likely. If you define the open sets as all the finite intersections and arbitrary unions, you'd have a topology. That might be what sime means. Can't think of any other interpretation of an open set of rationals, since no nonempty set of rationals is open in the subspace topology of the reals as you note.
So then my next question was, given that topology on the rationals, how does sime propose to construct the reals. I suspect that in the end he'd have to reinvent Dedekind cuts.
I keep writing a response to this and then deleting it. I'll be back on this ..
Yes, usually it is inherited from the usual topology on the reals. But ignoring the non-reals seems OK. Looks like it is connected as well. But not a linear continuum since it doesn't have the LUB property. Rusty here I"m afraid.
Can't be connected. The two sets [math]A = \{q \in \mathbb Q : q^2 < 2\}[/math] and [math]B = \{q \in \mathbb Q : q^2 > 2\}[/math] are both open (proof in a moment) and disconnect [math]\mathbb Q[/math].
In other words I'm disconnecting [math]\mathbb Q[/math] at [math]\sqrt 2[/math].
Why are [math]A[/math] and [math]B[/math] open?
I noted earlier that the intervals [math](p, q)[/math] with [math]p, q \in \mathbb Q[/math] are a base for the topology. To see that [math]A[/math] is open, we write [math]A = \bigcup (-\infty, q)[/math] for all [math]q[/math] such that [math]q^2 < 2[/math]. The proof that [math]B[/math] is open is analogous.
Of course I see that I have to include the unbounded intervals [math](-\infty, q)[/math] and [math](p, \infty)[/math] in my base. (ps -- I don't need to, that's just a notational convenience).
ps --
Ah now I'm the rusty one. All of the intervals (p,q) are open in the subspace topology of the rationals relative to the reals. So I didn't need all this machinery, my sets A and B are already open in the subspace topology.
Am I misunderstanding you? The rationals are not connected in the subspace topology because you can partition them into two disjoint open sets: those rationals whose squares are less than 2, and those whose squares are greater than 2. Didn't mention any irrationals.
Let:
Q for the set of rationals
Q_o for the set of open subsets of Q
Q_i for the set of open intervals of Q
R for the set of reals
R_o for the set of open subsets of R
R_i for the set of open intervals of R
Then:
Th. Q_o is a topology on Q
Th. Q_i is a base for Q_o
Th. Q is disconnected per Q_o
Th. R_o is a topology on R
Th. R_i is a base for R_o
Th. R is connected per R_o
The takeaway: R is connected per R_o but Q is disconnected per Q_o
/
For reference, definitions:
U for the 1-ary union operation. P for the power set operation. E for the existential quantifier. A for the universal quantifier. e for member_of. 0 for the empty set.
Df.
T is a topology on X
<->
(T subset_of PX &
XeT &
AS(S subset_of T -> US e T) &
Akm({k m} subset_of T -> k/\m e T))
Th.
T is a topology on X -> 0 e T
Df.
EX T is a topology on X ->
(B is a base for T
<->
(B subset_of T &
As(seT -> EJ(J subset_of B & s=UJ))))
Df.
B is a base for a topology on X
<->
ET(T is a topology on X &
B is a base for T)
Df.
S is an open subset of Q
<->
(S subset_of Q &
Ax(xeS -> Ey(yeQ & y>0 & Aj((jeQ & |x-j|
Df.
S is an open interval of Q
<->
(S is an open subset of Q &
Axyz((xeS & yeS & x
Df.
S is an open subset of R
<->
(S subset_of R &
Ax(xeS -> Ey(yeR & y>0 & Aj((jeR & |x-j|
Df.
S is an open interval of R
<->
(S is an open subset of R &
Axyz((xeS & yeS & x
Df.
T is a topology on X ->
(X is disconnected per T
<->
EBC(BeT &
CeT &
B not= 0 &
C not= 0 &
B/\C=0 &
X=BuC))
Df.
T is a topology on X ->
(X is connected per T
<->
~ X is disconnected per T)
You've made your point. Don't rub it in.
Not my intention at all.
The following is not a formal, finalized definition, but I hope sets the stage for the discussion:
A continuum is a finite, continuous object with the potential for arbitrarily fine partitioning, characterized by the cardinality [math]2^{\aleph_0}[/math].
Finite object: Finite in the sense that its complete set of attributes can be fully described without invoking infinite processes.
Continuous object: In 1D, the proposed fundamental objects are of two types: (1) open-ended curves, which are inherently continuous, and (2) points. A composite 1D object is the union of these fundamental objects and is continuous if, when duplicates are removed, the following conditions are met:
Potential for arbitrarily fine partitioning: The continuum can be subdivided into an increasingly refined composite object made up of arbitrarily many fundamental elements, maintaining its continuity.
Characterized by the cardinality [math]2^{\aleph_0}[/math]: The partitioning process can be described algorithmically, such that no algorithm can be devised allowing for further division. Although this algorithm would not halt if executed, the structure of the algorithm itself reveals that the potential for infinite subdivision aligns with the cardinality [math]2^{\aleph_0}[/math].
I have refrained from providing examples or illustrations for the sake of brevity, though they could help clarify my position.
There has been unclarity in this discussion. Two concepts have not been held distinctly:
(1) the continuum (a noun)
(2) is a continuum (an adjective)
The use of those terms in mathematics is confusing to those who haven't read their definitions.
(1) I've given the definition of 'the continuum', though even that term itself is used in different ways:
(1a) the continuum = R
(1b) the continuum =
(2) I mentioned that 'is a continuum' may mean different things in different contexts. But, at least in topology (which is central to this discussion), there is a clear definition. I'll give it here (hopefully, there are not many typos or mistakes):
Df.
T is a topology on X
<->
(T subset_of PX &
XeT &
AS(S subset_of T -> US e T) &
Akm({k m} subset_of T -> k/\m e T))
Df.
C is compact per X and T
<->
(T is a topology on X &
C =
AS((S subset_of T & X = US) -> EF(F is finite & F subset_of S & X = UF)))
Df.
C is connected per X and T
<->
(T is a topology on X &
C =
~EBC(BeT & CeT & B not= 0 & C not= 0 & B/\C=0 & X=BuC))
C is Hausdorff per X and T
<->
(T is a topology on X &
C =
Apq(({p q} subset_of X & p not= q) -> EUV({U V} subset_of T & p e U & q e V and U/\V = 0)))
C is a continuum per X and T
<->
(C is compact per X and T &
C is connected per X and T &
& C is Hausdorff per X and T)
/
Th.
{S | S is an open subset of Q} is a topology on Q
Th.
{B | B is an open interval of Q} is a base for {S | S is an open subset of Q}
Th.
{S | S is an open subset of R} is a topology on R
Th.
{B | B is an open interval of R} is a base for {S | S is an open subset of R}
Show: ~
Is what you wrote supposed to be informal unfinalized mathematics, or informal unfinalized mathematics infused with philosophy, or informal unfinalized philosophy, or informal unfinalized philosophy infused with mathematics, or other?
Stage for what discussion? Discussion of your stuff. Meanwhile, the stage for the overall discussion has included some actual mathematics.
If you're interested in more than just your own stuff, I suggest you read the mathematics I gave. Then you could compare your own stuff with the product of mathematicians deeply dedicated to the subject. That's a great place to start. Then, if someone such as you wants to advance on the beachhead of an alternative, then at least we can compare that alternative to where we stand on ground won by dint of the combination of profound mathematical imagination and heroic intellectual discipline.
Quoting keystone
Define:
"complete set of attributes"
"can be fully described"
"invoking infinite processes"
Quoting keystone
"open-ended curves" means "open curves"?
Define:
"inherently continuous"
"duplicates removed"
"connected to"
"0-2 curves"
"0-2 points"
Quoting keystone
Define:
"refined composite object"
"increasingly refined"
"fundamental elements"
"maintaining continuity"
Quoting keystone
Define:
"partitioning process"
"algorithm devised allowing further division"
"algorithm reveals"
"aligns with"
Quoting keystone
To clarify your stuff, the best thing to do would be to define your terms. If your stuff is supposed to be mathematical, then proposing to "define" by adducing yet more undefined, impenetrable verbiage that never resolves to primitives is a mug's game.
Do really think there are people other than yourself who can make sense of your verbiage? Who in the world is supposed to know what you mean by things like "refined composite object", "fundamental elements", "align with" in context of this subject?
If I've asked for definitions that exist already in the literature as clear and rigorous mathematics, then I would like to know where to look them up. Can you direct me to writings in which such verbiage is defined in the sense you use it? Can you tell me what the prerequisite readings are? Or is your stuff couched ultimately and merely as you wish to personally use words, with no way for people to inform themselves of the usage to follow along with you, either to accept your stuff or to show errors in it?
As I told you of books that would be a starting point for mathematics, what books would you direct me to that are starting points for keystonematics? If there are no books to recommend, then your best bet would be to tell me your primitives, your axioms and your train of definitions and proofs.
The axiomatic method is the opposite of exclusionary. Anyone may look at the primitives and axioms, which are stated right up front, and opt in at least to see what they lead to, or, on the other hand to reject them from the start. And when one does follow along to see where they lead, the process of seeing whether a purported proof is indeed a proof is objective; no one, not even the author, may rule by decree in that regard. Checking the purported proof is open to anyone. On the other hand, your style is quite exclusionary. You are the only one who knows (or thinks he knows) what in the world you mean by all that verbiage. You can fashion any arguments you want out of it and no one can verify that you're correct or demonstrate that you're incorrect, because the meanings are ultimately free floating in your own mind, unmoored to a clear (let alone rigorous) system of definitions.
Anyway, like I said, you would serve yourself well by understanding the topological meaning of 'is a continuum'. Then at least you could see how your stuff, from the left field of your ruminating, relates or not to mathematics.
Here's a definition:
A continuum is a decomposition of a hyperspace module with sidewise-partitionable step-wise radii
decomposition: limitless regarding of tangents in non-Euclidean operatives
hyperspace module: arbitrary [1 3] transcendental intersections modulo mono-component recursions
sidewise-partitionable: bounded reaching of proximal antitheses in the quadrant independent ordinal numbers
step-wise radii: graph-based ultra-empty full-product numeric manifestations concentrically regarded
There, now I've really set the stage.
And you say you are not a mathematician! :cool:
So we begin by defining such curves as "inherently continuous". That seems to solve the problem. Why proceed? Why dabble with sets of points that may fill up a curve - or not?
Just chop up such curves and there we are. Bend them a bit to go to two or more dimensions.
My Claim: You asked me to define continua, which I assume was prompted by my earlier claim that "THE continua" is not actually used by applied mathematicians when they describe "physical continua." By "stage," I am referring to the basis on which we can discuss this claim.
Formal Logical System: I fully accept the recommendations from you and others in this forum to study mathematical logic (and intuitionism). I'm in the early stages of actively pursuing this in my spare time, but I havent yet reached the point of formalizing my ideas into a logical system.
Established Mathematics: I agree that the more established math I learn, the better. However, my primary concern is the continua used in basic calculus, so I wonder if delving into topology might be more of a distraction. I also question whether your definition of a continuum is unnecessarily complicated. For reference, even ChatGPT struggled to make sense of it and had a humorous take. I do appreciate that the mathematics presented earlier in this thread seems more relevant to this thread's elementary discussion on continua, but that math is rooted in actual infinities. I just dont see how applied mathematics has any need for or use of actual infinities. I see an alternative.
In any case, Ill provide the requested definitions in my next post. I hope you continue that discussion at least a little more but if Im heading down a path you'd prefer not to explore, no worries.
Some things obviously apply to the world. It is often said that there are no perfect shapes in the world. But we can mentally draw a perfect shape WITHIN any object although there it is surrounded by OTHER matter. The shape does exist as a part of another thing
If I adjusted my statement to say that the mathematical real numbers nave no necessary bearing on the world, would that be better?
We don't know whether there's a real number-like continuum in nature. Is that better?
I can tell you've taken care to read my words closely. Even though we sometimes clash, I truly appreciate your effort.
My experience on this forum is that people don't like figures, but let me just include one as maybe it will help.
I detect some sarcasm here. Isnt the conventional approach to start with points, which are inherently discrete? How is that any less of a leap than starting with curves, which are inherently continuous? In fact, I would argue that you've encountered far more actual continua in life than individual actual points.
Why proceed? Because an unpartitioned continuum only holds the potential for usefulness. In its unpartitioned state, its effectively a void.
Just as a point cant be manipulated to form a curve, a curve cant be manipulated to form a surface. An open-ended surface is its own fundamental object...
Again, best to keep things straight:
(1) 'the continuum' is noun that names the set of real numbers.
(2) 'is a continuum' is an adjective that we talking about defining.
Also, the discussion had been, for a while, primarily (pretty much solely) mathematical. Of course, one should be allowed to discuss perspectives other than mathematics alone, but I suggest that one should be clear at each point as to what they are talking about - mathematics, physics, philosophy, or some combination that is explained.
Quoting keystone
Let alone that you don't offer primitives, axioms, definitions and proofs, I wonder whether you grasp the idea. Making definitions that rely on yet more undefined terms needs to finally arrive at the base set of primitives, which are undefined but inferences mentioning them are governed by axioms. Or, even if not that formal, at least to state starting notions that are so basic that they at least they can be comprehended intuitively at a basic level. Such things as "increasingly refined composite object made up of arbitrarily many fundamental elements" are so detached from a clear meaning that they are not at all comprehended at a basic intuitive level.
So, even if not formal mathematics, but rather as informal exposition of intuitions, ideas or philosophical views, I can't make heads or tails of whatever it is you're trying to say.
You mentioned providing examples. Ostensive indications and understandings not from explicit definition but rather from gleaning in context are fine and useful, mainly at the stage of basic intuitions. Indeed, I doubt that I could define such terms as 'is', 'in' or 'at' but rather I presuppose that people understand them however they contextually got to that understanding. On the other hand, your notions are not of that basic kind but rather are intricate enough that they need more than ostensive definitions.
Quoting keystone
Then that is mathematics. Or do you mean the study of physical phenomena that calculus is used for?
In any case, as far as I've ever seen, basic calculus doesn't mention the adjective 'is a continuum'. Rather, if at all, basic calculus mentions the continuum = the set of real numbers.
Quoting keystone
I wanted to provide a mathematical definition of 'is a continuum'. I find it in topology. People have been knocking around the term 'a continuum' in a math context. So what is a mathematical definition? I provided one. That's the opposite of distraction.
And topology is a study that informs analysis and put analysis in a broad context. Understanding the real numbers and the continuum in context of topology is definitely not a distraction. And why would topology be a distraction but your half-baked verbiage not be a distraction?
Quoting keystone
I provided a quite streamlined definition. If an equivalent definition could be simpler then you're welcome to state it. Less formally:
A topological space C is a continuum if and only if C is compact, connected and Hausdorff.
So I provided definitions of 'compact', 'connected' and 'Hausdorff'. Those definitions depend on the definition of 'a topology' so I provided that definition, which I couched with only these non-logical notions: 'element' (primitive), 'subset', 'power set', 'union', 'pair' and 'binary intersection', which are quite basic notions of mathematics (except for 'element', they also can be defined back to the sole non-logical primitive 'element').
Quoting keystone
What folly and how insulting!
(1) ChatGPT is so often stupid and dishonest about mathematics. When you posted a purported "proof" by ChatGPT, I even exactly showed you that ChatGPT doesn't know what it is doing*. And that is only one example and not even the worst instance of ChatGPT's stupidity and dishonesty. I could give you as many examples as requested (I'm not going to though since I waste my time only when I feel like wasting it).
* You didn't acknowledge that.
(2) Of course ChatGPT wouldn't be able to understand purely text-symbol rendering of mathematical symbols (for example, if you don't know that 'E' is to stand for the existential quantifier then 'Exy' would be gibberish to you).
My definitions are very straightforward use of just the basic logical symbols and a few basic symbols of set theory.
And if you don't understand the notation (though I specified most of it explicitly in a post only a few posts prior), then you can ask.
And, by the way, the definition I gave is especially efficient as you don't have to read all the chapters in the topology book that lead up to the definition; but instead my definition is extracted from that context while being standalone.
(3) But wait, out of curiosity, I put my definition to ChatGPT verbatim, and ChatGPT replied:
[quote=ChatGPT]This text outlines definitions related to topological spaces and various properties in the context of set theory and topology. Heres a breakdown of the key components:
Topology: The first definition establishes the criteria for a topology T on a set
X. It requires that:
T is a subset of the power set of X (denoted PX).
The whole set X and the empty set must be in T.
Any union of sets in T must also be in T.
Any finite intersection of sets in T must also be in T.
Compactness: The second definition describes compactness in terms of a topology T on a set X. A set C is compact if every cover of X by open sets in T has a finite subcover. This emphasizes the idea that even if you have a potentially infinite collection of open sets that covers X, you can always find a finite number of these sets that still cover X.
Connectedness: The third definition characterizes connectedness. A set C is connected if it cannot be partitioned into two non-empty, disjoint open sets. This property ensures that C is "in one piece" in a topological sense.
Hausdorff Condition: The fourth definition introduces the Hausdorff property, which states that for any two distinct points p and q in X, there exist neighborhoods (open sets) around each point that do not overlap. This separation property is crucial for many results in topology.
Continuum: Finally, the last definition states that a continuum is a set that is compact, connected, and Hausdorff. This is a significant concept in topology, as it combines these important properties into a single structure.
Comment
These definitions are foundational in understanding topological spaces and their properties. They help formalize concepts that are essential in analysis and topology. By establishing clear criteria for these properties, they pave the way for deeper exploration into how different topological spaces behave and interact. If you have a specific area you'd like to explore further or questions about these definitions, feel free to ask!" [/quote]
Quoting keystone
For Pete's sake again! The original post in this thread purported to prove that the continuum does not exist. The argument wasn't that it doesn't exist in nature but that it doesn't exist mathematically. The argument was bunk. Then there was discussion of the mathematics, so it was needed to give a mathematical definition of 'is a continuum'. And the definition itself makes no presupposition that there exists an infinite set. But I would bet that we can prove that if
Quoting keystone
Then I would be right to ask you to defined the undefined terminology in those definitions, and again until (1) You finally fail to reach primitives or (2) You end up in a circle or (3) You do reach primitives.
Not close enough. If I had, I'd have more bad things to say about them.
Is your stuff supposed to be mathematics, or mathematics infused with philosophy, or philosophy, or philosophy infused with mathematics, or something else?
"fundamental objects"
"by its very nature"
"multiple vertices representing the same node" Vertices are nodes.
"broken down into finer and finer elements"
"change the continuity"
"acts"
"algorithm describes"
"The cardinality of an algorithm is determined by comparing it to other algorithms whose cardinalities are already known" Circular.
"continua one partition at a time" You were supposed to be defining 'continua'.
"connected" (in your sense)
/
Maybe more simply, why don't you get back to me when you've figured out your definitions from a starting point rather than backwards from undefined to undefined?
Meanwhile though, still my question: Is your stuff supposed to be mathematics, or mathematics infused with philosophy, or philosophy, or philosophy infused with mathematics, or something else?
And when you cite ChapGPT for mathematics, I figure that you are as lacking in credibility as it is.
As I understand, you reject using infinite sets. But you say that we encounter continua. So continua are finite?
You've not defined 'a continuum' (a bunch of itself undefined verbiage is not a definition). So I don't know what you're talking about in mathematics when you use the word 'continuum'.
And every set has a partition. Many partitions for larger and larger sets.
And then straighten out a (continuous) curve and you have a continuum, which the OP argues does not exist. Perhaps you should start a thread entitled "The Continuum does exist".
I've never been able to see where it is you are going. Maybe it's just me, old and weary of days.
[quote=ChatGPT]This text outlines definitions related to topological spaces and various properties in the context of set theory and topology. Heres a breakdown of the key components:
Topology: The first definition establishes the criteria for a topology T on a set
X. It requires that:
T is a subset of the power set of X (denoted PX).
The whole set X and the empty set must be in T.
Any union of sets in T must also be in T.
Any finite intersection of sets in T must also be in T.
Compactness: The second definition describes compactness in terms of a topology T on a set X. A set C is compact if every cover of X by open sets in T has a finite subcover. This emphasizes the idea that even if you have a potentially infinite collection of open sets that covers X, you can always find a finite number of these sets that still cover X.
Connectedness: The third definition characterizes connectedness. A set C is connected if it cannot be partitioned into two non-empty, disjoint open sets. This property ensures that C is "in one piece" in a topological sense.
Hausdorff Condition: The fourth definition introduces the Hausdorff property, which states that for any two distinct points p and q in X, there exist neighborhoods (open sets) around each point that do not overlap. This separation property is crucial for many results in topology.
Continuum: Finally, the last definition states that a continuum is a set that is compact, connected, and Hausdorff. This is a significant concept in topology, as it combines these important properties into a single structure.
Comment
These definitions are foundational in understanding topological spaces and their properties. They help formalize concepts that are essential in analysis and topology. By establishing clear criteria for these properties, they pave the way for deeper exploration into how different topological spaces behave and interact. If you have a specific area you'd like to explore further or questions about these definitions, feel free to ask!"[/quote]
So, I don't know why ChatGPT "struggled to make sense of it" for you.
But ChatGPT got these wrong:
"A set C is compact if every cover of X by open sets in T has a finite subcover."
No, not a set C. Rather, it should be (in the manner of ChatGPT):
A set X is compact per a topology T on X, if every cover of X by open sets in T has a finite subcover.
"A set C is connected if it cannot be partitioned into two non-empty, disjoint open sets."
No, not a set C. Rather it should be (in the manner of ChatGPT):
A set X is connected per a topology T on X if it cannot be partitioned into two non-empty, disjoint open sets in T.
As of a recent amendment to site guidelines, using ChatGPT as an academic source is grounds for a warning. You're not being warned for it now, I just thought I'd bring it to your attention, so that you can avoid doing it again.
Engineering claringly uses math as if it applies to reality. You seem to be saying there is nothing contradictory about continuums or that there would only be such only if they were in the real world. So then there is something about physical matter that in its properties is not entirely mathematical as we understand that. That may be true, although I would like to hear reasons why some day. Where do we draw the line when applying math to matter? How do we know we've gone too far?
String theory vs loop quantum gravity. One has little points that are really strings (1 dimension in 0 dimension?) And the other discrete space. The biggest question in physics (quantum gravity) wants to settle the question of the continuum. They don't want to just throw their hands up
Agreed.
Quoting TonesInDeepFreeze
My focus is on the philosophy underlying the rigorous mathematical foundations that support the informal mathematics used in physics. I have no issue with the informal mathematics itself (e.g., the calculus of Newton and Leibniz), but I believe there is a disconnect between these informal methods and the rigorous foundations (e.g., Analysis of the continuum) meant to support them. This disconnect isnt due to flaws in the rigorous foundations but rather stems from a misinterpretation of their meaning and philosophical implications. Specifically, I argue that the concept of actual infinity, which plays a central role in these interpretations, should not be part of the framework. With that said, I want to stress that the work of Cantor, Cauchy, etc. play a central role in my philosophy.
Quoting TonesInDeepFreeze
I believe an ostensive definition with figures is exactly what you need to make sense of what I'm saying. I'll provide that in a separate post. I understand that this approach isn't for everyone, so no worries if you choose not to respond to it.
Quoting TonesInDeepFreeze
I want to learn topology, but in this thread I think it makes sense to focus on the topics that are absolutely essential for understanding continua (such as real numbbers). I dont believe topology, at least in the way youre suggesting, is necessary for this understanding. Of course, my current ideas are half-baked, but I believe the direction Im taking is key to grasping the true nature of continua.
Quoting TonesInDeepFreeze
I'm not going to challenge this definition because (1) I'm not well-versed in topology and (2) I still recognize its value. However, the philosophy behind applying this definition to describe non-trivial continua assumes the existence of infinite sets, which I believe is unnecessary.
Quoting TonesInDeepFreeze
I didn't? If I didn't, I accept that it gave an invalid proof.
Quoting TonesInDeepFreeze
A few questions wouldn't be enough. I suspect I'd have to read at least one textbook to truly understand what you mean by "A continuum is a decomposition of a hyperspace module with sidewise-partitionable step-wise radii."
Quoting TonesInDeepFreeze
Maybe I should do as jgill recommends and create a new thread. Thoughts?
Quoting TonesInDeepFreeze
Yes, but readers often offer a certain level of leniency to keep the conversation flowing smoothly. For example, you stated that 'is a continuum' is an adjective, which it isnt. I didnt want to nitpick that because, as a reader, I can easily fill in the gaps and understand what you meant.
Quoting TonesInDeepFreeze
Yes, that's the first part of my definition - continua are finite objects. I believe all objects (mathematical and physical) are finite. At the same time I believe that the rigorous mathematical foundations of set theory, including all the work of Cantor, are valid, and that my only qualm with set theory is the philosophy underlying it (centered around actual infinities). But perhaps we should focus on continua for this discussion...
The following is for your amusement only. It's not worth our time to discuss the validity of this response.
:up:
Consider the following as an ostensive definition. Is this not basic?
I will be interested in what younger and more agile brains make of this. :chin:
Quoting jgill
Me too. I often think about giving this up and just spending my free time like most people do. I'm truly open to being pushed in either direction.
Quoting jgill
It's not just you. My ideas are half-baked and I'm not great at communicating them in a way digestible to mathematicians. I appreciate you following along even though you don't see where it is I am going with this.
What misinterpretations of the meanings of foundational mathematics? What writings by mathematicians or philosophers are you referring to?
Quoting keystone
The mathematical definition is given in topology. How could the actual mathematical definition not be at the very heart of comparing the mathematical definition with alternative definitions? It seems to me that you're rationalizing your unwillingness to inform yourself on the subject.
Quoting keystone
What? You didn't immediately apprehend that was a spoof? But a spoof with a point; a point that you missed even though I mentioned it explicitly.
Thus you went past my point, which is: If you don't understand the actual definition I gave, then you can ask me. It's curious that you wish to engage people with your notions, to provide them with, at best, vague definitions that they should take time and energy to understand or persist asking for explanations to then also take time and energy to understand, but you haven't the slightest inclination to even glance over a mathematical definition given to you by a person who has, at extreme length and in extreme detail (in at least two other threads) engaged your notions. Why is that? Could it be in your personal characteristics? (Some variation of being so overly infatuated with your own mind that there's little intellectual juice left in you to bother learning much about the mathematics that other people have given lives of intellectual labor to?) But I should hush myself lest one of the Pecksniffian pearl clutchers here accuses me of "ad hominen".
Quoting keystone
What? You started your post by agreeing that it is an adjective. It is an adjective, a predicate in this case.
You said you agreed but then you disagree. So I doubt you got the point at all:
'the continuum' is a noun phrase that designates a particular mathematical object, viz. the set of real numbers.
'is a continuum', formally, is an adjective phrase, a predicate, that designates a property of certain mathematical objects, viz. the property of being a compact, connected, Hausdorff topology (or, put another way, being a compact, connected, Hausdorff set relative to a topology on that set).
'is a continuum', informally, is an adjective phrase, a predicate, that designates a property of certain things, e.g. a rainbow.
But in this discussion, we see people refer to both 'the continuum' and 'continua', so we should be careful not to conflate those terms.
Quoting keystone
So you have no objection to the axiom of infinity itself, only with philosophizing that there exist "actual" infinite sets? And what do you mean by "actual"? If one views mathematical sets to be mathematically actual but one does not opine as to whether there are physically actual sets, is that okay with you? If one holds that abstractly there are infinite sets but one does not opine that physically there are infinite sets, is that okay with you?
Claringly, not a word I know. Typo for something else? Clearly? Clarity?
In any event, many disciplines from physics to engineering to biology to economics to baseball statistics use math as a tool. Math itself is studied by mathematicians for its own sake, without regard for utility. But I think you are conflating using math as a tool to study or model an aspect of reality, with reality itself. An error you will repeat below and that I'll try to place in context.
Quoting Gregory
If standard set theory, ZFC, is consistent, then there is nothing contradictory about the mathematical real numbers; as they are a construction within ZFC.
As Gödel showed, ZFC can not prove its own consistency. So we can never be sure if ZFC is consistent, without assuming the consistency of even stronger axiomatic systems. But if ZFC is consistent, then the real numbers are not contradictory. That's as far as I can go. I can't tell you for sure if ZFC is consistent, and I can't claim with certainty that there isn't some terrible contradiction lurking within our conception of the mathematical real numbers.
I have no idea what's true in an absolute sense about the real world, nor does anyone else. We have some fabulous mathematical models of the real world that predict the outcomes of experiments to ten or twelve decimal places (the magnetic moment of the electron being a famous example). But that just says that we have a pretty good mathematical model. It doesn't tell us what nature is all the way down.
Science is about models, not ultimate truth. I think that's pretty well understood these days, but also widely misunderstood.
Quoting Gregory
Other way 'round, I would say. There is something about mathematics that's not necessarily entirely physical. Mathematics has a true continuum. It's unknown whether any such thing exists in nature.
Mathematics has infinite sets; the natural numbers {0, 1, 2, 3, 4, ...} being an example intuitively plausible to almost everyone. You can "always add one more." But there are no infinite collections in nature, as far as we know.
So mathematics has many objects, ideas, and gadgets, that as far as we know, have no correlate or instantiation in nature.
Quoting Gregory
We apply math to nature to the extent that it's useful. Riemannian geometry is just the thing for general relativity. Functional analysis of Hilbert spaces is just the thing for quantum physics.
We've "gone too far" when we start believing that our mathematical theories ARE reality, as opposed to merely MODELING reality. As you are consistently doing, and as you are about to do in your next paragraph.
Quoting Gregory
This is exactly wrong. In fact you mentioned this wrong idea a while back, in a post that was filled with so many misconceptions and errors (IMO of course, nothing personal) that I didn't bother to answer it, lest I appear to be piling on. Perhaps I should get to it.
String theory and LQG are competing mathematical models, I repeat, mathematical models, that are proposed to be able to predict the results of experiments that we can carry out with our historically contingent experimental apparati.
I don't think anyone claims that they are competing metaphysical theories, claims that nature is "really" that way. Or if they do, they are making a category error, confusing physics for metaphysics.
There is not a dispute between string theory and LQG as to how nature "is." There are two competing ideas for how nature should best be modeled.
Or to put this another way: To the extent that physicists argue about the best mathematical model of the world, they are doing physics. To the extent that they argue about how the world really "is," they are doing metaphysics.
Physics is not metaphysics. Physics is about models. Metaphysics is about the (possibly unknowable by we finite, fallible humans) way things really are. Plato's cave and all that. All we can see is shadows.
I hope this point is clear. Physics is about building models that explain the experiments we can do, up to the limits of precision that we can measure. It is NOT any longer the theory of what is "true" in any absolute sense. This is summed up in the famous phrase, shut up and calculate. Meaning, don't ask what's truly going on. Just use the theory to analyze and predict the results of experiments.
We should remember not to confuse the map with the territory.
Are you mixed up? You said ChatGPT struggled to make sense of my actual definition, not my parodic definition. When I put my actual definition to ChatGPT, it replied as I posted. Meanwhile, it seems that the point of my parody went past you, even though I said what the point is. And ChatGPT listed possible purposes of that whacky definition but overlooked parody or put-on as one of them.
AI is remarkable. The ability, on the fly, to compose text, to code programs, to create images, to make spreadsheets, etc. is amazing. But it's not intelligence. It can't discern truth from falsehood even in a million different simple cases. It fails often to prove even simple mathematical theorems. Etc. And it can't explain its own errors. It's a source of gross misinformation and misunderstanding. It is to be vigilantly mistrusted. Humankind doesn't deserve to degrade itself by blind acceptance of every inferior and dangerous stage of technology. And AI doesn't have a sense of humor at a level greater than a five-year-old.
Should be Wikipedia too.*
* Actually, I don't favor censoring references to the output of AI bots or publicly edited encyclopedias. And Wikipedia is magnitudes better than AI bots (which is not saying much for Wikipedia). But that doesn't mean I shouldn't say how dangerously unreliable they are.
Who do you think they are comprehensible to, other than yourself?
I'll look at this later, if my time, patience and supply of snacks is adequate.
For example:
Quoting TonesInDeepFreeze
Why is it that the intro to calculus/analysis textbooks Ive read never mention topology? Is it because these texts dont need a general definition of continua since they only work on the continuum, whereas topology is needed for a defining continua? Keep in mind, youve already given me a reading list that Im just a few pages into. Adding topology isnt a problemId even prioritize it if it made sense. But I think its fair for me to question whether expanding my reading list is really necessary.
Quoting TonesInDeepFreeze
Ha! No, I didnt. But I was being honest with my response. There was so much technical jargon that I had no idea what you were talking about, so I asked ChatGPT. That only made me more confused, so I stopped. And I told you that. The same goes for topologyI stopped because Im not informed on the subject, and I told you that too.
Quoting TonesInDeepFreeze
I think the main reason we're not fully connecting is that Im not presenting my points in a way thats suitable for a mathematician, and Im not fully understanding some of your points because theyre not framed in a way thats accessible to a non-mathematician. Ive been using ChatGPT as a tool to help me grasp the more complex ideas, but you discourage that. To be fair, my difficulty responding to your posts on other threads is what motivated me to start studying logic on my own. However, given life's complexities, I only have a limited amount of intellectual juice to dedicate to this. That said, Im doing my best. But I understand if you decide that we cannot have a worthwhile discussion.
Note: I am passionate about my idea but I don't think that's the main factor here.
Quoting TonesInDeepFreeze
I agreed with the meaning you intended but not with the exact words you used. Thats my point. I noticed a flaw but filled in the gap to keep the conversation moving. 'Is a continuum' is indeed a predicate, and 'continuum' is a predicate noun. If you had written 'is continuous,' then 'continuous' would be a predicate adjective. But 'is continuous' is neither a verb nor an adjective.
Quoting TonesInDeepFreeze
Yes I get that. I've been consistent with this.
Quoting TonesInDeepFreeze
My view is that the Axiom of Infinity represents an inductive algorithm for constructing the inductive set, which is said to have a cardinality of aleph-0. I dont believe the inductive set itself exists; instead, the inductive algorithm is all we need, and it carries the same cardinality. When someone invokes the Axiom of Infinity, theyre really presenting the inductive algorithm, which is what I believe in. No one has ever exhibited an actual inductive set abstractly in the way that a finite set can be exhibited abstractly. And to be clear, Im not referring to its physical existence.
Quoting TonesInDeepFreeze
I must admit that it did.
Quoting TonesInDeepFreeze
But should it be mistrusted in 2-5 years?
Quoting TonesInDeepFreeze
A mathematician with plenty of patience and an open mind...so far it's only been ChatGPT...
Quoting TonesInDeepFreeze
The next bag of Sweet Chili heat Doritos is on me. :P
https://m.youtube.com/watch?v=Jl-iyuSw9KM&t=235s&pp=2AHrAZACAcoFCkNjYyBzYWJpbmU%3D
Thanks for your response. The above video is very interesting but it's minute 2 I'm concerned with. This is how i see all geometric objects, and all objects in general actually. It's not as if i recoil in horror before matter itself, but i don't understand why something in mathematics so simple cannot be explained to me as if I were 8. Maybe I'm just neurally divergent. I've teased apart the finite from the infinite in an object, and in putting them together I find them contradictory, as have many philosophers in history, Hegel being one of them. Good day
Over the years colleges have designed their curricula to suit the levels of abstract thought students can bring to the classroom. Calculus is taught in high schools at minimal levels of sophistication and during the first year or two at college with a bit more rigor. But even there the emphasis is on understanding applications of the subject. Most students in these classes are not math majors. Analytic geometry is part of this instruction. The student progresses to a higher level at the junior or senior years. At this point they are usually capable of the sort of abstract thinking that underlies calculus in an advanced calculus course, or an introduction to real analysis course. As for topology, I taught an introductory course at the junior/senior levels.
All of the above may not be true at a sophisticated university with high entrance standards.
And the above are generalizations. Individual students may be more or less proficient than I have indicated. For example, a number of years ago a young freshman at my school registered for my advanced calculus course - and received an A (he was also a talented climber and we became friends). Then I have had seniors who barely passed.
(I began college at Georgia Tech in 1954, and was fortunate to be one of the few incoming students who scored high enough on the entrance exam for me to start anywhere I wished, so I was placed in an experimental class of beginning calculus, immediately taught with epsilon-delta precision, with rigorous proofs. For the first half of the semester I had hardly a clue what was going on, while some of my classmates seemed to understand the material. Then halfway through all of it suddenly made sense. After that introduction, when I got into the regular curriculum for the next semester it seemed almost trivial)
Ok, conformal rescaling. Conformal means "angle preserving." So they're mapping the infinite plane onto a finite disk by projecting it through a sphere.
That's an interesting topic. I'm not sure how it relates to the subject of our conversation but if it's meaningful to you, all to the good.
Quoting Gregory
That's good, since matter is all around us.
Quoting Gregory
Am I failing to explain a mathematical question?
Perhaps you can phrase your question in a sentence or two, clearly, and I'll do my best to explain.
It might be that I'm not quite understanding your question.
As it is, I don't know what mathematical question you are asking.
Quoting Gregory
Well let me know if I can answer any specific questions.
You said you have no objection to set theory itself but that you object to misinterpretations of it.
The first in your list there is just an observation that the main areas of interest for prominent set theorists concern infinite sets. That's not a misinterpretation of set theory.
The second is just a theorem of set theory. A theorem of set theory is not a misinterpretation of set theory. So, if you approve set theory itself, then you approve its theorems, including the one you just mentioned.
The third is not something I recall ever reading.
The fourth is just an observation that is seen to be true by opening a textbook. It's not a misinterpretation of set theory.
Quoting keystone
First off, why do textbooks for courses in U.S. Civics not mention John Locke, William Blackstone, the Federalist Papers, John Marshall or Plessy v Ferguson?
Next, my guess is that, yes, analysis textbooks don't usually define 'continua' for the reason you just mentioned. However, as to topology in general, many analysis books do delve into some topology too. Analysis and topology go hand in hand. It's arguable whether it's better to learn topology first for its generality and elegance that applies to analysis (though that lacks analysis as a motivation and source of examples) or analysis first for its motivation and source of examples (though that lacks the generality and elegance of topology). While, of course, both are a gas to study in and of themselves.
Quoting keystone
Ha! Blatant strawman! I didn't say you have to study topology to understand the definition of 'continua'. It's the opposite since I gave you a definition that is self-contained and needs only understanding of a handful of first chapter set theory concepts. That was the very point, as I mentioned it: to give a definition that needs nothing other than that handful of chapter one set theory concepts.
Quoting keystone
Next time I'll make it even more outlandish for you so that it is inescapable. I'll include Madame Chiang Kai-shek, Krusty The Clown, a wig factory in Duluth, Minnesota and a toothpick on the ground as examples, respectively, of a homeomorphism, isometry, generalized convergence and a product space.
Quoting keystone
And I told you that you don't need to know topology to understand the definition. You don't need to have a prior understanding of one single sentence of topology to understand the definition I gave. And you would see that even just by reading the definition itself. But, of course, prophets such as yourself need not bother with the writings of others.
Quoting keystone
I asked you already: Who do you think it's suitable for? Especially if not for a mathematician, then who? Who do you think would read your stuff and understand it at any level that qualifies as grasping and absorbing a reasonable version of what you have in your own mind? I don't think you're sincere in wanting to communicate. If you were, you would give people the consideration of clearly articulated concepts. Anyway, I speculate that the reason you won't read the substantive material in my posts is psychological. You divert to the false claim that the definition I gave is too specialized. I'd rather stay on the subject of you. Whatever psychology is going on in you that makes you your own special variation of a crank is more interesting than your malformed, ersatz math musing.
Quoting keystone
You're talking about mathematics. Of course discussion then is going to include mathematics.
And, again, the definition I gave requires only familiarity with a few first-chapter, easy concepts in set theory.
You are so busy espousing that you don't read that to which you respond.
Quoting keystone
How many examples do you need to appreciate that that bot flat out lies, pretends to explain when it is terribly botching the subject, and is incapable of, or unwilling to, check its misinformation, confusion and illogic. It is really really pathetic that you would so seriously louse up the integrity of your study by reference to a source so inimical to truth, understanding and clarity.
And as if it's not enough that the Internet is inundated with cranks, now we have to contend with bots that are cranks!
Quoting keystone
You don't ration wisely.
Quoting keystone
Contrary to evidence.
Quoting keystone
That deserves a fact check, which I'm too lazy now to research.
Quoting keystone
That's not the axiom of infinity! It is nonsense to say that you don't object to set theory by recourse to agreeing not with the axiom of infinity but with something very very different! How stupid do you take people to be? How stupid do you think people are not to see the sophistry you just pulled? You're insulting.
Quoting keystone
I don't know. It could be magnitudes worse - fully Orwellian and dystopian (if it continues on the path it's now on) or it could be profoundly better - paradisiacal, or anywhere in a continuum ('continuum', nice word that) between. But I'm pessimistic.
But so what? We're not dealing with what it will be. We're dealing with the toxic, pernicious thing it is now.
Quoting keystone
So a hypothetical being. Not any actual mathematician you've ever met? Not even evidence that there is such a mathematician. And not evidence that even the most patient and open-minded Bodhisattva of a mathematician wouldn't tell you, "Get back to me when you've worked out some math".
And you think a mathematician would not reply in terms that presuppose, at the least, basic undergraduate mathematics?
Quoting keystone
I can't handle the pathos of that now.
Quoting keystone
Leave it to you to offer the worst junk food.
But I've been down this road already with you in another thread. I took a lot of time and effort to turn your gibberish into communicative mathematics. Then, all along the way, you revised your idea, so I revised in response, which is fair. But eventually, your proposal came to an impasse of illogic, yet you wouldn't budge and merely insisted on your notions though they had been shown inconsistent. A dead end with you. But maybe this time it could be different. Hope springs eternal.
I'm not a big fan of matter. How nice it would be to exist without being subject to the vicissitudes of objects - massive, medium size and subatomic - clashing and banging all around you, wantonly careening at you, and roiling inside you without regard for the effect it all has on you. Matter doesn't care at all about me, so why should I respect it? Well, I do respect some of it - nice people, a lovely beach, a perfect avocado, and some jazz records and math books. But most of the rest of it, phooey! One thing for sure, no one ever involved in a head on automobile accident ever said, "Thank the universe for the laws of physics".
Same for me. But it was in 7th grade in sex ed.
Would it be mathematically possible to project an infinite plane unto a "discrete chunk" (to use QM language)? To me this sounds like a contradiction, but "discrete space" seems like a contradiction to me as well. If it's spatial it has parts. Is discrete defined well in mathematics? Again, they use it in QM.
An incredible paragraph. You're not just a mathematician
That should be: I'm not just a non-mathematician.
If I had to guess where you are headed, I might say that taking a continuum (a line,say) as axiomatic somehow you are cutting it into a fine mesh using the S-B Tree. But how this has a bearing to elementary calculus is a bit foggy. Perhaps Farey sequences to partition Riemann integrals. Just guessing.
I'm going to put 'k-' in front of words to be clear it's your terminology not to be conflated with the usual mathematical usage.
So far, you haven't defined 'is a k-point' and 'is a k-curve', so, at least for now, I'll take them as primitive, with the axiom: No k-point is a k-curve.
It seems that a k-continuum is a certain kind of finite(?), undirected(?), loopless(?) graph whose k-vertices are either k-points or k-curves. And (I surmise) no k-vertex is connected to itself.
You should state definitively what kinds of configurations are k-continua and are not k-continua. Here's what I have so far:
These are k-continua:
one k-curve
one k-curve connected to one k-point
one k-point connected to a k-curve connected to another k-point
These are not k-continua:
a graph in which occurs a k-point not connected
a graph in which occurs a k-point connected to another k-point
a graph in which occur two or more k-curves but at least one of them is not connected to another k-curve (I surmise)
a graph in which occurs a k-curve connected to more than two k-points
Is there a natural number n>1 such that there is no k-continuum such that there occurs n number of k-curves connected to one another? (In other words do you disallow that a k-continuum may have arbitrarily finitely many connections of curves from one to another?)
Are these disallowed from being k-continua?:
a graph in which occurs a k-curve connected to a k-point connected to another k-curve
a graph in which occurs a k-curve connected to another k-curve connected to a k-point
I'm not sure that exhausts all possible configurations. You should figure it out to define 'is a k-continuum'.
What logic do you use? Classical? Intuitionistic? Other? If you don't state your logic then I will take it to be classical.
What set theory axioms do allow? Perhaps Z set theory without the axiom of infinity and the axiom of regularity (though, I predict you will soon enough be stumped without the axiom of infinity). Note: Just saying "finite processes, no infinity" is hand-waving unless you mathematically define 'is a process' without recourse to infinite sets. If you mean Turing machines or equivalents, then let's see how you actually couch your definitions and inferences with them.
For reference, here are the axioms of Z set theory without the axiom of infinity and the axiom of regularity (a symbol guide for the text symbols I use is in my forum About panel):
(1) extensionality:
Axy(Az(zex <-> zey) -> x = y)
For any sets x and y, if they have the same members, then x = y.
(2) separation schema:
If F is a formula in which x does not occur free, then all closures of
ASExAy(yex <-> (yeS & F))
are axioms
If F is formula, then for any set S, there is the subset {x | xeS & F}. Even more informally, for any formalizable property R and set S, there is the set of elements of S that have the property R.
(3) union:
ASExAy(yex <-> Ez(zeS & yez))
For any set S, there is the set of all and only the members of members of S.
Notated as US = {y | Ez(zeS & yez)}.
(4) pairing:
ApqExAy(yex <-> (y = p v y = q))
For any sets p and q, there is the set such that both p and q are in the set and nothing else is in the set.
Notated as {p q}.
(5) power set:
ASExAy(yex <-> y is a subset of S)
For any set S, there is the set whose members are and only the subsets of S. Notated as PS = {y | y is a subset of S}
Summarizing:
If x and y have the same members, then x is y.
For any formalized property and for any set, there is the subset whose members have that property.
For any set S, there is the union of S.
For any sets x and y, there is the set {x y}.
For any set S, there is the set of subsets of S.
(If we add the axiom of infinity and the axiom of countable choice, then we can derive classical analysis. cf. 'Notes On Set Theory' - Moschovakis pg. 116)
G is a graph
<->
EVDf(G =
V not= 0 &
V/\D = 0 &
f is a function &
dom(f) = D &
Ab(beD -> Epq(f(b) = {p q} v f(b) =
)))
&
&
In other words: a graph is a non-empty set of vertices, a set of edges disjoint from the set of vertices, and a function that assigns to each edge either an unordered or ordered pair of vertices. (An unordered pair of vertices has no direction; an ordered pair of vertices has a direction.)
G is an undirected graph
<->
EVDf(G =
V not= 0 &
V/\D = 0 &
f is a function &
dom(f) = D &
Ab(beD -> Epq(f(b) = {p q})))
In other words: an undirected graph is a graph such that there are edges with direction.
G is a loopless graph
<->
EVDf(G =
V not= 0 &
V/\D = 0 &
f is a function &
dom(f) = D &
Ab(beD -> Epq(p not= q & (f(b) = {p q} v f(b) =
))))
In other words: a loopless graph is a graph such that there is no edge between any vertex and itself.
Once you satisfactorily cover https://thephilosophyforum.com/discussion/comment/935856 I'll see about rigorously defining 'is a k-continuum' as a graph. But I see trouble ahead anyway with your personal notion of partitioning. I pessimistically predict that ultimately you won't be able to get past handwaving with it.
That seems to be the way with a lot of things. I'm certainly hoping to finally reach such moments of clarity in logic, and eventually topology.
Let me restate the examples I mentioned:
Naïve infinite set theory is thought to be about actually infinite sets when I think it is really about potentially infinite algorithms for constructing the infinite sets. (I want to stay clear of axiomatic set theory since I haven't read the required material.)
Cauchy sequences are thought to be sequences of actually infinite terms when I think they are really about potentially infinite algorithms for constructing the infinite sequences.
When we draw a cartesian plot it is thought that there exist actually infinite points in the plot when I think there really are only finitely many continua, each having infinite potential for partitioning.
Quoting TonesInDeepFreeze
...not an American.
Quoting TonesInDeepFreeze
All right, I gave up too soon. I'm not going to be able to respond to everything tonight as I'm running out of time and there's a lot to respond to (which I'm greatly appreciative of). Tomorrow is looking like it will be a busy day for me as well but I do plan to respond to everything. Included in that, I'll spend time trying to understand your topological definition of continua and provide a response.
Quoting TonesInDeepFreeze
Sadly, sometimes I need people to explain jokes to me....
Quoting TonesInDeepFreeze
Mathematicians hold a high bar for clarity. Might it simply be that I'm not a mathematician?
Quoting TonesInDeepFreeze
At this point, a mathematician who can piece together informal ideas. At a later point (once I've read more), a mathematician.
Quoting TonesInDeepFreeze
I admit that sometimes when it gets too heavy I glaze over the details. But have I really not adequately responded to many of your points in this thread?
Quoting TonesInDeepFreeze
It works well sometimes though. I see it moreso as a handy tool to use with caution.
Quoting TonesInDeepFreeze
I haven't studied axiomatic set theory but I have taken axiom of infinity to mean that there exists an inductive set. Is that not it? What I want to reinterpret this as is 'there exists an algorithm to construct an inductive set'.
Quoting TonesInDeepFreeze
Well, that's basically where things ended with you, fishfry, and jgill in the last thread I was active on. That seems to be where things are heading in this thread. I'm starting to get the point.
I see you've already followed up on this. I haven't read it yet as I really want to spend sufficient time digesting it and responding. As I mentioned in my last post, my time is short tomorrow so it may be a couple of days before I respond but I do play to respond to that and the topology comments you made in earlier postings. At this point I'll just say a big thanks to you!
I think the S-B tree is just one particularly pleasant way to cut a continuum.
Quoting jgill
...we never did get to calculus in the last thread. I don't know whether we'll get there in this thread either. Let's see.
Sure. I'll show the procedure in a moment. But what does that have to do with QM? You continually conflate math with physics and I continually note that this is a category error. Do you mean discrete energy levels? That's not any more mysterious than the discrete natural numbers 1, 2, 3, 4, ...
Quoting Gregory
I will show the procedure in a moment.
Quoting Gregory
Here's the Wiki article on discrete space.
A topological space is discrete if all its points are isolated.
A point is isolated if you can draw a little circle around it that doesn't contain any of the space's other points.
Example of an infinite, discrete space: The integers. Think of the integers on the number line:
......-2...-1...0...1...2...3...4.....................
You can see that around each integer, I could draw a little circle of radius 1/4, say, and that circle would not contain any other integer. So each point of the integers is isolated. And since all of the points of the integers are isolated, we say the integers are a discrete space.
We can make any set into a discrete topological space by simply declaring that every subset of the space is an open set.
Alternately, we can declare the set to have the discrete metric, in which the distance between a point and itself is 0, and between any two distinct points is 1.
With this definition we can make the real numbers into a discrete topological space. I know this is counterintuitive, but it only involves an abstract definition. It's a logic game more than anything else, but it's a fact that we can turn the real numbers into a discrete space simply by giving it the discrete metric.
And then the identity function, which maps each real number to itself, is a function that maps the continuous reals into the discrete reals.
That is, we take two copies of the real numbers. One is given the usual topology, which makes it a continuum; and the other is given the discrete topology, which makes it a discrete space.
Then the identity function from the continuous copy of the reals to the discrete copy maps a continuous set to a discrete set. It's even a bijection, which is extra counterintuitive. But that's one way to do it.
Note that there is no physics involved. This is a purely mathematical exercise.
Quoting Gregory
Yes, in math a discrete space is any space where every point is isolated, as I noted.
You might be a little put off by using "space" to mean any old set with some kind of topological structure. It has nothing to do with space as in physics or cosmology. And nothing to do with QM. Just math.
ps -- You asked about mapping an infinite plane onto a discrete space. In that case just do the same trick with two copies of the Cartesian plane, one with the usual Euclidean metric and the other with the discrete metric. It's just a math trick, maybe less to it than meets the eye. But we can definitely map continuous spaces into discrete ones.
What is your trip, man?
You said, "my only qualm with set theory is the philosophy underlying it (centered around actual infinities)".
So that would be taken to mean that you have no objection to set theory itself but only to certain philosophy about it. But set theory itself is the set of theorems derived from the axioms. So you would have no objection to the axioms. But you do object to the axioms (and not just philosophy about them). For example, you do object to the axiom that there exists a successor inductive set while instead you approve the very different principle "there are potentially infinite algorithms for constructing infinite sets".
News flash: "there are potentially infinite algorithms for constructing infinite sets" is NOT the axiom of infinity. Saying you approve "there are potentially infinite algorithms for constructing infinite sets" but that you don't approve "there exists a successor inductive set" is NOT saying you don't object to the axiom of infinity but only object to philosophy regarding it; rather it is saying you DO object to the the axiom of infinity but approve a very different principle. Again, you take me for stupid if you think I don't see that. It is a childish game you play.
Same for Cauchy sequences and the Cartesian plane.
Quoting keystone
You're as bad at history as you are at honest, logical thinking. The Federalist Papers, John Marshall and Plessy v Ferguson are American. John Locke and William Blackstone are English but at the deepest foundation of American principles of government and law.
It seems you missed the analogy. You asked why topology is not very much featured in calculus and analysis books you've read. Putting aside that topology is often part of analysis books, the answer is that a calculus book is about setting forth the most basic mathematics that is put to use in different fields of study; it's not about very much understanding the foundations of that mathematics or its broader mathematical context. An analysis book is more foundational than a calculus book but still mainly about diving right into the real and complex numbers as opposed to getting deep into the details of the set theoretic foundations and the topological generalizations.
By analogy, a book for a course in U.S. Civics is mainly concerned with describing the federal government, its organizations and functions, relationship with the states and with its residents and citizens, with some of the philosophical and legal background but not deep into those.
Quoting keystone
I think it's more likely that you're a self-infatuated poseur.
Quoting keystone
In other words, a mathematician of a certain temperament and then later a mathematician. Huh?
Quoting keystone
Your idea of a "detail" is a one-liner axiom. You would glaze over just about anything. With all the glazing over you do, you should be in the business of hams or donuts.
Quoting keystone
A quart of raw milk with listeria works well to quench your thirst. A handy beverage to drink with caution.
Quoting keystone
That is it.
Quoting keystone
Make up whatever principles you want. But "there exists an algorithm to construct an inductive set" is not the axiom of infinity and it's not a "reinterpretation" of the axiom of infinity. It's a decidedly different idea.
Again, when you say you don't object to set theory but you do object to one of its axioms while preferring to plunk down for a different principle, you make no sense! Would you please stop it? Have at espousing whatever alternative principles you like, but it makes no sense to say you don't object to set theory when explicitly you do!
/
And you skipped recognizing that you strawmanned when you said you'd have to study topology to understand my definition.
If you wish to engage me with this, then know that first I need for you to determine what are all the possible configurations and then to say exactly which are a continuum and which are not, as I mentioned.
I thought I indirectly addressed this when I said I was going to go back to your earlier messages on topology and respond to them. But if you're looking for something explicit, yes I unintentionally strawmanned you.
Quoting TonesInDeepFreeze
I do wish to engage you with this. On top of being busy these days, I expect it to take me a few days to read, digest, and respond to everything you've said. Please stay tuned.
As an aside, in our previous discussions I felt hurt and turned off by your tone but for some reason in this thread I'm actually quite appreciative of our interactions. Thanks!
You suggested that topology might be a distraction and that my definition might be unnecessarily complicated
I replied:
Quoting TonesInDeepFreeze
and
Quoting TonesInDeepFreeze
You replied by saying again that topology is not necessary, as if I had said it is necessary, as you were replying to a post in which I said that there's nothing in my definition that requires knowing anything more than a few basic set theory concepts. So, clearly, you ignored what I said and made it appear that I had been saying that you'd have to study topology to track my definition - the opposite of what I said. That is bad faith, whether intentional or a product of not bothering to read.
After a lot of that kind of thing with you, and much worse instances, in this thread and previous ones, I eventually get to the point of feeling that what is most interesting about you is not the particulars of your confusions about mathematics but what goes on in the head of person who is so wrapped up in himself that he wants other people to take their time and labor to understand his own personal, malformed musings while he pretty much just blows off the substantive and informative information and explanations he gets in reply.
The most salient instance lately is your ridiculous claim that you don't object to set theory but only to philosophy about it, yet you do object to the axiom of infinity, while evading that to do so is indeed to object to set theory, not just philosophy about set theory, and then you so sophistically evade that point too by endorsing not the axiom of infinity but a principle very different from the axiom of infinity! Sheesh!
And you claim not to object to set theory but to "misinterpretations" of it, yet your examples of supposed misinterpretations are not misinterpretations! Sheesh and double sheesh!
Quoting keystone
You're a saint.
The show I'm interested in seeing is a hoped for episode in which you account for all the configurations to determine which are a continuum and which are not. If that goes well, then the studio might consider extending the series into another season.
I read this same argument in Kant recently. He wants mathematics to come from our intuition of the world yet doesn't believe the second antimony must apply to appearance. The only reason you don't want math to fully apply to reality is because you suspect a problem with infinite divisibility, right? Is not 5 yards minus 3 yards 2 yards? Always, forever? Is not 5 feet minus 3 feet 2 feet? I can get smaller and smaller. There is no reason it should end. You want math to apply to the world when they build bridges but won't go all the way, saying instead there is some invisible indeterminate line across which we can't do math. And you say this without a supporting argument. I don't buy it
I've never known a fellow mathematician who would have agreed with this. A mathematical philosopher perhaps. Let's see what @fishfry has to say. I always enjoy his commentaries.
Not at all. Infinite divisibility is not in question within math. But there's no evidence for it in physics. That's my only point.
Quoting Gregory
In physics there is a minimum distance, below which we can not sensibly apply our laws of physics. There is no infinite divisibility in physics.
https://en.wikipedia.org/wiki/Planck_units
Quoting Gregory
No, there's a measurement boundary below which we can't do physics. And it's not something I say. It's something Max Planck said, and that a hundred years of physics has found no exception to.
Quoting Gregory
You deny the Plank length? I don't follow your point at all. Physics is very clear on this matter. Nature has a minimum length, below which we can't reason sensibly about. That doesn't mean that infinite divisibility isn't part of nature; it only means that infinite divisibility is not a part of our best theories of physics.
That's why I say that infinite divisibility is part of math; but as far as we know, and until some future genius not yet born comes up with a new idea, it's not part of physics.
For me the world is as mathematical as geometric imagery. The world is mystical, nah, miraculous in how it is woven together. Maybe mathematics gives you that sense too. Thanks for the conversation!
On this we fully agree.
Quoting Gregory
Likewise, thanks.
I'm going to respond in two posts. This post covers topics not directly related to my ideas and my next post will cover topics more directly related to my ideas. I hope this covers the important points we've been discussing but let me know if I skipped over something important.
Topology: I now have a basic understanding of the definition of a continuum as compact, connected, and Hausdorff. Connectedness seems straightforward, and I believe its a necessary part of any definition. Compactness is more intriguing because "the continuum" itself isn't compact, which is surprising since it means "the continuum" isn't a (topological) continuum. While I understand that compactness is important for results like the Extreme Value Theorem, I dont yet fully grasp why it's a crucial property of continua. Given that I want define the sort of object suitable for calculus, I'm inclined to not include this property in my definition of continua.
The Hausdorff property, on the other hand, is a more subtle aspect I hadn't considered, likely because most (if not all) spaces Ive encountered are already Hausdorff. I think it's important and makes sense, but I am unsure how it fits in with my view. This is something Ill need to explore further in the future. You're right that topology is relevant to this discussion.
Philosophy vs. mathematics: Based on your criticisms, I need to rethink my position. I believe that basic mathematics operates on two parallel tracks: (1) the objects and operations themselves, and (2) the descriptions of those objects and operations. To use a Matrix analogy, this would be like (1) the lived experience within the matrix, and (2) the underlying code or digital rain that describe/define it. Both tracks are fundamental. In basic mathematics, the objects are what we depict, such as points, continua, etc., while the descriptions are the formal logical statements we write down.
Though Im still learning the basics of logic, I find intuitionism most appealingespecially since you've made it clear that ZFC leaves no room for flexibility when it comes to the ontology of infinite sets (and my focus is on transitioning from actual infinities to potential infinities). To be frank, I'm not willing to accept infinitely many axioms or statements holding a truth value so it's not just a matter of avoiding the Axiom of Infinity. So you're right, I do have an issue with ZFC.
However, my primary focus isnt the description itself (or the supporting philosophy), but rather the objects and operations they describe. Specifically, Im interested in the philosophy that justifies the existence of mathematical objects and our operations on them. The orthodox view is a whole-from-parts (point-based) philosophy, which aligns well with ZFC. I, on the other hand, propose the opposite: a parts-from-whole (continuum-based) philosophy. Perhaps this view might align better with CZF, I don't know.
Infinity plays a central role in both (1) mathematical objects and operations, and (2) logical descriptions. I realize now that Ive been unclear in distinguishing between the two. For instance, my concerns with infinite sets and Cauchy sequences likely fall under (2), the realm of logical descriptions. To address these issues properly, I recognize that I need to deepen my understanding of logic and intuitionism. However, my primary focus is on the objects themselves, such as the Cartesian coordinate system. I believe this system needs a parts-from-whole, continuum-based reinterpretation, as the current understanding relies heavily on the notion of actual infinity.
In short, I would like to see the replacement of the actual infinity based whole-from-parts + ZFC mathematics with the potential infinity based parts-from-whole + ?CZF? mathematics. However, I'm in no position to talk about ZFC vs. CZF.
Quoting TonesInDeepFreeze
The idea behind these axioms seem sensible yet I'm wary of schemas which imply infinitely many instances of those axioms.
Quoting TonesInDeepFreeze
Understood.
Not sure what you mean by actual infinity. Are you speaking of infinity as a sort of number that can be arithmetically manipulated, or infinity as unboundedness? I have always used the concept of the latter rather than the former. But set theorists use both I think. Please provide an instance of "actual infinity" in the Euclidean plane. A projection onto a sphere is not allowed.
CONNECTED: I've realized that I was loose with my use of the term 'connected'. I see that 'connected' means that there is a path between vertices (which can consist of one or more edges). Moving forward, I'll use 'adjacent' to indicate that two vertices are connected by a single edge.
1D Continua: I just want to highlight that all of the following is in the context of 1D continua.
The following are not k-continua:
-a graph in which occurs a k-point not connected (Not connected)
-a graph in which occurs a k-point adjacent to another k-point (Not connected in the topological sense, even if connected in the graphical sense)
-a graph in which occurs a k-curve adjacent to another k-curve (Not connected in the topological sense, even if connected in the graphical sense)
-a graph in which occur two or more k-curves but at least one of them is not connected to another k-curve (Not connected)
-a graph in which occurs a k-curve adjacent to more than two k-points
-a graph in which occurs a k-curve adjacent to another k-curve adjacent to a k-point (Not connected in the topological sense, even if connected in the graphical sense)
The following are k-continua:
-one k-curve (Not compact, but I'm inclined to include to capture the essense of the real line)
-one k-curve adjacent to one k-point (Not compact, but I'm inclined to include to capture the essense of the real line)
-a graph in which occurs a k-curve adjacent to a k-point adjacent to another k-curve
Quoting TonesInDeepFreeze
A k-continuum can be composed of arbitrarily finitely many vertices.
If there are n k-curves (where n is a positive natural number), there must be between n-1 and n+1 k-points.
If there are n k-points (where n is a natural number), there must be between max(1,n-1) and n+1 k-curves.
Quoting TonesInDeepFreeze
Does this cover it?
For k-continua:
1) All graphs are finite, undirected, and loopless
2) All graphs are 1D drawable (in that each can be embedded in a circle without any of its edges crossing) (Since I'm focused on continua used for calculus, perhaps this should be a line)
2) All k-vertices are connected
3) All k-points are adjacent to 1 or 2 k-curves (and no k-points)
4) All k-curves are adjacent to 0, 1, or 2 k-points (and no k-curves)
Quoting TonesInDeepFreeze
[EDIT: I've realized that in the following response I made a bad move in fast forwarding to numbers and intervals. Please ignore for now. I'd rather discuss continua in a simpler sense, free of numbers for now. However I'm leaving it in this post since JGILL has already responded.]
IGNORE-->
[i]I need to rethink the use of the word 'Partitioning'. That word suggests that k-objects can be divided into smaller, perhaps more fundamental, objects. I want to treat k-objects are indivisible wholes. Let me try to explain my view of partitions by means of an example. I'm going to utilize numbers and intervals, even though we haven't discussed it yet.
I have two continua described by Graph 1 and Graph 2, respectively.
Graph 1: A-B-C where
A is k-curve (-inf,5)
B is k-point 5
C is k-curve (5,+inf)
Graph 2: E-F-G-H-I where
E is k-curve (-inf,5)
F is k-point 5
G is k-curve (5,10)
H is k-point 10
I is k-curve (10,+inf)
These are distinct graphs, but Graph 2 is a partition of Graph 1 because it captures all of it's structure. In other words, Graph 2 can behave exactly like Graph 1 if E acts as A, F acts as B, and G-H-I acts as C.[/i]
<--IGNORE
I will use the Euclidean line instead of the plane, since we're currently focused on 1D.
I'm interested in distinguishing between objects and operations, particularly in relation to infinity.
In the orthodox view, the Euclidean line is thought to be fully formed, with all (an infinite number) of its points existing simultaneously. This kind of infinity, which counts objects, is what I refer to as actual infinity. (Note, Im not referring to infinity as a coordinate, which I believe is what you're asking about.)
In my view, however, there is no such thing as a complete Euclidean line. Instead, we start with a single k-curve (-?,+?) and can partition it as much as we like, without ever exhausting it. The continua we construct in this process are always finite (in the sense that they are represented by graphs with a finite number of k-vertices). It is endless operation of partitioning that is infinite, and this infinity is a potential infinity. I don't believe in supertasks, so I would argue that all infinities related to operations are potential infinities, never fully realized.
I do think infinite cardinals and ordinals are very valuable, but not at describing the size of completed objects (e.g. sets) but rather at describing the potential of endless operations.
The article mentions several applications connected to data processing and CS. But calculus approached this way is obscure and unlikely to replace elementary calculus as it it is currently taught. Just my opinion. You are probably not pursuing this line of thought.
The biggest hurdle for an intelligent but amateur mathematician is rediscovering a result established some time ago. Hence, my words of caution.
The link you provided is fascinating, especially since calculus and graphs are central to my perspective. However, I believe the connection stops there. While I appreciate discrete calculusafter all, numerical methods are what I work with professionallythe concept Im developing here is rooted in continuous calculus.
To clarify, discrete calculus deals with finite objects like graphs, while the approach I'm pursuing involves potentially infinite operations. These operations, however, are applied to finite structures, like graphs.
In this context, a mathematical Platonist wouldnt need to focus on operations, as mathematical objects are seen as complete entities. But for a constructivist, operations are essential, as constructing these objects is at the core of the philosophy.
Quoting jgill
These are wise words of caution.
I seem to lack your insight in this example. It appears you simply take a real line and divide it into several line segments by inserting "k-vertices". You are assuming the existence of these points on the line. Indeed, the line segments are continua. In the example A-B-C what if instead you used the square root of two as the dividing k-vertex? You seem to be assuming the common notion of the real line. Maybe if you extend your ideas into 2D they will seem to be more than trivia? As a constructivist, what are you constructing other than a few line segments?
Why don't you jump right into calculus concepts in 2D instead of dwelling on the trivial, incredibly boring 1D case. Either that or make the 1D case something interesting, to capture the attention of a reader. Just a suggestion.
I was wrong to mention numbers/intervals without properly establishing the notion of a continua. I'm going to leave that message there, but please allow me to refrain from responding here as I believe a discussion involving numbers will not help. Months back you said:
Quoting jgill
I'm quoting you here because your recommendation is exactly the path I endeavor to go own.
Quoting jgill
Constructing 1D k-continua from (indivisible) k-points and k-curves.
Quoting jgill
I've come to understand that although 2D visuals may seem more impressive than 1D, the core of my perspective can largely be conveyed in 1D. Introducing calculus and 2D concepts too early might disengage the reader, as it would lack the foundation of a fully developed idea.
Let's have another take at this. In this take, I will not mention numbers. I also will not use 'partition' anymore as it suggests that indivisible objects can be decomposed. Rather, I'm going to use unify/unification.
In the above figure, I have 3 independent k-continua:
BaC can be unified into A because BaC can be treated as a whole that captures all the structure of A. Perhaps I can write it as follows:
BaC = (BaC) = A
BaDbE can be unified into BaC because DbE can be treated as a whole that captures all the structure of C. In other words,
BaDbE = Ba(DbE) = BaC
A is a unification of BaC. A is also a unification of BaDbE.
BaC is a unification of BaDbE.
A is not a unification of BaC because its elements cannot be unified to capture the structure of BaC.
Thank you. You saved me a lot of time and effort. Because my prediction that you would resort to half-baked handwaving is confirmed, so I am done with trying to help you formulate your stuff into mathematics.
A circle is an infinite set. But you say you're going to do this without infinite sets. You don't know what you're doing and you don't know what you're talking about. And if you were to say something like "a circle that's only a potential infinity" or whatever, then that would not cut it, since you haven't given a mathematical definition of such a thing. Just as I mentioned, you start to give mathematical definitions, but they finally end up relying on even more complex notions that are themselves presumptuously undefined and only gestured at with half-baked handwaving. You're a vortex.
What structure? A line segment has structure? One line segment has the same "structure" as another? You must see something there that eludes me. But I am old and a lot gets past me.
You seem not to understand how the mathematical method of handwaving works. It's not ZF or PA or one of those; it's the theory BS. You need to familiarize yourself with its advanced techniques as exemplified by @keystone.
True enough. But I keep hoping there is something profound I am missing in all this. :roll:
What I termed '1D drawable' is intended to be the exact 1D analogue to a planar graph, nothing more. I'm presenting my ideas informally, but certainly not BS'ing.
Are you saying that one can't use the concept of planar graph until a plane is defined? Are you saying that such finite planar graphs necessarily require the existence of infinite sets? I think one can capture the essence of planar graphs without explicitly mentioning planes in the definition or asserting the existence of actual infinity. If so, then perhaps I could do the same with '1D drawable'.
I would have hoped that this disagreement didn't justify ending the conversation. I kindly ask that you give me another chance.
It's not you, I should have provided a description of the structure I was referring to. I intend to reply to you but I likely will not find time for several days. Please stay tuned.
There's an important distinction between handwaving and BS. Handwaving involves vagueness or imprecision, where the core idea might be sound but lacks detail or rigor in its current form. BS, on the other hand, is fundamentally incorrectan argument that doesn't hold up under scrutiny and lacks substance from the start. A good idea often begins with some handwaving as it's being formed, but through refinement and rigorous thought, it can mature into a precise and well-supported explanation. In contrast, a bad idea rooted in BS remains flawed because it has no solid foundation to build on. While both terms carry subtleties, I hope this captures the essence of the difference.
I believe that my core idea is sound, but it needs further refinement and I'm not communicating well. Your frustration is justified. But I could really use your help.
True. I hope there is something of interest coming from this discussion. But we've been through metric spaces and topology and now are venturing into graph theory with some sort of hope of connecting that with calculus. I have my doubts, but am trying to keep an open mind.
I share your sentiment on all accounts - from the hopes to the doubts. Thank you.
k-continuum 1
k-continuum 1 is represented by the 3-vertex graph: a - - b, where:
Properties:
1. k-point a identifies location a.
2. k-point b identifies location b.
3. k-curve continuously connects a and b without including the endpoints a and b.
In this model, k-points and k-curves are indivisible objects, and k-continuum 1 has no additional properties beyond these.
k-continuum 2
k-continuum 2 is represented by the 5-vertex graph: a - - c -
Properties:
1. k-point a identifies location a.
2. k-point b identifies location b.
3. k-point c identifies location c.
4. k-curve continuously connects a and c without including the endpoints a and c.
5. k-curve
Again, k-points and k-curves are indivisible objects, and k-continuum 2 has no additional properties beyond these.
Relationship between k-continuum 1 and k-continuum 2
k-continuum 2 can represent all the structure of k-continuum 1.
Here's how:
1: Both k-continua identify location a in the same way.
2: Both k-continua identify location b in the same way.
3: The k-curve in k-continuum 1 can be represented as the union of 3 vertices in k-continuum 2, namely: = U c U
This means that the combination of the k-curves and
Therefore, k-continuum 1 can be seen as a unification or simplified version of k-continuum 2, where the additional structure in k-continuum 2 can be collapsed into the simpler form of k-continuum 1.
I am doing quite the opposite. K-continua can only be partitioned at the edges connecting the vertices. As such, k-continuum 1 cannot be transformed into k-continuum 2. Instead, k-continuum 2 can be manipulated (entailing unions) to reflect the structure of k-continuum 1. This distinction is philosophically significant because it enables us to work with and construct continua without the need to invoke actual infinities.
Quoting jgill
I understand this is a fairly basic concept, but it sets the foundation for a constructive approach to mathematics. Instead of working with objects that have 'actual' infinite divisibility, we will focus on constructing k-continua (which are by definition finite in every way). With this view, the essence of calculus lies not in the properties of any single k-continuum, but in the properties of the processes designed to construct particular endless sequences of k-continua. You might notice a resemblance here to Cauchy sequences.
...the first step is to accept that k-curves are indivisible. k-vertices in these graphs cannot be partitioned. I was presenting my position in a very different way back in our previous thread with fishfry, but this was the main sticking point.
OK, you have a line that is indivisible. But it has k-vertices that "cannot be partitioned". Can a vertex be partitioned? Like saying a point can be partitioned. Concise language is very important in math, not so much so in philosophy.
No, k-vertices cannot be partitioned. The only objects in these graphs that can be 'cut' are the edges.
Why resort to graph theory and call a simple line an edge? Is this an effort to enhance an almost trivial concept of line and point? Again, why not go to 2D? Maybe your ideas will make more sense in that context.
The edges do not represent objects like lines; rather, they signify connections. When I say that edges can be cut, I mean that these connections can be severed. The edges themselves do not contribute to the 'extension' of k-continua. In the 1D case, the only fundamental objects that have extension are k-curves and they cannot be partitioned. Within graphs they are represented by k-vertices.
Quoting jgill
This approach aims to elevate lines to the status of fundamental, indivisible objects, thereby eliminating the need for actual infinity. As long as points remain the foundational geometric elements, an actually infinite number of them will always be required to describe continua and paradoxes such as the Dartboard Paradox will remain.
Quoting jgill
Well, I think it's important to first establish the meaning of real numbers within a 1D context with my view. But you've asked for 2D multiple times so let me give it to you with no numbers or equations.
The image below depicts a 2D k-continuum, composed of 31 fundamental objects: 10 k-points, 15 k-curves, and 6 k-surfaces. (If it's not clear what each of the 31 fundamental objects are, let me know and I can create an 'exploded view' image.) In the case of k-curve 7, it continuously extends from k-point 14 and k-point 2 without including the endpoints. While greater mathematical value could be extracted by associating each of these objects with specific equations, we are not at that stage yet. For now, the value lies in its particular structure.
The image below is a mathematical graph representation of the k-continuum depicted above capturing its structure (i.e. connections). The edges connecting the vertices indicate the adjacencies between fundamental objects. For instance, since k-surface 6 is adjacent to k-curve 7, there is an edge linking them. Importantly, these edges represent only the connections between objects, not the objects themselves.
How all this simplifies normal calculus is questionable.
Thank you for taking the time to try to understand the figure and for sharing what didnt make sense. I now realize that I skipped too many steps and made the figures more complex than necessary. Please allow me to make another attempt, and I hope this version is clearer. Also, Ive decided to stop using the terms 'k-vertex' and 'k-edge.' The graphs Im working with follow standard properties, so Ill simply refer to them using the usual terms: vertex and edge.
Quoting jgill
I was initially planning to respond with a high-level overview of how this relates to calculus, but I realize that discussing it now would likely cause more confusion until the figures are clear. On further reflection, I think it was really helpful that you asked for the 2D explanation, as the 1D version is too simplistic and can easily lead to confusion between edges/vertices and k-curves/k-points. Introducing k-surfaces forces me to better distinguish the fundamental objects from the graph objects.
I understand that you prefer not to lead the conversation, but I want to sincerely thank you for asking thoughtful questions that have helped me better articulate my perspective. I hope it's now in a form that TonesInDeepFreeze will be willing to engage with.
@TonesInDeepFreeze, would you consider taking a look at my recent message to jgill? The graph I described there represents a k-continuum, partly because it is a planar graph. For instance, if there were an edge connecting vertex 1 to vertex 8, it would no longer be planar and, therefore, wouldn't describe a k-continuum.
Instead of discussing 2D continua and area, lets simplify by returning to 1D continua and length. Length is not a property of an infinite collection of k-points, but rather an intrinsic property of a single k-curve. This should become clearer once we introduce rational numbers into the discussion.
Quoting jgill
Irrational numbers will require special treatment, but I believe a treatment inspired by Cauchy sequences will largely address the challenge.
Quoting jgill
By introducing the fundamental k-objects (such as k-points, k-curves, k-surfaces, and so on), I've laid out the fundamental building blocks of the top-down approach. I acknowledge that these ideas so far may seem unremarkable, akin to someone attempting to build bottom-up mathematics by focusing solely on the successor function and not doing anything with it. However, if my latest figures made sense, the mundane part is behind us, and we can now move on to more interesting territory.
Quoting jgill
My discussions here rarely go as planned, so please take this plan with a grain of salt:
1. Rational Numbers Describing any arbitrary 1D k-continua entirely using rational numbers.
2. Real Numbers Part 1 Describing potentially infinite sequences of 1D k-continua using rational and irrational numbers.
3. Real Numbers Part 2 Shifting focus to the algorithm for constructing sequences rather than the impossible task of constructing a complete sequence.
4. Real Numbers Part 3 General definition of a real number
5. Cardinal Numbers Applying transfinite cardinal numbers to describe potentially infinite processes, avoiding the need for actually infinite sets.
6. 2D Part 1 Extending the 1D concepts to their 2D analogues.
7. 2D PT 1 - Derivative and Reinterpreting Motion
8. 2D PT 2 - Integral and Reinterpreting Length
9. Ordinal Numbers Offering a reinterpretation of ordinal numbers in the context of potential infinity.
GOALS:
1. To provide a top-down foundational framework for basic calculus that avoids reliance on actual infinities.
2. To argue that the philosophical issues in quantum mechanics arise from bottom-up mathematical intuitions. Physics at a foundational level is inherently top-down, and by developing new intuitions grounded in top-down mathematics, these philosophical issues in QM can be resolved.
Quoting jgill
I'm eager to move forward with this plan if you're open to it. There's no commitment to a lengthy discussionwe can take it one step at a time, and you're free to end the conversation at any point along the way. Of course, if you'd prefer to wait for someone else to potentially lead the discussion, I fully respect that decision as well.
I thought you were defining these lines as continuous. Fundamental objects.
Perhaps I'll head in this direction and see what you think...
Quoting jgill
I dont have much experience with logic yet, but from what I know, my perspective seems to align well with intuitionism. My plan is to begin by learning classical logic as a foundation and eventually explore intuitionism.
Quoting jgill
Contrary to what my last post may have suggested, in the 1D context, there is always a k-curve between neighboring k-points (i.e. k-points are not densely packed) and k-curves are indeed continuous. Please allow me to clarify:
Consider the following 3 example k-continua (please note that I'm using 1/0 to denote infinity):
Every possible 1D k-continua can be described using a combination of rational numbers and k-intervals.
ASIDE: When I label a k-continuum using rational numbers and k-intervals, I'm not merely assigning arbitrary strings of characters, but rather indicating a specific structure/orderingplease forgive mederived from the Stern-Brocot (SB) tree. In fact, the three examples above correspond to the top three rows of the SB tree. I understand youd prefer not to delve into the SB tree, and as long as you don't question the meaning behind my rational labels, I think we can steer clear of it.
That's BS. BS includes nonsense, doubletalk and falsity. And handwaving is not necessarily just lack of rigor to be supplied later. And you presume that your "core ideas" are "sound".
I said I'd be willing to check you out to the extent that we could turn your ruminations into primitives, definitions and axioms. I predicted that right after the first round you would resort to yet more undefined handwaving and I said that I would drop out when that happened. Indeed, with the very first predicate 'is a k-continua' still not fully defined, you've piled on a big mess of more of undefined terminology and borrowing of infinitistic objects while you claim to eschew infinitistic mathematics. You disrespect my intellectual interest that way, just as occurred several months ago with a different half-baked and self-contradictory proposal of yours. You are a sinkhole of a poster. You need to obtain an understanding of the basic concepts of primitive, definition, axiom, and proof. I'm done with providing you assistance of this kind.
You raised a single issue with my response, which I immediately clarified-specifically, that by "1D drawable," I simply meant a 1D analogue of the established term "planar diagram". You haven't given me a good reason for you to drop out. If your offer to help was sincere, you wouldn't back out the moment I sneezed.
Since you've been gone, the discussion with jgill has allowed me to clarify my position to the point where (I think) he understands what I mean by k-continua. I am not spouting nonsense or doubletalk. You haven't identified any falsity in my current position. Please, give me a chance.
You need to define "1D analogue of the established term "planar diagram"" in terms that don't presuppose any mathematics that you have not already defined and derived finitistically and such that it justifies such verbiage as about "embedding in a circle".
But don't bother if it is to re-enlist me. I was willing to take it step by careful step with you. But you can't discipline yourself to do that, as instead you just jump to whole swaths of handwaving. I said that at the very first point you invoked anything not previously justified by you then I'm out. I don't need to waste my time and energy on you. You are BS.
You haven't identified any falsity in my current position.
You haven't even defined enough to get the stage of consideration of truth or falsity.
Quoting keystone
I have! Many times! And previously too. But you abuse my time and effort. I'm done.
Quoting keystone
How dare you question my sincerity that has been demonstrated over and over in careful attention to details, in my labor to explain things for you, in this thread and in one several months ago? Get a load of your narcissistic self. You are full of yourself and full of BS ... though that is redundant.
It's ironic that you got cold right after I went back, carefully studied, and addressed your comments on topology. That feels harsh, but I suppose I shouldnt be surprised. In any case, I appreciate the times when you were helpful. We all have limited time, and its important not to spend it on things we don't want to do. Wishing you all the best.
Apparently, you don't recall the post in which I said that I'm willing to indulge you only up to the point that you go past the process of definitions.
You don't need to concern yourself with my decisions about how I spend my time and energy. Instead, you need to start by at least getting a grasp of the basic ideas of primitive, definition, axiom and proof.
I'm working with standard finite graphs, nothing unorthodox about my use of them. As such, I don't need to produce an original definition of them. If you don't like how the informal definition of 'planar graph' uses the word plane then you can instead use Kuratowski's theorem. Admittedly, I haven't studied Kuratowski's theorem...
I didn't ask for a definition of 'planar graph'. You didn't read what I said about this a few posts ago. You are a sinkhole.
Actually, I think you're the sinkhole. You seem to enjoy destructive conversations.
First, I'd like to point out that this part (Part 2) takes some liberties with actual infinities for explanatory purposes (and to keep my individual posts sufficiently small), but these will be addressed and resolved in Part 3. Let's explore the meaning of the real number 0.9 repeating from my perspective. For now, let's set aside equivalence classes and represent 0.9 repeating as the following Cauchy sequence of k-intervals:
Term n in this sequence is defined according to the following equation:
As depicted below, term 1 describes a k-curve in k-continuum 1, term 2 describes a k-curve in k-continuum 2, term 3 describes a k-curve in k-continuum 3, and so on. Generally speaking, term n describes a k-curve in k-continuum n.
A real number, such as 0.9repeated, doesnt correspond to a single k-point (as a bottom-up view would have it) but rather 0.9repeated corresponds to an infinite sequence of k-curves, shrinking in size as you progress deeper into the sequence in the spirit of Cauchy. [In Part 3, Ill adjust this explanation to avoid implying the existence of actually infinite sequences].
Keep digging your sinkhole deeper.
Apologies for the typo. Also, I initially used -1/0 to represent negative infinity because thats how it appears in the Stern-Brocot tree, but since weve skipped over discussing the SB tree, Ill switch to the more familiar notation.
Quoting jgill
I intentionally kept things uninteresting to maintain a sense of familiarity. Now, I'll begin to diverge from the familiar, which will hopefully make things more interesting. Here's part 3...
In my view, 0.9 repeating does not actually correspond to all the infinite highlighted k-curves in the image below, simply because no k-continuum beyond 3 actually exists, as none have been constructed yet. In the spirit of constructivism, one is not justified to use ellipses to represent the completion of infinite work.
Instead, 0.9 repeating represents the following highlighted object in the generalized diagram.
I cannot call that highlighted object a k-curve because, until n is assigned a specific natural number, the object it describes is not yet a k-curve. The same applies to the other objects and labels in the figure, so I will introduce some new terms (nothing fancy, just adding "potential" in front).
Essentially, Im proposing that 0.9 repeating corresponds to a potential k-interval, which describes a potential k-curve. What I'm leading towards is framing calculus not as the study of actual objects (such as fully constructed k-continua and its constituents), but as the study of potential objects (such as potential k-continua and its constituents), where some or all of the labels remain in algorithmic form for as long as possible. Of course much more is needed to be said about this.
But first, I've been overlooking the fact that real numbers are typically defined as equivalence classes of Cauchy sequences, not just individual Cauchy sequences. In this context, equivalence classes introduce another actual infinity which needs reinterpretation, but let's save that discussion for a future post.
(Aside: If I had the opportunity to redo some earlier posts, instead of k-objects vs. potential k-objects, I would use actual objects vs. potential objects, getting rid of the k- prefix altogether. But I suppose it's too late to make that change now...)
Cauchy sequences themselves are infinite sets.
I agree. However, the main point of my post was to clarify that I'm not working with Cauchy sequences themselves, but with the algorithm used to construct any arbitrary term. In my figure, I highlighted the Cauchy sequence and noted, '0.9 repeating is not this.' In the subsequent figure, I highlighted what I believe 0.9 repeating actually represents and following that I expanded on this in bold.
For the moment, please treat 1/1 and 1.0repeating as distinct objects. Without bringing in the SB tree, let me just say that the former is a fraction represented by a string with finitely many (three) characters, while the latter is a real number represented by a string with infinitely many implied characters.
In my view, when numbered, k-points must have fractional values without exception. It is meaningless to speak of a k-point with a real number value because such a k-point cannot be defined within this framework. Therefore, it is incorrect to claim that a sequence of k-curves converges to a real-numbered k-point.
In my recent posts, I have been establishing that real numbers instead describe potential k-curves. These can be thought of as k-curves yet to be constructed, which, when constructed, will have the potential to become arbitrarily small (but not zero length).
This shift in perspective moves the focus away from a philosophy centered on the destinationlimit objects like irrational pointsand instead emphasizes the process itself, described by algorithms. By shifting from the destination to the journey, the need for actual infinity disappears. Our discussion sets the necessary groundwork for establishing a calculus that operates without invoking actual infinities.
By "actual infinity" I suppose you mean a kind of number that can be manipulated by arithmetic processes.
Quoting keystone
This is either very deep - or shallow gobblygook.
I view transfinite cardinal and ordinal numbers as crucial for understanding the nature of infinity, and, as you know, they can be manipulated through (special) arithmetic processes. However, I take issue with using transfinite numbers to describe actual abstract objects rather than potential abstract objects. For instance, (assume we live in an infinite world*) and consider a computer program designed to input any natural number, n, and output the set of the first n natural numbers. (In an infinite world*) the program has the potential to output a set larger than any natural number so the potential output has a cardinality of [math]{\aleph_0}[/math]. But that program (even in an infinite world*) cannot actually output a set with a cardinality of [math]{\aleph_0}[/math]. Potential is important and I feel like it's been forgotten in our Platonist world.
*I dont actually believe in an infinite world, but Im suggesting that mathematics allows us to speak in general terms without assuming any specific limits.
Quoting jgill
Deep gobblygook is not an option? :razz: If you're following what I'm saying, a discussion on calculus is not that far off. I just need one more post to provide a formal definition of a real number and then we can advance to 2D. Id be very interested to hear your thoughts on whether my view contains an implicit actual infinity or if it might be insufficient as a foundation for basic calculus. I'm certainly benefiting from this discussion but I understand that one should not entertain gobblygook for too long.
Elementary calculus does not require "actual" infinities. It gets along quite well with unboundedness, or what you might call potential infinity. As I have said before, I have written many papers and notes without ever becoming transfinite.
I agree that calculus can work quite well with the concepts of unboundedness and potential infinity, but 'actual' infinities are implicitly assumed throughout the standard treatment. The standard treatment is built on [math]\mathbb{R}[/math], the complete set of real numbersimplying an actual infinite amount of numbers and points. As a result, when interpreting the notion of a tangent, one is inevitably led to paradoxical ideas like instantaneous rate of change. When interpreting the notion of area, one is inevitably led to the paradoxical idea that events with zero probability can still occur (dartboard paradox).
Ultimately, calculus is currently treated as the study of objects at the limit, rather than the unbounded process of "approaching" the limit (I use approaching in quotes because that word suggests that there's a destination which I do not believe in). I aim to establish a foundation focused on the journey rather than the destination (i.e. the algorithms themselves rather than their output).
Quoting jgill
Have you written calculus papers/notes that are not (implicitly or explicitly) built upon infinite sets like [math]\mathbb{R}[/math]?
I was speaking of ordinal numbers beyond the naturals. Our definitions of "actual" infinities differ. No big deal.
Quoting keystone
Of course I have used R, but not a transfinite number. Unless I occasionally use the "point at infinity" in complex analysis. Which I rarely do since it is a projection upon the Riemann sphere. It might appear that you are moving in the direction of Discrete calculus. But go ahead. I am curious.
Natural number arithmetic does not involve infinities, yet natural numbers are inseparably tied to [math]{\aleph_0}[/math]. In a similar vein, I argue that real calculus is inseparably tied to [math]2^{\aleph_0}[/math]. My interpretation of the orthodox philosophy is that both [math]{\aleph_0}[/math] and [math]2^{\aleph_0}[/math] represent "actual" infinities because they are used to describe complete objects, such as sets. It is in this sense that I refer to orthodox calculus as being tied to "actual" infinities.
Quoting jgill
The ideas I'm proposing are fundamentally centered on continuous calculus. Concepts like continuity, real numbers, and limits are crucial to my perspectiveI simply interpret them through a different lens.
Quoting jgill
Great! I'll continue in my next post, though it might not be today as I'm starting to feel tired.
: While working on my response, I realized it made the most sense to start from the beginning, using clearer and more descriptive terms and definitions. Looking back, I believe this post aligns with the kind of response that @TonesInDeepFreeze and @fishfry were looking for in this thread and in our previous thread, respectively. I hope the length is balanced by enough clarity to make for a fast read. I believe this sets up the foundation for a calculus free of any connection to actual infinities. I propose that continuous calculus is not the study of continuous actual structures but rather the study of continuous potential structures.
Definition: Ideal Point
In 1D, a ideal point is -? or ?, such that -? is less than any rational number and ? is greater than any rational number.
Definition: Actual Point
In 1D, an actual point is a rational number.
Definition: Actual Curve
In 1D, a actual curve is doubleton set {a,b}, where a and b are either actual or ideal points.
Definition: Simple Functions on Actual Curves in 1D
Lower bound function, L: Actual Curve {a,b}?min(a,b). The lower bound of actual curve {a,b} is min(a,b).
Upper bound function, U: Actual Curve {a,b}?max(a,b). The upper bound of actual curve {a,b} is max(a,b).
Length function, d: Actual Curve {a,b}?|b-a|. The length of actual curve {a,b} is |b-a|.
Definition: 1D Actual Structure
A 1D actual structure is a finite, undirected graph in which each vertex represents an actual point, ideal point, or actual curve. Ideal point ? and ideal point -? must be included. Edges connect these vertices to indicate adjacency between the objects.
Definition: Continuity of 1D Actual Structures
A 1D actual structure is continuous if it satisfies the following continuity requirements:
1. Connections Involving Actual Points:
Each vertex representing an actual point q must be linked with exactly one vertex representing an actual curve for which q is the lower bound and one vertex representing a actual curve for which q is the upper bound.
2. Connections Involving Actual Curves: Each vertex representing a actual curve {a,b} must be linked with exactly one vertex representing actual/ideal point a and one vertex representing actual/ideal point b.
3. Connected: There exists a path between any two vertices.
Definition: Convergence
Convergence of a function: The function x(n): Natural number ?Actual Point converges if a constructive proof demonstrates that for any ?>0, natural number N can always be found such that for any n>N, |x(n)-x(N)|.
Convergence of a function to actual point a: The function x(n): Natural number ?Actual Point converges to actual point a if a constructive proof demonstrates that for any ?>0, natural number N can always be found such that for any n>N, |x(n)-a|.
Convergence of a function to rational number a: The function x(n): Natural number ?Rational Number converges to rational number a if a constructive proof demonstrates that for any ?>0, natural number N can always be found such that for any n>N, |x(n)-a|.
Convergence of a function to another function y(n): The function x(n): Natural number ?Actual Point converges to y(n): Natural number ?Actual Point if a constructive proof demonstrates that for any ?>0, natural number N can always be found such that for any n>N, |x(n)-y(n)|.
Definition: Potential point (reinterpretation of a real number)
In 1D, a potential point is a function p(n): Natural number?Actual Point such that p(n) converges.
Definition: Potential curve (alternate reinterpretation of a real number)
In 1D, a potential curve is a function c(n):Natural number ? actual curve such that L(c(n)) and U(c(n)) converge, and d(c(n)) converges to rational number 0.
Definition: 1D Potential Structure
A 1D potential structure S(n), where n is a natural number, is a finite, undirected graph whose vertices represent:
Edges connect these vertices to indicate adjacency between the objects.
Definition: Continuity of 1D Potential Continuum
A 1D potential structure is continuous if it satisfies the following continuity requirements:
1. Connections Involving Actual Points: Each vertex representing an actual point q must be connected to two vertices: One for which q is the lower bound, either:
And one for which q is the upper bound, either:
2. Connections Involving actual curves: Each vertex representing a actual curve {a,b} must be connected to two vertices: [u]One for its lower bound, either:
And [u]one for its upper bound, either:
3. Connections Involving Potential Points: Each vertex representing a potential point p(n) must be connected to two vertices: One for which p(n) is the lower bound, either:
And one for which p(n) is the upper bound, either:
4. Connections Involving Potential Curves: Each vertex representing a potential curve c(n) must be connected to two vertices: One which bounds L(c(n), either:
And one which bounds U(c(n)), either:
5. Connected: There exists a path between any two vertices in the graph.
Good luck with that. Probably of more interest to CS people.
Quoting keystone
A sequence of rationals I assume. If you had two functions on Q then a suitable metric would be the supremum.
Quoting keystone
How do you define "continuous"? Are you sure it is indivisible?
Sorry, but your list of definitions is mind-numbing. Your top down is becoming way more complicated that bottom up, IMO. And the irrational numbers have yet to appear.
Your best bet would be to find a mathematician willing to deal with your arguments and pay him/her a fee to do so.
Yeah, my view leans heavily on algorithms.
Quoting jgill
Yes. I have since edited the post to clarify this.
Quoting jgill
Suitable for what?
Quoting jgill
Good point. I have since edited the post to clarify this. When defining an actual curve I was providing an informal intuitive explanation where I carelessly used 'continuous'. Ultimately an actual curve is simply an object having an actual interval.
Quoting jgill
Isn't anything communicated with absolute precision a bit mind-numbing? Not that I achieved that level of precision, but it was trying to be more precise. I find logic much more mind-number, but that's just me...
Quoting jgill
Ultimately, it all reduces to the same calculus used by applied mathematicians today. However, building a foundation on constructive philosophy is likely to introduce more complexityat least that's how it plays out in logic. Actual infinity is certainly simpler to work with, but is it truly sound? Newtonian mechanics is simpler than relativity, which is simpler than quantum mechanics. So, what should be the foundational choice for physics - the simplest? There's an elegance to QM and I believe the same can be said about the top down view of mathematics.
Quoting jgill
Ive already outlined the framework for irrational numbers. Both potential coordinates and potential intervals are reinterpretations of real numbers, including irrational ones. If we get past the list of definitions then the next step is to present an example that demonstrates how irrational numbers come into play.
Quoting jgill
I've tried in the past, but nowhere else has been as beneficial as here. That said, Im open to recommendations. Its challenging for an amateur mathematician to find someone with the right skills and interests. I primarily used Upwork.com.
Quoting keystone
Careful. I would not compare if I were you.
Quoting keystone
For defining "distance" between functions. When I dabble in the complex plane contours become points in the metric space and the distance between them is the Sup|f(t)-g(t)| over 0
If I were younger I might have more time to try to unravel your presentation. You have wandered from metric spaces to topology and now graph theory, with that dreadful SB-table trailing along. Then you have all these definitions which a mathematician is unlikely to find of interest.
Quoting keystone
Try a nearby university where a grad student might want a little extra cash.
Good point. I've needed to learn this lesson too many times.
Quoting jgill
Since the functions I'm working with all converge, I don't believe the supremum is necessary for distance, but it might be necessary for other purposes.
Quoting jgill
I'll look into this. Thanks for the suggestion.
Quoting jgill
Ive just revised the post to remove unnecessary mention of objects, making it shorter. If you skip the sections on the definitions of continuity, the post is only 444 words. I mention the continuity section because it's wordy but obvious. For instance, we already know that the interval ?0 5? linked with coordinate 10 cant be continuous, as 10?0 and 10?5, implying a gap between them. I just explicitly lay out all scenarios to capture the obvious. I hope you might reconsider giving it another look, but I completely understand if you choose not to continue. This discussion has already been incredibly helpful to me.
Quoting jgill
Ive admittedly wandered off track at times, and you've been patient with the many detours along the way. However, Im a bit surprised that once I introduced a more mathematical approachlike discussing the Stern-Brocot tree and providing proper definitionsyou felt the discussion was becoming less interesting to mathematicians. I had expected the opposite.
@fishfry: Would you be open to re-engaging with me on this topic (that we discussed months back)? I believe the post quoted here will give you something much more concrete for you to chew on. Plusno figures this time! I'd really appreciate your advice.
A vertex represents an actual curve?
Quoting keystone
In fact, I had never heard of the S-B tree before it was introduced on this forum. It is not true that every mathematician will find every math topic interesting. (Wiki lists well over 25,000 if I recall). Had I been a number theorist or a CS person I may have known of it. I see it averages about 47 pageviews per day on Wiki, and classed as low priority. But that's not trivial by any means. My own page gets only 15.
Each indivisible object, whether potential, pseudo, or actual, is represented as a vertex within a structure, regardless of its dimensionality. This approach underscores the fundamental indivisibility of these objects. The only object that is divisible is a structure.
Quoting jgill
Imagine how fortunate I (an amateur) feel to have stumbled across it (and Niqui's paper on arithmetic based on it)! :razz:
Quoting jgill
Agreed.
Quoting jgill
Sometimes the significance of a discovery isn't recognized until many years later.
It's had 164 years. We'll see. :cool:
Quoting keystone
(Wiki)
OK. But it gets a bit anti-intuitive. Hard to imagine a curve is a vertex. But then, I treat curves in the CP as points in a metric space.
I wish other mathematicians would chime in on this thread. I am very old and have forgotten what I didn't learn. :roll:
An actual curve in 1D is unique in that it is fully defined by its endpoints. However, in 2D and higher dimensions, a curve is determined not only by its endpoints but also by an equation. Perhaps incorporating that equation into the vertex might make the concept more digestible.
Quoting jgill
Yeah, that would be nice, but I really do appreciate you taking the conversation this far. You got me thinking!
I explore various properties of contours in the complex plane, defining a metric space whose "points" are contours. [math]z(t)=u(t)+iv(t),\,\,\,0\le t\le 1[/math].
"Distance" is defined [math]d\left( {{z}_{1}},{{z}_{2}} \right)=\underset{t}{\mathop{Sup}}\,\left| {{z}_{1}}(t)-{{z}_{2}}(t) \right|[/math]. But here t is a positive real number, which you have not defined yet. Usually, the u(t) and v(t) are differentiable, giving a smooth curve. So incorporating this sort of thing into the definition of vertex assumes what you will probably wish to prove. I wonder what an "edge" in your graph would be?
A real number corresponds to a specific subgraph within a potential structure. In the 1D case, this is represented by a potential curve and the two potential points that are directly connected to it.
Quoting jgill
Incorporating differentiability?
Quoting jgill
An edge signifies adjacency between objects. For example, in conventional interval notation, an edge would exist between the curve (0,5) and the point [5,5] due to their direct adjacency. In contrast, the curve (0,5) is not adjacent to (5,10) because a gap exists between them (at point 5), so no edge would connect the vertices representing the two curves.
You've lost me. Guess it's time for me to quit. Overall, I think you have started down a path that is far too complicated for the desired result. However, if the result you seek is more philosophical than mathematical you may have something.
Someone could say the same thing about the epsilon-delta formulation of a limit, which was introduced to give calculus a more rigorous foundation. After all, infinitesimals produced the desired results and were simpler to work with.
Quoting jgill
No worries. Thanks for the discussion!
At first I thought this is not true, but that is because I followed a learning curve that incorporated analytic geometry before calculus, and this allows clear illustrations and examples en route to limits. You work in CS, however, and what seems like unintuitive definitions to me probably make more sense to you. Had I taken a course in graph theory what you are proposing might seem less opaque.
Continue, if you like, and I will comment from time to time as I learn more about graph theory.
I suggest this thread be placed in the Lounge since it obviously has limited appeal to the general audience, but has merit in philosophy of mathematics. Just my opinion.
I work in simulation as an engineer, not a computer scientist.
Quoting jgill
I'm new to graph theory and currently only utilizing the basics. Even though the concepts I'm using are fundamental, they require a significant paradigm shift, which can be challenging to adapt to.
Quoting jgill
Ill give others some time to respond, which will also allow me to think through my ideas as I work on generalizing them across higher dimensions. But since my big post from 6 days ago was opaque to you I'm quite discouraged from continuing on.
Quoting jgill
When you mention "the Lounge," are you referring to an actual place, or do you mean taking a pause in the conversation until others join in?
The "Lounge" is part of TPF.
The one trait I share with Feynman is understanding a concept through examples. You started with the imagery in 2D. Maybe more?
I share that trait as wellI need some time to fully generalize my ideas. As I work through this, Im noticing that some adjustments are necessary, so Id like to minimize making changes mid-conversation. I really appreciate your continued interest, and Ill be back soon. Thanks!
That is BS. Handwaving to the extent you execute it is very much BS. The category of BS includes both falsehood and nonsense. EDIT: And the falsehood coming from you is the pretense that you're providing grounded definitions and that you're not resorting to infinite sets. Just saying [paraphrase:] "Oh, it's not really infinite, I'm talking about the process not the object" is just abracadabra BS. If you were sincere and credible then you'd state the algorithms as actual algorithms and not resort to infinite sets such as circles. Meanwhile, there is a ton of work that's been done in constructive mathematics and other alternatives, but you don't take lesson from it as you march head forward to glory.
Quoting keystone
Your k-BS is "working" with still undefined terminology with your special k-BS flavor of handwaving.
Quoting keystone
Transfinite cardinals and ordinals are infinite sets.
Quoting keystone
aleph_0 is an infinite set. Of course, there is also an algorithm that, at any stage, outputs up to any natural number, so that for any natural number, if the algorithm runs long enough, that natural number will be outputted, but you mention "transfinite numbers" plural, so let's look at the next one, viz. aleph_1. There is no algorithm for aleph_1 such as for aleph_0, and same for even greater cardinals. You don't need to wave transfinite cardinals around as if they have anything to do with k-BS.
Quoting keystone
You believe incorrectly, and it would be better that you not wiggle in suggestions such as that your k-BS "aligns" with what I am "looking for".
I took you at face value and mistakenly entertained the idea that you could be in good faith with all the business about graphs. But then, to complete your definition, you invoked the existence of circles. But circles are infinite sets and you provided no justification for invoking them, completely out of the blue. It turned out, yet again, that indulging you is a dead end in a sinkhole - an insult to the time and thought I wasted in this and other threads with you.
Quoting keystone
Who needs for you to "argue" that? We already know it.
On and on with yet more pseudo math k-BS definitions when you never finished the very first definition you started.
Quoting keystone
Get a load of you, bestowing yourself with "absolute precision". Your k-BS is as much absolute precision as a mound of moldy fish gut is a sashimi delicacy.
Quoting keystone
Yes, your Fields Medal is just a matter of time. Just have to get all those stuck-in-the-mud, imagine-less, establishment math fools to come around to open their eyes to your brilliance.
Calculus uses infinite sets on day one. Even before a student gets to calculus, with analytical geometry we're using infinite sets. The real line and the real plane are infinite sets.
But it seems you mean that calculus doesn't usually mention transfinite ordinals (though the set of natural numbers is a transfinite ordinal), which is true.
Quoting jgill
Points at infinity are not required to be transfinite numbers.
I'm done with our conversation.
You would do a lot better not to reference me by saying that you believe that what you posted is what I am looking for.
That's what I meant. To think of infinity itself as some sort of number. Trying to keep the discussion at an amateur level.
Quoting TonesInDeepFreeze
I'm thankful for that. Why resurrect this dead thread?
The BSer mentioned me in a misleading way, and posted more BS.