Proof that infinity does not come in different sizes
If I count 1, 2, 3, 4 ad infinitum, will I reach infinity? One cannot count to infinity, and even if something like a number sequence goes on forever, it will not reach infinity. To call {1,2,3,4,...} an infinite set is to imply that {1,2,3,4,...} consists of an infinite number of numbers. No doubt, even if 1, 2, 3, 4 goes on forever, an infinite number of numbers will never be reached. So the question that must be asked now is whether there is any meaningful difference between 1, 2, 3, 4 ad infinitum and [1,2,3,4,...}.
One might argue that the latter encompasses imagining that the count to infinity is complete, but one cannot imagine such a thing. Perhaps one might argue that there is no count involved with regards to the latter and that it's just a fact that Infinity encompasses an infinite number of natural numbers. But if that's the case, then Infinity also encompasses an infinite number of possible real numbers and possible letters or possible x. But where there is no counting involved, all infinites are of the same size/quantity (or rather, infinity is one quantity as opposed to different quantities). How would a difference in size be established between them when there is no counting involved? And if there is counting involved, how would infinity be reached?
One might argue that the latter encompasses imagining that the count to infinity is complete, but one cannot imagine such a thing. Perhaps one might argue that there is no count involved with regards to the latter and that it's just a fact that Infinity encompasses an infinite number of natural numbers. But if that's the case, then Infinity also encompasses an infinite number of possible real numbers and possible letters or possible x. But where there is no counting involved, all infinites are of the same size/quantity (or rather, infinity is one quantity as opposed to different quantities). How would a difference in size be established between them when there is no counting involved? And if there is counting involved, how would infinity be reached?
Comments (306)
So you have gone astray somewhere.
https://www.cantorsparadise.com/
Any infinite sequence is equal in terms of number of elements to any other infinite sequence, but i do not think they are equal in terms of magnitude or value.
Considering these two infinite sequences:
sequence 1 = {1, 2, 3, 4, ...}
sequence 2 = {.5, 1, 1.5, 2, 2.5, 3, 3.5, 4, ...}
Take for instance the first 4 elements of any 2 infinite sequences, and observe the last number. If the last number of one series is bigger than the last number in the other series then that sequence has a larger magnitude. In this case then sequence 1 is larger in magnitude than sequence 2.
sequence 1 = {1, 2, 3, 4, ...} = 4
sequence 2 = {.5, 1, 1.5, 2, ...} = 2
Therefor sequence 1 is greater than sequence 2 in terms of magnitude or value.
Alternatively, if one selects two arbitrary numbers that are common to both sequences such as the numbers 1 and 4 then sequence 2 is larger than sequence 1 because more numbers are included by sequence 2 between values 1 and 4.
sequence 1 = {1, 2, 3, 4, ...} = 4
sequence 2 = {1, 1.5, 2, 2.5, 3, 3.5, 4, ...} = 7
Therefor sequence 2 is greater than sequence 1 in terms of magnitudes or values represented. It can be said that sequence 2 has higher resolution than sequence 1.
The size of an infinity is determined by the size of its elements, not the size of the collection (viz., it is not determined by how many elements the collection has but, rather, how many elements, including recursion, the elements have).
As a basic example, it is clear that a set of {1,2,3,...} is smaller than {{1,1},{1,2},{1,3},...}. Likewise, the largest infinity is the one with infinities as its elements all the way down recursively, which I cannot shorthand accurately, due to its nature, than {{*},{*},{*},...} where * represents an infinitely recursive set of collections.
You do not consider the possibility that Cantor is wrong?
Suppose someone brought proof. How will you recognise it?
Quoting punos
Suppose two things are travelling at two different speeds. One is faster than the other. Both are set to go on forever. Would you say something like the value of the faster one is greater than the value of the slower one in terms of distance covered? Or would you say they are both of equal value? Or would you say both are set to go on forever but neither will reach an infinite amount of distance covered and so one will have travelled farther than the other if a measurement was to be taken of how much distance it has covered in comparison to the other.
Generally the problem physically exists in this form,
Brain; (Abstraction)
More specifically,
Brain; (Abstraction 1, infinite set 1)
Brain; (Abstraction 2, infinite set 2)
And a relation,
Brain; (Abstraction 3, the relation of sets)
The problem I see is that any element of an infinite set is a finite number and can be reached by finite means. So those numbers that are finite can be subject to logic because they are defined.
Infinity as undefined is off limits to logic.
If you introduce infinities into the elements then again you are using undefined terms.
I do see some logic in the OP arguments.
It seems possible to map a smaller infinity, one to one, on a larger infinity simply by freezing the larger infinity and letting the smaller one catch up.
Since we set imaginary parameters anything goes. This is not based on anything physical at all.
How would you respond to this:
How would a difference in size be established between two infinite sets when there is no counting involved? And if there is counting involved, how would infinity be reached given that one cannot count to infinity?
Quoting Mark Nyquist
Whilst I believe it's possible for two different things to go on forever, I don't believe it's possible to have two different sized infinities because even if the two things (such as two number sequences) go on forever, infinity will not be reached (we cannot count to infinity). So when you say "freeze the larger infinity" I assume you mean something like stopping it from continuing to go on forever. But my whole argument is that if something goes on forever, it does not make it infinite precisely because one cannot count or expand to infinity.
Quoting Mark Nyquist
Some things are imaginary, but some things are truths about the nature of Existence/Being (such as triangles have three sides or one cannot count to infinity).
Actually, I think you have the better grasp of this problem being that the extended nature of infinities is off limits to logic.
Well, what you call truths I call Abstractions and the parameters can be anything we choose. Again, no physical basis so variation in opinion is expected.
Quoting Mark Nyquist
No one our earth has ever seen a physically perfect triangle (because perfectly straight lines are impossible in our universe as far as I'm aware). Yet we know that the angels in a triangle add up to 180 degrees. What I'm trying to say with this example is that the parameters cannot be anything we choose. If what we choose or say is contradictory (such as triangles have four sides or one can count to infinity) we cannot meaningfully/rationally say it. As for what determines what's meaningful/rational and what's not, I believe that is Existence. If x is true of Existence, it is rational/meaningful (for example triangles have three sides is true of Existence). If x is contradictory or not true of Existence (such as one can count to infinity), it is not true of Existence (as in the nature of Existence is not such that triangles have four sides or that one can count to infinity).
Okay. Just giving another perspective.
A lot of interesting math in the subject.
I agree the abstractions should conform to the subject matter.
A couple examples,
Central banks where a potential infinite supply of currency is mapped to a populations finite physical resources.
Zelenski-ism where infinite military wants are mapped to a coalitions finite resource base.
See Cantor's diagonal argument, which proved that there are higher-order cardinal numbers.
You've already admitted that you're not a mathematician, so it's strange that you think you know mathematics better than Cantor (and Russell).
I've seen cantor's diagonal argument and the following objection applies to it:
How would a difference in size be established between two infinite sets when there is no counting involved? And if there is counting involved, how would infinity be reached given that one cannot count to infinity?
Quoting Michael
Reason is accessible to everyone (not just mathematicians). I try to focus on the argument at hand as opposed to who is doing the arguing.
It doesn't. If you were a mathematician then you would understand it. Your question simply shows your ignorance of mathematics. You're really in no position to argue against Cantor.
Can you answer the following:
Can we establish set x as being bigger than set y without counting the number of items in x and y? If yes, how? If no, what do we do with the problem of "one cannot count to infinity"?
Cantor's diagonal argument.
That is not an answer.
It's like me asking "can you count to infinity?" where the answer should be no, but someone responding with "Jack's diagonal argument" implying you can without actually showing how.
Again, I've seen Cantor's diagonal argument. It does not answer the questions I asked you in my last post to you.
It is an answer. You just don't understand it because you're not a mathematician.
Quoting Philosopher19
Can we establish set x as being bigger than set y without counting the number of items in sets x and y? If yes, how? If no, what do we do with the problem of "one cannot count to infinity"?
Peace
Yes, we can establish set X as being "bigger" than set Y without counting the number of items in X and Y. We can establish this by using Cantor's diagonal argument. It is a well-accepted mathematical proof. If you were a mathematician you would understand it.
Quoting Michael
If you used reason you'd know that you cannot count to infinity and that you cannot say x is bigger than y without some measurement/count involved to compare the sizes of the two.
If you were a mathematician then you would know that this is false.
You're just in no position to argue against Cantor.
http://godisallthatmatters.com/2021/05/22/the-solution-to-russells-paradox-and-the-absurdity-of-more-than-one-infinity/
I read your work.
I can't give an opinion but you have put some work into it to your credit.
I think my issue is in mapping an infinity to a known finite. A one to one mapping will use up the finite and end. The unmapped trailing infinity becomes a useless appendage.
And there is the issue of logic working for the finite but failing in the infinite.
Interesting to see other people's opinions.
I might be thinking mapping an infinity to a larger infinity also leaves this useless appendage....not sure, just my instinct not real math. Still...parameters are arbitrary.
And if you want a better understanding of the issues here, see Chapter Four of Open Logic.
Unlike Philosophy Forum, it's guaranteed free of psychoceramics.
Thanks Mark
I think it's clear that one cannot count to infinity So one cannot say that x is an infinite sequence of numbers just because it goes on forever. If I count forever I will not reach infinity. I cannot say assume I completed my count with this set of numbers and that set of numbers and then argue that that set is a bigger infinite set than the other.
Just think: how can one infinite quantity be bigger than another when the quantity of infinity is one quantity?
How does one not laugh at this?
How would a difference in size be established between two sets when there is no counting of the number of items in the sets involved?
If there is counting involved, how has one reached an infinite number of items?
I also asked an additional question:
If infinity is a quantity, how is it more than one different quantity?
If I ask how many items in that set and the answer is infinite and I ask how many items in that other set, it is surely contradictory for someone to say to me and even bigger infinity. There is no beyond one infinity for there to be the possibility of a bigger infinity.
Quoting Philosopher19
By bijection. See Open Logic Ch.4.
Quoting Philosopher19
"Counting", and ill-defined notion, is not involved in bijection, although "enumeration", a well-defined notion, is.
Quoting Philosopher19
See Cantor's diagonal argument.
A one to one to correspondence implies a count of one side compared to the other. But infinity is not reached or exhausted and cannot be counted to
Quoting Banno
Is it not? Do you not count how many maps onto how many?
Quoting Banno
I have already seen. Tell me what about it suggests that infinity is more than one possible quantity despite it being the case that infinity is one semantic as opposed to two. Note that 5 is one semantic as opposed to two.
Yes, if you were to measure both distances at a specific point in time, but outside the context of a finite time measurement, the distance is probably equally infinite for both.
I think that to make sense of infinities, one has to have a system for extracting their finite properties, as I mentioned in my prior post, or by looking, for example, at the difference between one element in a sequence and the next, which has a specific finite value. This specific value for example can be considered a fundamental component of a periodically regular sequence, by which any periodically regular sequence can be constructed, including infinite ones.
I think that's good advice for me. Seems like the direction of the discussion and the actual math departed ways.
The issue here is not one of logic, but of pedagogy. The logic is clear, there are multiple infinities. The issue is why some folk cannot see that to be the case, even when presented with the proof.
Consider:
Quoting Philosopher19
Is it that Philosopher19 has a picture of infinity such that, since one cannot count to infinity, one cannot have a grasp of infinity?
One way infinity is introduced to children by showing them that for any number, we can construct a bigger number - by adding one, or some other finite number. Then comes "Infinity plus one!". The child will have understood infinity not as something one counts to, but as the ability to carry on in the same way...
In a way, Cantor showed the child's "infinity plus one" to be a reality... :wink:
So Philosopher19 it seems has a notion of infinity that is dependent on actually counting to infinity, rather than "carry on in the same way...", and hence takes it as granted that a one-to-one correspondence must involve counting. Two approaches occur to me, when I put on my long-discarded teacher's hat: to show a variety of infinite one-to-one correspondences, making the point that we do not need to count them all, or even at all, to see that they go forever; and to look at infinity in other contexts - art, perhaps - in order to show that one can understand infinity apart from counting.
Anyway, we are not being paid to teach Philosopher19, so that goes by the by.
Rather, we use the adjective 'is infinite'.
To define 'is infinite' we may:
First, define 'finite'. That can be done in various ways. One way is to define:
x is an ordinal if and only if (x is membership-transitive and x is well ordered by membership)
n is a natural number if and only if (n is an ordinal and n is well ordered by the inverse of membership)
x is finite if and only if there is a natural number n and a one-to-one function from n onto x
x is infinite if and only if x is not finite.
/
In mathematics, 'equinumerous' is defined:
x is equinumerous with y if and only if there is a one-to-one function from x onto y.
That corresponds to the utterly basic intuition that sets have the same number of elements if and only if they can be put in one-to-one correspondence. For example, proverbially, there are the same number of sheep in the flock as there are stones in the pile if and only if for each stone there is a corresponding sheep and no stone corresponds to more than one sheep.
Set theory uses this definition for both finite and infinite sets.
Given this definition, we have prove that the set of natural numbers is not equinumerous with the set of real numbers.
Now, if one has different intuitions about equinumerosity, then one may imagine set theory not to use the word 'equinumerous' but instead 'zequinumerous' or whatever, for mathematics does not depend on the words it happens to use but rather on the formal relations, no matter what natural language words we use to nickname those relations.
/
* But there also are the points of 'infinity' and 'negative infinity' on the extended real line. But these are not indications of CARDINALITY. They regard two elements in addition to the set of real numbers and an ordering that such that infinity is greater than every real and greater than negative infinity, and negative infinity is less than every real and less than infinity. The points themselves may be of any cardinality.
* We also use phrasing and/or the lemniscate, for example, "as n goes to infinity". But this is a facon de parler that can be explicated in various ways such as "n rangers over the natural numbers" or "the domain of the function is the set of natural numbers".
If you start with a set of integers 1 to a million and another set of integers one to infinity and pair one to one up to a million then the set of infinity unpaired is infinity minus one million which is meaningless and undefined.
That is still an issue not answered. Can logic apply to an undefined set.
I'm rusty at this but does someone know?
You can't apply natural language words to mathematics. Things like "goes on" and "be reached" mean nothing in mathematics, those phrases can only informally refer to real mathematical concepts such as addition or limits, otherwise you end up with gibberish like here.
1, 2, 3, 4... goes on forever. The verb go implies movement. Therefore the natural set of numbers moves through time and space!
Infinity minus any finite number is still infinity. Doubtless others might make this informal answer rigourous.
I don't see what is "undefined" here, let alone "meaningless".
So infinity minus one million is defined?
No it is not defined.
There is not a "minus" operation involving infinite cardinals in the same way as with integers.
Rather:
For any set x, there is its cardinality called card(x).
Now let '\' stand for the complement operation:
x\y = {p | p in x and p not in y}
If x is infinite and y is finite then
card(x) = card(x\y)
That is meaningful and correct.
Saying x-y is not even meaningful.
Rather, there are various sets that have the property of being infinite.
And there is no "minus" operation as being bandied here.
Okay these are really just mental abstractions and it looks like your framework really is arbitrary.
I'm sure it's standardized academically but there are still problems.
Rather, if x is an infinite set, and y is a set with n number of elements, then there are n number of elements that are in x but not in {p | p in x and p not in y}.
Casually speaking you might say "there are n less elements" but to be mathematically accurate, we need to not use 'less' in that imprecise way and instead say " there are n number of elements that are in x but not in {p | p in x and p not in y}."
'less than' has a mathematically exact definition, and it is not used in the way being bandied here along with 'minus'.
Again, the point is pedagogic, not logical. Here's the question:
Quoting Mark Nyquist
One might understand this as: What is the cardinality of the integers that come after one million? It's still ??.
One can have whatever formalization of mathematics one wants to have and any definitions one wants to have. But if we wish to know exactly what is the case with the usual formulations and definitions in mathematics then we need to discuss them as they actually are formulated and defined in mathematics.
That's helpful.
And it first starts with using 'infinity' as a noun in a context such as this. That just sets up all kinds of misunderstandings.
And, yes for any natural number n, the cardinality of the set of integers greater than n is aleph_0.
No I think we have a grasp of infinity or an awareness of the semantic. Some are more focused on this awareness than others. Some are more sincere to this awareness than others. Part of that awareness entails one cannot count to infinity. You can add one to any quantity, except of course infinity. Such is the nature of infinity. Yet, it seems to me that some seem to believe "beyond infinity" is meaningful.
What more can I say?
I believe I've said enough in this discussion and that beyond this is time not well spent.
Peace
But to be positive about this, I would suggest: What definition of cardinal subtraction does anyone here have in mind? That is to provide the formula P in the following (where P has no free variables other than x, y and z; and for each
If x and y are cardinals, then x-y = z <-> P.
Anyway, did someone say "beyond infinity"? 'Beyond infinity' has no apparent meaning. First, again, 'infinity' should not be used as a noun in this context. Set theory does not define 'infinity'. Second, the set theoretic fact is that for any infinite cardinality there is a greater infinite cardinality. So, yes, in that sense, there is an infinite cardinality "beyond" the cardinality of the set of natural numbers, but there is no "beyond infinity".
A quick look will tell you that there are twice as many feet as there are people. You do not need to count the number of people to know this to be true; just check for amputees...
Bijecting two feet for each person.
Which is the equivalent of saying beyond the quantity of infinity, there is a greater quantity of infinity (which is contradictory to the semantic of infinity). Again, you can add one to any quantity except of course the quantity of infinity.
Quoting TonesInDeepFreeze
I don't believe I'm the one saying untrue things.
And for about the fifth time, there is no object named by 'infinity'. So there is no object named by 'the quantity of infinity'.
There is least infinite cardinal, which is the cardinality of the set of the infinite set of natural numbers. And there are cardinals greater than the least infinite cardinal. Moreover, for each cardinal, whether it is a finite cardinal or infinite cardinal, there is a greater cardinal.
And when someone says "contradictory to the semantic of infinity" here it indicates that one did not read, or chose to ignore, what was written about that.
I'll repeat: Study of set theory is not a commitment to adherence to all the many meanings of 'infinite' and 'infinity' in everyday discourse, in philosophy and even other science. Rather, the adjective 'is infinite' has a special and very particular definition in set theory. It some ways it is compatible to other non set theoretic meanings of notions, but is not compatible with certain other non set theoretic meanings or notions.
No one disputes that the set theoretical definition might not accord with anyone's other notions. But that does make set theory inconsistent. A theory is inconsistent if and only if it proves some sentence P and not-P.
"There is a cardinality greater than the cardinality of the set of natural numbers" and your notion "There is no cardinality greater than the cardinality of the set of natural numbers" is not a contradiction of set theory, because set theory does not prove "There is no cardinality greater than the cardinality of the set of natural numbers" despite that that sentence is one you believe in your non set theoretic notions.
Banno humor...
The benefit to me of what you've posted here is that I now reject the following from the OP and would change the last part of it in the link I provided to my post:
Quoting Philosopher19
I still hold the belief that saying 1,2,3,4 ad infinitum or {1,2,3,4,...} does not mean one has shown an infinite number of natural numbers. One has essentially suggested a number sequence goes on forever. But since one cannot count to infinity, it is the case that the total number of items in that sequence will not be infinity. If I do not do this, I will hit contradictions. If I do this, I will avoid contradictions.
Seeing as your post benefited me, I should thank you. So thank you.
To Infinity and Beyond!
Quoting Philosopher19
Hey, no problem. Start with a definition of "untrue".
It seems to me that you think I'm not paying attention to what you're saying and I think you're not paying attention to what I'm saying. I think we should end our discussion.
Peace
The only reason something like a sequence of numbers can go on forever, is because of Infinity. It is not because the sequence of numbers are Infinite.
Two different things can go on forever at different speeds, but this does not mean that one will go farther than the other when both are set to go on forever. It may look that way if you were to try and "map the distance covered by one to the other", but neither will ever cover an Infinite amount of distance for one to be able to conclude something like "this Infinite distance covered is greater than that Infinite distance covered". Of course, this is not the same as saying something like "this amount of distance covered in Infinity is greater than that amount of distance covered in Infinity".
And if you want to discontinue your posting in this thread, then you can discontinue it. No one is stopping you.
"saying 1,2,3,4 ad infinitum or {1,2,3,4,...} does not mean one has shown an infinite number of natural numbers."
Yes, saying that for every natural number there is a greater natural number does not in and of itself imply that there is an infinite set of natural numbers. But {0 1 2 3 ...} is not notation that for every natural number there is a greater natural number, but rather it is an informal notation to stand for the set of all and only the natural numbers.
The way set theory proves there exists a set with all and only the natural numbers is by an axiom from which we prove that there exists a set with all and only the natural numbers.
So the objection that we can't extrapolate from "for every natural number there is a greater natural number" to "there is a set with all the natural numbers" is true but a huge strawman since set theory does NOT claim that we can extrapolate that way.
Again, such objections are a product of sheer unfamiliarity with the subject matter.
It is by tolerance that there are forums such as this that allow such spreading of confusion.
Quoting TonesInDeepFreeze
We are in disagreement right there. You say you can have an infinite number of natural numbers. I say this statement will lead to a contradiction. That contradiction being "this infinite set is bigger than that infinite set". This is a contradiction because infinity is that which you cannot add to or have more than of. If you don't believe in this then how can we possibly agree?
This is why I said:
Quoting Philosopher19
I don't deny that there is such a set, but I deny that the total number of natural numbers in this set reaches infinity. Imagine you have all the natural numbers in {1,2,3,4,...}. Can you show what number comes before infinity to be able to meaningfully assert something like {1,2,3,4,...} consists of an infinite number of natural numbers?
You can't just say it has all the numbers and all the numbers amount to infinity
An example:
Banno takes infinity minus one million and gets infinity.
You say you can't subtract from infinity.
I say an infinite set of integers minus the first million integers is a set with the first million integers removed and I could list them.
1, 2, 3, 4......
I'd say pot calling the kettle black.
Now for the third time:
Anyone can have whatever concept of mathematics they want to have. But having a different concept from set theory doesn't entail that set theory itself proves any contradictions.
As to subtraction with infinite cardinals, again I say, just start by defining it.
And you skipped what I said about removing.
Suppose something goes on forever such that it covers more and more distance as it goes on. So it covers 5km, 10km, 15km ad infinitum. I can't say {5km, 15km, 20km, ...km} is an informal notation to stand for all the distance it covered and that that distance is infinite. Do you see what I'm saying? You can't just say {1,2,3,4,...} is an informal notation to stand for the set of all and only the natural numbers and that the total number of natural numbers in that set is infinity.
I believe in the same way that I can't say the total distance covered is infinite, you can't say the total number of natural numbers in that set is infinity.
But you are such a pot, as recently you ignored what I said.
I am addressing your point. I believe you are not reading all of it. See my last post to you.
A separate question is whether in mathematics there exists such a set. And I addressed that, but you skipped what I wrote.
I don't know what more to say. When I use the label/word "infinity", I'm not sure you're focused on the same semantic that I'm focused on.
Quoting TonesInDeepFreeze
I'm sorry if discussing with me has been a negative experience for you.
I don't begrudge anyone from having whatever concept and definition of infinitude they wish to have.
But having a different concept and definition of infinitude doesn't thereby entail that there is a contradiction in set theory or mathematics.
Again, yes, there may be a contradiction between set theory and certain other formulations. But that does not entail that there is a contradiction within set theory.
Again, for emphasis yet again, since you keep skipping this point, no one should deny you from having whatever concepts and definitions you would like to have, and if thereby set theory does not suit you or does not make sense to you, then so be it, but that doesn't entail that set theory leads to any contradiction in itself.
Yes, set theory does not have the same concept of infinitude that you have. As well as, which you also keep skipping, set theory does not refer to an object named 'infinity' but rather to the property of being infinite, which is a crucial distinction.
In this thread, there was discussion about set theory and that discussion had important errors. So I provided a systematic synopsis of the area in discussion as that synopsis corrected the errors and explained why they are in error. Then replies came to my posts, but certain of those replies still had misconceptions about set theory.
Again, espouse whatever concept of infinitude you wish. But that does not justify an incorrect and misinformed critique of set theory.
And whether the thread is or is not a negative experience for anyone, it still stands that your posting has been an absurd loop.
The answer to your problem is quite simple. In mathematics things are done by axiom. If you want to count to infinity and beyond, simply produce an axiom which allows you to do that, and bingo the infinite is countable, and you're ready to go beyond. Look closely at the following:
Quoting TonesInDeepFreeze
Quoting Banno
Actually, the inverse is what is the case, counting is a form of bijection. But this does not necessarily imply that all bijections are a form of counting. And, some might still argue that there are forms of counting which wouldn't qualify as bijections. It all depends on how one might restrict these concepts though definition.
Also, AGAIN, there is no object in set theory called 'the infinite'.
And it is not the case that set theory says that every infinite set is countable.
And a bijection is a certain kind of function. And set theory doesn't have a term 'counting' so set theory does not say anything about whether bijections are a form of counting.
Definitions:
f is an injection iff f is a one-to-one function. We may also say 'f is an injective function'.
f is an injection from x into y iff (f is an injection and the domain of f is x and the range of f is a subset of y).
f is surjection from x onto y iff (f is a function and the domain of f is x and the range of f is y).
(So every function is a surjection from its domain onto its range.)
f is a bijection from x onto y iff (f is an injection and f is a surjection from x onto y).
(So every injection is a bijection from its domain onto its range.)
/
It seems to me that your feet-head is just an example of multiplication. Two feet for each of n number of people is 2*n feet.
Or it could be a bijection between the number of pairs of feet and number of people. There are the same number of pairs of feet as there is the number of people.
A clearer example would be just two lines of people such that we could see that there is a one-to-one correspondence between the people in one line and the people in the other line.
This comparison illustrates that the "distance" covered between 0 and 1 in the first example is different from the "distance" between 0 and 1 in the last example. It can be concluded that the first example which includes the rationals and irrationals represents a larger infinity than simply an infinite series of natural numbers.
Does this explanation make sense? I admit that I haven't spent much time studying the intricacies of infinities and am not completely familiar with the technical terms and notations that mathematicians typically use to discuss these concepts.
The set of rational numbers between any two natural numbers is not sequenced by the ordinary less-than relation on rational numbers.
Between any two natural numbers there is a denumerable sequence of the set of rational numbers between the natural numbers, but it is not isomorphic with the ordinary less-than relation on rational numbers.
The set of irrational numbers between any two natural numbers is not sequenced by any countable ordinal.
The set of irrational numbers between any two natural numbers is sequenced by some uncountable ordinal if we have the axiom of choice.
The set of rational numbers between any two natural numbers is equinumerous with the set of all natural numbers.
The set of irrational numbers between any two natural numbers is not equinumerous with the set of natural numbers.
It is not the case that there are more rational numbers between 0 and 2 than between 0 and 1.
It is not the case that there are more irrational numbers between 0 and 2 than between 0 and 1.
The set of rational numbers is equinumerous with the set of natural numbers.
Any infinite subset of the set of rational numbers is equinumerous with the set of natural numbers.
Any infinite subset of the set of rational numbers is equinumerous with the set of rational numbers.
The set of irrational numbers is not equinumerous with the set of natural numbers.
The set of irrational numbers is not equinumerous with the set of rational numbers.
There are infinite subsets of the set of irrational numbers that are equinumerous with the set of natural numbers.
There are infinite subsets of the set of irrational numbers that are equinumerous with the set of irrational numbers.
Any infinite subset of the set of natural numbers is equinumerous with the set of natural numbers.
If x is an infinite subset of the set of natural numbers and y is a finite subset of the set of natural numbers, then {n | n is in x and n is not in y} is equinumerous with the set of natural numbers.
If x is an infinite subset of the set of rational numbers and y is a finite subset of the set of rational numbers, then {r | r is in x and r is not in y} is equinumerous with the set of rational numbers.
If x is an infinite subset of the set of natural numbers and y is a finite subset of the set of natural numbers, then the union of x and y is equinumerous with the set of natural numbers.
If x is an infinite subset of the set of rational numbers and y is a finite subset of the set of rational numbers, then the union of x and y is equinumerous with the set of natural numbers.
'distance' is defined by the absolute value of the difference between points, not by cardinality. The distance between 0 and 1 is 1, no matter what about the cardinalities of the set of irrationals and the set of irrationals between 0 and 1.
That's all ordinary mathematics, proven from the ordinary axioms.
One is free to propose different axioms that prove differently.
Since there is a whole lot of difference between the different types of numbers you outline, I think it a very good idea for a mathematician to look for a whole new set of axioms to better deal with the problem of having different types of numbers. This could avoid the problem of needing principles to relate the different types of numbers to each other, in an attempt to reconcile the sometimes irreconcilable difference between them. Attempting to reconcile the incompatibility between them tends to create a new type of infinity. So every time a new type of number is produced to deal with a specific problem that has arisen, a new type of infinity is produced. One could get rid of a whole lot of unnecessary complexity with a more comprehensive set of axioms..
My belief is that we can't just produce axioms. We can only recognise truths about Existence such as 1 add 1 equals 2 or the angles in a triangle add up to 180 degrees or one cannot count to infinity.
Quoting TonesInDeepFreeze
You see, my position is that there is only one semantic/definition for the label "infinity". If we are focused on different semantics, we are not talking about the same thing. I think it would then help if we don't use the same label for that thing so as to make it clear that we are talking about different semantics.
Quoting TonesInDeepFreeze
I think a belief or theory has to be consistent with Existence as a whole, and not just consistent in isolation. To me, by definition, any theory or belief that encompasses the following belief "the set of all sets is contradictory" is a contradictory belief. It would be like any theory or belief encompassing the belief that "triangles are not triangular", which is contradictory belief to encompass.
Quoting TonesInDeepFreeze
Again, I think you are focused on a different semantic to me. To me, the semantic of infinity is one quantity or quality. It is a quantity or measure or quality that can never be reached. My position is best summed up with the following:
The only reason something like a sequence of numbers can go on forever, is because of Infinity. It is not because the sequence of numbers are Infinite. The only thing I view as Infinite, is Existence.
You should see non-Euclidean geometry where the angles in a triangle can be more or less than 180 degrees.
Compare the following:
1) There are an infinite number of elements between 0 and 1
2) There is no end to the number of elements between 0 and 1
If there is no end to something, how can another thing with no end be twice as large as it? Don't they both have no ends?
This is why there is a distinction between something that can go on forever and something that is infinite. Infinity allows for things like a number sequence to go on without end, but the thing that goes on without end is not infinite, it just goes on without end without actually reaching infinity just as one cannot count to infinity and reach infinity even if one was projected to count forever (so the number sequence is not infinite).
Quoting Michael
Imperfect triangles are imperfect by definition. I'm focused on absolutes.
Axioms are simply produced, created. The ones which prove to be useful are put to use, and they persist by becoming conventional. "Truths about Existence" is irrelevant to the mathematicians who create axioms.
In binary, 1 + 1 equals 10. In mod1 arithmetics, 1+1 equals 0.
Suppose someone produces an axiom. Will it not be the case that that axiom will either be contradictory in relation to certain truths or consistent in relation to certain truths? Existence determines what is true and what is false. Whether any belief or axiom highlights truths or is contradictory to truth is determined by Existence/Truth. If not, there is no truth or semantics to work with to deduce further truths.
What do you mean by an "imperfect" triangle?
Quoting Michael
What you may call a non-euclidian triangle, I call an imperfect triangle. A perfect triangle has perfectly straight lines and its angles add up to 180 degrees. Another shape may resemble this without actually perfectly being this (like an imperfect triangle whose angles don't add up to 180 degrees but is near)
The angles in a true triangle add up to 180 degrees because that is the nature of Existence. It is not because someone said it or highlighted it.
What is this supposed to mean?
Quoting Michael
I think it's clear enough, therefore, I don't want to clarify further.
I believe the the concept of infinity is often misunderstood because it can be applied to different contexts, such as time and space, which are not necessarily equivalent. To explore the differences in sizes between different infinities, let's consider a few thought experiments that illustrate how infinity can vary in magnitude.
First, imagine you have achieved immortality and are presented with two options: to receive $1 every day forever or $1 every year. Intuitively, you would choose $1 every day because, over the same infinite duration, you would accumulate more money. This illustrates that while both options extend to infinity in time, the rate at which you receive money differs, leading to a larger "size" of wealth in one scenario over the other.
Now, let's consider a spatial analogy. Imagine two pipes, both of infinite length, but one has a diameter of 1 inch and the other has a diameter of 10 inches. Despite their lengths being equally infinite, the pipe with the larger diameter has a greater volume. This demonstrates that even with one dimension being infinite, other finite dimensions can contribute to a difference in "size" or capacity.
Interestingly, if we were to expand the diameter of the pipe to infinity as well, we would lose the essence of what makes a pipe a pipe. To maintain its identity, certain characteristics, like diameter, must remain finite. This constraint allows us to differentiate between pipes of different diameters, even if their lengths are infinite.
Lastly, consider an infinite number of pencils, each 6 inches long, laid end to end to form a line of infinite length. If we compare this to another line composed of an infinite number of 3-inch pencils, both lines would stretch to infinity. However, if you were to take one pencil from each line, there would be a clear difference in their lengths. This paradox highlights that while the total lengths of both lines are infinite, the "size" of their components is different, and this difference is observable when comparing individual elements.
So, the concept of infinity can indeed vary in magnitude depending on the context. Temporal infinity can differ based on the rate or frequency of an event, while spatial infinity can vary when other dimensions are considered. These examples show that not all infinities are created equal, and it is the nuances in their properties that allow us to distinguish between them.
You did a very good job covering the details of the formal math. About 8 hours ago. Again, helpfully.
I'll give my perspective. I think the way this math physically exists is only by physical brain state that is able to support it. So not everyone is going to be at the level of the math people.
Given there is a lot to know, my approach is to model it as mental algorithms to get a bird's eye view:
Brain; (Algorithm 1)
Brain; (Algorithm 2)
Brain;.(Algorithm 3)
And so on.
Obviously the math people pick up on a lot of these that the rest of us don't have, but for all of us, picking up on as many of these little recipes as we can can be a good strategy.
If this is so, then none of these concepts have any existence outside our brains. What we see and should expect is a lot of variation in approaches to problems unless they are standardized such as in formal math.
You cannot start counting 1,2,3,4,... ad infinitum and reach somewhere, anywhere. Infinity has neither a start or an end.
Then, counting (natural) numbers you can never reach infinity because that infility would be also a number, and infinity is not a real or natural number.
Quoting Philosopher19
A set is a collection of objects (elements, members). I'm not sure if we can talk about an infinite set, although there are some theories about it (e.g. ZermeloFraenkel).
As I see it, an infinite set cannot be defined as one consisting of infinite numbers, because only the fact of being defined (limited) makes it (de)finite. An infinite set would be something limitless, hence undefinable.
All this raises questions about the infiniteness of the Universe, whether it started (created) from something or it always existed, etc. And, as I see it, since we don't have a proof that it is created from nothing, it must have always existed, even in the form of extremely high density and temperature, which at some point exploded (re: Big Bang), or in any other form. But I'm not the right person to talk about these things.
And, of course, I'm well aware that all I said is subject to debate ...
But Euclidian triangles don't exist in nature.
Consider a hotel with an infinite number of rooms, all of which are occupied. Due to the infinite guests, there are no vacant rooms. However, if each room in the hotel were to magically double into 2 rooms, the hotel would then have an additional infinity of rooms to accommodate an extra infinity of guests. Although the number of rooms seems the same in both cases, the capacity differs in some sense. In the first case, no more guests can be accommodated, while in the second case, an additional infinity of guests can be accommodated. This doubling (spacial sense) can continue (temporal sense) indefinitely in both time and space.
Yes, i am aware of that, but i didn't see the point in describing something one could just read anywhere. I was trying to show a different way of conceptualizing different sizes of infinities. That's all, but i'm more interested in if my example is a reasonable one or not.
Is it?
Yea, like i said im not up on all the terminology. I'm a little bit motivated now to look a little deeper into it, because i do find it interesting. I'm going to look up some of these concepts you mentioned like transfinite cardinals, and ordinal arithmetic. But i'd like to ask.
What was i describing in my last example about the infinite hotel. What is the correct terminology for what i described?
The statement that some infinities are bigger than others comes from set theory. The OP talks about infinities in the context of counting procedures. These are two different concepts of infinity.
In set theory, infinities are just infinite sets. In turn, infinite sets are those equinumerous to one of its proper subsets. That is, they have the same cardinality, which is defined by the existence of a bijective map between the two.
In set theory, an infinite set is bigger than another when there exists a surjective function from one to the other, but not vice-versa. What Cantor has proven is as simple as that: one cannot construct a surjective map from ? to ?.
Perhaps you would like to work with a different definition of cardinality, infinite set, or infinity (scraping sets altogether). That is fine, but keep in mind that you would be changing subjects, rather than disagreeing with set theory in general or Cantor in specific.
Most of what i think about infinities comes from my own intuitions, so forgive me if i sound a bit ignorant of the well-established terms and procedures involved.
The "counting procedure" aspect is what i relate to the temporal sense of speaking or thinking about it. The other side seems to be more spatial in character, which instantiates an infinity all at once, outside time, so to say. It's just something i noticed recently and thought it might be useful to know when thinking about infinities. There are probably proper terms for these distinctions, and if there aren't then there should be.
Welcome to TPF! :smile:
Suppose that the universe has infinite space, and let's also say that there is an infinite number of particles in this space. For there to be space between the particles, would that not make space a bigger infinity than the infinite number of particles in the infinite space?
Notation I'll use:
'iff' for 'if and only if'
df. x is equinumerous with y iff there is a one-to-one correspondence between x and y
df. x is countable iff (x is finite or x is equinumerous with the set of natural numbers)
df. x is denumerable iff x is equinumerous with the set of natural numbers
'*' for cardinal multiplication
Your scenario can be boiled down to this:
There are denumerably many rooms. And if we multiply the number of rooms by 2, then there are still denumerably many rooms.
That reflects the set theoretic fact that if H is a denumerable cardinal and K is a countable cardinal, then H*K = H.
In this case, H is the number of rooms in the original hotel and K = 2.
Thank you for that clear and easy to understand example.
When you say "extend to infinity in time", I assume you mean go on forever. It follows that both will forever add to their money. It also follows that one will always have more than the other. But it also follows that neither will ever have an infinite amount of dollars precisely because their wealth will not amass to infinity dollars. To say that it would is to say that one can count to infinity.
Quoting punos
I believe there is no contradiction in saying that something can go on forever. So I believe a pipe can go on forever. But to me, Infinity is the reason something can go on forever or be endlessly added to. It is not the measurement of the thing that goes on forever.
I think Infinite and Infinity should be exclusively used to refer to Existence, and a part of Existence is not equal to the whole of Existence. Trying to divide Infinity into parts seems contradictory to me. Another reason for why I think a pipe that measures infinite in length is an impossibility.
Quoting Alkis Piskas
Agreed (especially with "Infinity has neither a start or an end").
Quoting Alkis Piskas
I don't believe we can talk about different infinite sets because it will lead to contradictions. But I do believe Infinity and Existence denote the same Entity. I see Infinity/Existence as the set of all existents. I see there being no end to the number of existents purely because the nature of Infinity/Existence allows for such possibilities.
Quoting Alkis Piskas
It is clearly contradictory for something to come from nothing. And since I have heard some say that the universe is expanding, my view is that the universe is not infinite (if it's expanding, it's not infinite). But I do view Existence/Omnipresent as Infinite. I see Infinity as the reason for why an endless number of things can be imagined or thought about or experienced (dream or otherwise). Infinity has Infinite potential, therefore, an endless number of things can be imagined or thought about or experienced (we and our minds are wholly contingent on Existence. There is no non-Existence for us or our minds to draw anything from).
To my knowledge they don't exist in our universe due to gravity. But I see our universe as just a part of Existence/Nature/Infinity. Something has to account for why we are aware that "the angles in a triangle add up to 180 degrees". To me, the nature of Existence/Infinity accounts for this awareness.
Existence can accommodate both perfect and imperfect triangles. We have experienced imperfect triangles (as in we have visually seen them), we have not experienced perfect triangles. But somehow, we have the knowledge that the angles in a triangle add up to 180 degrees. This is the awareness we have got in Existence and have gotten from Existence.
How?
It would not! Are you familiar with injective, surjective, and bijective functions?
Suppose there are two sets of objects, A and B, whose size (cardinality) we wish to compare. That is, we want to know which is bigger (or equal): size(A) or size(B)? This is also written as card(A) and card(B).
If there is an injective function f : A ? B mapping the objects of A into the objects of B, then for every distinct object in A one can find a distinct object in B. This entails that size(A) ? size(B).
If there is a surjective function f : A ? B, then for every distinct object of B one can find a distinct object in A. This entails that size(A) ? size(B).
If there is a bijective function f : A ? B, this just means that "f" is both injective and surjective, so that there is a one-to-one correspondence between elements of A and elements of B. This entails that size(A) = size(B).
This is the toolkit that defines the notion of size (cardinality) in set theory, and it must be used to compare sizes among sets, including infinite sets.
So take the positive natural numbers ? = { 1, 2, ··· }, which is infinite. Also take the non-zero integers ? = { ··· 2, 1, +1, +2, ··· }, which is also infinite. I have excluded zero for convenience.
Now, ? might seem bigger than ?. However, one can construct a bijective map between the two, proving that they have in fact the same size. There are many such possible maps. Any one of them suffices.
? One such map begins by mapping all odd numbers in ? to the positive numbers in ?, like so: 1 maps to +1, 3 maps to +2, 5 maps to +3, and so on. The mapping rule is: 2k+1 ? k+1 (where k starts from 0).
? It continues by mapping all even numbers in ? to the negative numbers in ?, like so: 2 maps to 1, 4 maps to 2, 6 maps to 3. The mapping rule is: 2k ? k (where k starts from 1).
This covers all numbers both in ? and in ?, so they have the same size.
Infinite sets have this weird property where one can rearrange their items in many different ways, leading to surprising one-to-one correspondences. This is well illustrated by Hilbert's hotel.
In the case you provided, we could have 100 units of empty space for every 1 particle, but both would still be equinumerous, for there would still be a map between them. Just map the first 100 particles to the first 100 units of space, then the next 100 particles to the next 100 spaces, and so on. (This is an abstract mapping: you are not in fact shuffling particles around.) You will never run out of particles to map to some unit of space. Every particle will be mapped somewhere; every spatial unit will be designated a particle. So they might well be infinities of the same size.
This would be different if the space in question were continuous (like ?). Since particles are discrete (countable), they have the same cardinality as ?, which is strictly less than that of ?.
I really don't think "truth" in this way is relevant. This is more of an issue of pragmatics, mathematics is a tool. You wouldn't say that one saw is more truly a saw than another saw, or on shovel is a more true shovel than another. So the axioms which are accepted, "which are bought", are the ones which mathematicians like to use. It may well be the case that existence determines truth, like you say, but that's not relevant to the selection of mathematical axioms.
I do believe we can bring "truth" into the picture in a different way though. Since mathematicians can choose to use whichever axioms they feel comfortable with, we can say that the axioms follow use. That means that they are a reflection of what mathematicians are doing. Therefore we can say that they are descriptive rather than prescriptive. The axioms do not give mathematicians rules for how to do things, because the mathematicians get to create and choose their own axioms. So the axioms simply provide a representation of what mathematicians are doing. Since they are descriptions, "truth" is to be found in how well the axioms represent what the mathematicians are actually doing. As an analogy, consider looking at a dictionary and judging how truthfully the definitions represent how people are actually using the words which are defined there.
Right.
Quoting Philosopher19
I see what you mean. Well, the words "exist" and "existence" can be used in different ways. And it can be used strictly (substantially, concretely) and loosely (insubstantially, abstactly). And I guess the second form applies to what you say above.
Quoting Philosopher19
Right. Or, impossible. Yet, we stiil can use the expression "something from nothing" loosely or figuratively. But there are always some conditions (something) that allow the creation of some other thing (something). This is the universal law of cause and effect.
Quoting Philosopher19
But doesn't an expanding universe mean that this process is infinite and thus the universe itself is limitless? It is not much different than if we consider the universe as being static, in which case it can also be infinite.
Well, that's why the infiniteness of the Universe is still debatable today! :smile:
Infinitiness and expansion are independent. The Universe could be infinite and expanding; finite and expanding; infinite and static; and finite and static. All combinations are possible.
It is important to understand what it means to say that the Universe is expanding.
If the Universe is infinite, such expansion does not mean that the Universe is increasing in cardinality (set-theoretic size). Infinity is infinity (of a given size: aleph-0, aleph-1, aleph-2, and so on).
What the Universe's expansion means, whether it is infinite or not, is that its local energy density is decreasing. In other words, there is more spatial structure between each of its internal field excitations (particles, energy).
(Note that, if the Universe were infinite, its global density would remain constant. This is because global density would be calculated by dividing two infinite sets of the same cardinality.)
This touches a related point: the Universe is undergoing an internal expansion. It is not expanding *into* something. There is no space external to the Universe. The Universe is just acquiring more internal structure. The cosmological details are sure to be complicated and relevant (and I'm no cosmologist), but that is the general gist.
There is also a related point: the Universe could be finite but still unbounded, without an edge. It could just be twisted unto itself, as in a loop, like the surface of the Earth. On our planet, if you keep going North, you'll eventually just change hemispheres and start going South.
Hi Dan, I see you're new here, so welcome to this space.
I don't think it's proper to say that expansion means "more spatial structure" between internal field excitations, unless you are speaking of a "spatial structure" which is other than Einsteinian space-time.
Doesn't infinity mean endless? i.e. unreachable eternal continuation in concept?
If it was reachable, then it wouldn't be infinity. Any set or size would be unknowable, if it were infinity. Therefore talking about different size, set or number of infinity, is it not a nonsense?
Thanks for the warm welcome and the thoughtful reply
What is the proper interpretation of the cosmological constant ?? I understand that it corresponds to a vacuum energy density, pervading all reality. Such energy is called dark energy, I gather. Since I'm sketchy on field theory, I don't know how this goes, but somehow this energy density produces a repulsive force beween any two objects in spacetime (within each other's lightcones?). Matter remains cohesive because ? is very small compared to other forces, so that its effects really only show at an intergalactical scale (megaparsec).
Now, somehow this leads to the expansion of the Universe even in the case where the Universe is finite and bounded, which is a possibility considered by cosmologists. In this case, the Universe is increasing in total size, but not increasing *into* anywhere, so it becomes bigger because it has more internal spatial structure. This is what I meant. Why do you think this is incorrect?
One can talk about infinity conceptually, as one does in mathematics, without reference to its empirical verifiability.
When it comes to the empirical application of the concept of infinity, it is indeed reasonable to think that it is fundamentally unverifiable whether something is infinite. So we couldn't know whether spacetime is continous or discrete, because our measurements have finite resolution. The same would go for whether the Universe is infinite in extension or not.
However, humans can be quite ingenious, and we shouldn't rule out any possibility apriori, just from armchair thinking. Perhaps the supposition that the Universe is continuous rather than discrete has different consequences for our finite observations; I don't know. The same goes to the cosmological hypothesis where the Universe is infinite, which is thought to hold in case the Universe's matter density equals its dark matter density, a possibiity referred to as ? = 1. (I'm pretty much quoting Wikipedia on the expansion of the Universe.)
Infinity by size, set, numbers, zero to infinity, infinity with no beginning and no end, infinity applied as mental constructs, infinity applied to physical matter, pairing of infinities, pairing infinite sets to finite sets, infinity as a simple concept of continuing without limit. Advanced math concepts of infinity......
Some of the advanced math theories are maybe just some mental showboating of things the math people can do with their brains.
Actually the original attempts are the most interesting from a philosophy perspective.
"Let us not forget: mathematician's discussions of the infinite are clearly finite discussions. By which I mean, they come to an end." - Philosophical grammar, p483. Wittgenstein.
Welcome to TPF~ :cool: :up:
A person hasn't studied the pertinent mathematics, doesn't know anything about it, doesn't understand it. So their response to it is to say that it might be just a bunch of "mental showboating" anyway.
The 'showboating' is actually impressive to me so don't take it the worst way.
Well, there are a lot of scenarios on the table. Let astronomers and cosmologists debate about them ...
Welcome to TPF! :clap:
That's kind of funny...just like it is
By Being. Existence just Is. It just is the case that triangles are triangular or that Existence is Infinite or that 1 plus 1 = 2. Or if you're interested in more on Existence, it just is the case that Existence indubitably exists and is Perfect. I won't go into detail with regards to how Existence is Perfect and indubitably exists. I'll just provide the link to the argument: http://godisallthatmatters.com/2021/05/03/the-image-of-god-the-true-cogito/
Again, in mathematics, the concept is 'is infinite' as an adjective, not 'infinity' as a noun. And continuity is a different idea, while the idea of "size" is approached by the formulation of the idea of cardinality.
Nothing at all like it is.
Quoting Metaphysician Undercover
Of course, it is possible, for example, for mathematicians to be using the label "infinity" to refer to a semantic that is different to the semantic of infinity. But from what I've seen of mathematicians, they either have no part for infinity, or they're using infinity wrong. I believe they're doing the latter which leads to the former (which I think is why I have heard it said before that "maths is incomplete")
Quoting Metaphysician Undercover
I just think if mathematical axioms are to be selected, they have to be such that they do not lead to what is contradictory to Existence/Truth (or just semantics in general).
Quoting Metaphysician Undercover
If a mathematician or a philosopher decides on an axiom or theory that requires belief in the following (or at least logically implies it or leads to it): Nothing can be the set of all things (which logically implies Existence is not the set of all existents), or one infinity is a different bigger than another (or is a different quantity than another), I believe that axiom or theory should be disregarded or at least viewed as contradictory to Existence/Truth (or at least contradictory to the semantic of infinity).
So your method of conversation is to ignore when someone informs you nearly a dozen times on a point:
In this context, mathematics doesn't use 'infinity' as a noun, as if there is an object named 'infinity', but rather 'is infinite' as an adjective to name a property. That distinction is crucial to understanding the subject matter.
Quoting Philosopher19
What you've seen is what you've allowed yourself to see, which is virtually nothing about the actual mathematics you've not even bothered looked up.
I don't mean to use Existence loosely/abstractly. By "Existence" I mean that which encompasses all things physical or otherwise (if otherwise is possible). So dreams (which some may view as non-physical) are clearly a part of Existence. The term universe seems limited to me in terms of accounting for all that exists (I cannot comfortably say something like "the universe has a space for dream worlds" whereas I can comfortably say "Existence has a space for all worlds including the universe and dreams"). To me, Existence/Infinity clearly fits the bill of 'encompasses all things/existents' whilst universe does not.
Quoting Alkis Piskas
If the universe is expanding, it is expanding in something. Some thing has to be Infinite to allow for the possibility/potentiality for the universe to forever expand. But it is also the case that even if the universe expands forever, it will not become infinite (this is not unlike me saying even if I count 1, 2, 3 ad infinitum, I will never reach infinity).
So to me, Infinity/Existence is the reason that something can expand forever or go on forever. As for the thing that expands (like the universe), it is a part of the Infinite. It is not itself infinite.
Thanks for the welcome!
As regards Wittgenstein's remark, we use finite statements to fixate reference on infinite objects and work out their properties. There is no contradiction in that.
Here is a finite definition of an infinite set: "A given set S is infinite iff there exists a bijective function between S and a proper subset of S." Furthermore, such a bijective function can be stated finitely.
Here is an example. Take the set of natural numbers ? = { 0, 1, ··· }. Now take a proper subset of ? containing only even the numbers, ? = { 0 , 2 , ··· }. These two are equinumerous because there is a bijective function f : ? ? ?, given by f(n) = 2n.
The proof that "f" is bijective is finite. So is the proof that ? is a proper subset of ?.
I get your point with regards to empirically verifying infinity, but I believe the a priori is superior to the a posteriori in that whatever observation we make (scientific or otherwise), has to be interpreted in line with the dictates of pure reason. It also has to be such that it does not contradict the semantics that we are aware of (for example, we must not have a theory that amounts to saying or logically implying that triangles don't have three sides because that contradicts the semantic of triangle) .
If a scientist says something like "I have observed something pop in and out of Existence" because it may have looked that way to him, we have to reject him because 'something popping in and out of Existence' is clearly contradictory. Non-Existence does not Exist for something to pop into or come out of. Things can be turned on and off but this is not the same as things popping in and out of Existence.
In the context of the discussion and the differing opinions I might have been suggesting a while back that the mathematicians here should (occasionally)take their metaphorical pen from the mathematics page to a brain theory of mathematics page.
If you understand that our brains are churning out stand alone theories that work fine in a certain context but don't all work together in every context you will better understand why we disagree.
Is that reasonable? Keep doing what you are doing.
It makes no difference. Existence is Infinity (here it is a noun). Existence is Infinite (here it is an adjective). You cannot become Infinite (adjective) even if you expand forever. You are not Infinity (noun) if you are not Infinite (adjective).
You keep saying I ignore your points, but rightly or wrongly, I also think you have not read or considered what I've written with sufficient attention to detail.
Quoting TonesInDeepFreeze
I am not claiming to have seen everything. But (again, rightly or wrongly) I think I've seen enough to say:
Quoting Philosopher19
In any case, in the event that you have made good points and I have failed to give them the right amount of attention, I apologise. I do think that I am being sincere and honest in this discussion (as well as not closed-minded).
I previously corrected that:
The set of real numbers between 1 and 2 has the same cardinality as the set of real numbers between 1 and 3.
The set of rational numbers between 1 and 2 has the same cardinality as the set of rational numbers between 1 and 3.
Quoting Vaskane
Those other ways haven't been the context in which 'infinity' is used as a noun here. The gravamen of the original poster has been that there are not different "sizes of infinity". The notion that there are infinite sets with different cardinalities is a set theoretic notion in which context it is crucial not to speak as if there is an object named 'infinity'.
It makes a real difference. By saying 'infinity' as a noun and then that there are different sizes of infinity is to picture an object that has different sizes. There is no such object in mathematics.
Quoting Philosopher19
I've answered that. I have read and re-read and thought about what you've posted. What you have posted is in ignorance of the mathematics you criticize, mixed up, and dogmatic. You commit a non sequitur by inferring from the fact that I have corrected you on certain crucial points that I haven't read and considered what you posted.
And even if I had not read, re-read and thought about what you've posted, it would not change that you have continued to ignore the information given you. Please stop saying 'infinity' as you do in context of the mathematics you're criticize. The unthinking and habitual use of 'infinity' in that context both reflects a misunderstanding of that which you criticize and contributes to even more misunderstanding of it.
Quoting Philosopher19
Good faith in posting a critique of mathematics would entail at least knowing something about it.
Set theory is not standalone in the sense that it has no application outside itself.
I am very well aware that that set theory does not account for all contexts of human knowledge. I have posted about that previously. And it is a point that is obvious most especially even to those who work with set theory.
OK.
Quoting Philosopher19
I guess this "something" is "space", right? Like a balloon ...
But this seems impossible since space is part of the universe itself; it cannot be larger than it. E.g. like the space around a balloon that is inflated ...
Quoting Philosopher19
OK.
As I said, astronomers and cosmologists are more suitable for answering these questions ...
In ordinary mathematics the idea of size is formalized as cardinality.
I've said that one may propose whatever other concept or alternative mathematics one wishes to propose. But when you say "on the basis of mathematics" that would ordinarily be understood not to be some unspecified personal alternative you have but rather ordinary mathematics.
So for you, the nature of existence accounts for the awareness that Euclidian triangles sum up 180º degrees. That is what you said in your two comments.
You have not specific what the nature of existence is besides a brute fact. If so, how do brute facts account for our knowledge of something that is not verified empirically Euclidian triangles?
You said here:
But you did not say how.
Can you provide a formal criterion for what constitutes the size of an infinite set, beyond its cardinality?
When taking about intervals in ?, the cardinalities of the [1,2] and the [1,3] intervals are exactly the same, namely, the cardinality of the continuum. The reason is that there are bijective functions linking the two. Take, for instance, a function f : [1,3] ? [1,2] with the rule f(x) = (x+1) / 2.
Hey, to you I'm just words on a screen, but I'm an actual person. Sorry if I misunderstood you, there are a lot of comments in this thread and I'm not up to speed with the whole context.
It's fine, I can see you're a nice guy.
Quoting Vaskane
Unfortunately for you, you have conflated distance of an interval with the size of the infinite set of numbers in that interval.
The distance between 1 and 2 is smaller than the distance between 1 and 3.
But the cardinality of the infinite set of numbers in those intervals is the same.
First, you took exception to me taking 'size' as cardinality, as you said that cardinality is not the only sense of 'size'. So I pointed out to you that you yourself said the context was mathematics. Then you didn't recognize that but instead ...
Second, you conflated distance of an interval with the size of the infinite set of numbers in that interval.
It stands that "the set of numbers between 1 and 2, is a smaller set of infinite numbers between 1 and 3" deserved being corrected. Or, sure, shoot the messenger if you prefer.
First, incorrect objection to cardinality.
Second, conflating distance with cardinality.
Third, citing area when there is no area involved.
It's a line. Area is not involved.
Your claim "the set of numbers between 1 and 2, is a smaller set of infinite numbers between 1 and 3" is plainly wrong and deserves being corrected by whatever "dude" extends the favor of correcting it.
https://jlmartin.ku.edu/courses/math410-S09/cantor.pdf
I think Infinity is why something can go on forever. But if something goes on forever (or keeps going without end) it will not become infinite (just as if I keep counting without end, I will not reach Infinity)
To me, the only thing that is Infinite, is Existence. And Existence has always Existed and will always Exist (so It has no beginning and no end whilst all ends and beginning are within It. And if something goes on forever within It like a number sequence or a forever expanding universe, then that thing will never reach Infinity/Infiniteness
Quoting Corvus
I think it's nonsense to say Infinity comes in various sizes. But the semantic of Infinity itself is not nonsense because it is clearly meaningful. As for sets, the only thing that can be the set of all cardinalities or houses or any other meaningful/imaginable/understandable thing, is Existence/Infinity. Since Existence is Infinity, it allows for there to be no end to the number of numbers possible (because you can always add one and this can go on forever without Infinity being reached or exhausted).
I said nothing about rows and columns.
First, you incorrectly objected to taking size as cardinality when you said yourself that the context is mathematics.
Second, you conflated distance with cardinality.
Third, you claimed it's about area, though area is not involved.
Fourth, you were a Navy cryptologist and something about me regarding rows and columns, though I said nothing about rows and columns.
Will there be a fifth attempt to evade the plain fact below?:
"the set of numbers between 1 and 2, is a smaller set of infinite numbers between 1 and 3" is incorrect.
I would think it would be much less wear and tear on your credibility to instead just think about the correction given you and then recognize it.
He did it again! He still persists in mischaracterizing mathematics as claiming that there is an "Infinity" [capitalized, no less] that has different sizes. That is after the mistake of that has been pointed out at least a dozen times, and as he protests that he is posting in good faith.
Then we have people saying that there are contexts other than set theory so one should be tolerant not to demand that set theory is the only context we may consider. Quite so. But the context of the original poster is not just a proposal for another concept but a claim that the mathematical set theoretic concept "is nonsense". And that is intolerance. To know nothing about the mathematics behind the concept of cardinality but instead just call it nonsense. That is egregious intolerance.
I would just say if the universe is expanding, then it is expanding in Existence (as opposed to 'space-like-the-space-in-our-universe')
If scientists have in fact measured the size of our universe, then our universe is finite (meaning that our universe is not all there is to Existence. I hear there's been more talk of parallel universes lately).
We know by way of pure reason that the universe cannot be expanding in non-Existence. Such a thing is not conceivable at all, therefore, it is not observable at all. And this is not an unknown like a 10th sense which some superior being may have that we can't comprehend. This is a clear case of something contradictory (like a round square) that no being would make sense of because it is a known contradiction as opposed to an unknown.
Whether you intend it or not, that is a trollish question.
Of course, lines have length. That is implied by my talking about distance.
I already explained to you that there is a difference between the distance of an interval and the cardinality of the infinite set of numbers in the interval. But instead of recognizing that information, you hit me with a mindlessly posed quiz.
So I suppose I need to lay it out for you in even greater detail:
The real line is the set of numbers ordered by the usual less-than relation on the set of real numbers.
The intervals [1 2] and [1 3] are segments of that real line.
For any pair of numbers there is the distance between them, which is the absolute value of their difference. The distance between 1 and 2 is 1, and the distance between 1 and 3 is 2.
But the intervals are a set of numbers, not just their max and min. The interval [1 2] is the set of real numbers that are greater than or equal to 1 and less than or equal to 2. The interval [1 3] is the set of real numbers that are greater than or equal to 1 and less than or equal to 3. Both of those sets have the same cardinality, i.e. both those intervals have the same cardinality.
Your non sequitur is to infer that intervals having different distances implies that they have different sizes. (And I say that because you still have not said that you didn't mean what you said when you said the context here is mathematics.)
Since this keeps getting lost with you, for the third time:
Intervals may have different distances (lengths, if you prefer) but the same cardinality.
Quoting TonesInDeepFreeze
To my understanding, mainstream maths claims:
There are infinites of various sizes (or at least infinite sets of various sizes, but that amounts to the same thing)
The set of all sets is contradictory
Is my understanding wrong?
Quoting TonesInDeepFreeze
If someone came to me and said I've seen a triangular square, I would say to them that that's nonsense to me and that it is impossible for them to have seen such a thing. It's in the semantic of square that it can't be triangular. I would not call this intolerance, but perhaps I could be more tolerant by trying to understand the person better. Perhaps what they really mean is that they saw some shape, that the best way that they could label it was "triangular square". Maybe they saw some kind of trapezium and did not know the label/word for the semantic of trapezium.
Similarly, if someone came to me and said they have demonstrated how infinity comes in various sizes, I would say to them that that's nonsense to me. It's in the semantic of Infinity that It does not come in various sizes.
I don't think I'm picturing an object. I think I'm just focused on the semantic of Infinity.
Quoting TonesInDeepFreeze
I think it is from all that I have seen and heard that I said the following:
Quoting Philosopher19
Whether all that I have seen or heard is enough, is another matter. You don't think I have. I think I have.
If I've understood him right, Cantor treats a number sequence that goes on forever as being infinite. But something going on forever does not make it infinite (if my counting to infinity goes on forever, that neither makes my counting infinite, nor does it mean I will eventually reach infinity). It also makes no sense to say something like "assume that your counting to infinity is completed such that you have counted the set of all natural numbers and have successfully proven that there are an infinite number of natural numbers" and then label this as {1,2,3,4,...}
If Cantor did not do this, he would not then be forced to conclude as he did with his diagonal argument. Now I feel the following is relevant:
Quoting Philosopher19
(1)
There is no object called 'Infinity' in the sense you have been using it.
Here is a way to say what you want to say:
In mathematics, there are sets that are infinite but that have different cardinality from one another.
Better yet:
If x is infinite then there is a y that is infinite and y has greater cardinality than x.
(2)
The statement that there exists a set z such that every set is a member of z is inconsistent with the axioms of set theory; and set theory proves the negation of the statement that there is a set z such that very set is a member of z.
Better yet:
'There exists a z such that for all y, y is a member of z' contradicts this instance of the axiom schema of separation: For all z, there is a x such that for all y, y is a member of x iff (y is a member of z and yis not a member of y).
And we prove that easily:
First, we prove:
There is no x such that for all y, y is a member of x iff y is not a member of y. Proof:
Suppose, toward a contradiction, that there is a such and x. Then x is a member of x and x is not a member of x. Contradiction. So there is no such that x for all y, y is a member of x iff y is not a member of y.
Next, Suppose, toward a contradiction, that there is a z such that for all y, y is a member of z.
Then from separation we have an x whose members are all and only those y that are both a member of z and not a member of y.
But since every y is a member of z, we have x whose members are all only those y that are not a member of y. That contradicts that there is no x whose members are all only those y that are not a member of y. So there is no z such that for all y, y is a member of z.
And it is fine that the concept is nonsense to you. That is not at issue. But that something strikes you as nonsense doesn't thereby render it nonsense.
You don't know anything about the mathematics, so you are in no position to try to convince other people that it is nonsense. The only thing you could fairly say is, "I don't know anything about the mathematics, so I can't fairly opine on it, but my own concept of infinity does not allow that there are sets of different sizes of infinity".
But then you keep harping on this "semantic" issue you have. If I hadn't said in this thread, I'll say what I've said in other threads:
(1) Words have different meanings and senses in different areas of study. In biology the word 'cell' means one thing and in criminology it means another. Even when the senses are closely related and in closely related or even overlapping fields, the senses can be different. When mathematicians talk about 'infinite' they don't thereby declare that that mathematical sense applies to all other areas including philosophy, cosmology or theology. However, yes, the mathematical sense may be applied in other areas, but it's up to the author to make clear what sense they mean. At the very least, mathematicians ought to be allowed to talk about 'infinite' in the mathematical sense without someone who knows virtually nothing about the mathematics arrogating to convince people that it is "nonsense" and even to claim to "prove" that it is nonsense with unstated standards of "proof" quite different from mathematical proof.
(2) Even if you don't have the least bit of reasonability to grant (1), then we could say, "Fine, from now on consider every book and article and post in mathematics as if the word 'infinite' were replaced with 'zinfinite'. The formal mathematics would be just the same except the word you arrogate to your own meaning would not be used and you could rest easy that mathematics does not impinge on you own concepts.
(3) Again, formal contradiction is any statement P asserted along with the assertion of the negation of P. There is no known contradiction in set theory. A DIFFERENT matter is that in certain senses set theory is not compatible with your own philosophy, and again that a certain word is used in mathematics differently than you use it.
Yes, it's the colloquial part that is so often abused by people who know virtually nothing about the subject. Especially among beginners in the subject, if we refrain from that unfortunate usage, then (1) We avoid having set theory look ridiculous as if it claims that there is an object that has different cardinalities. (2) We adhere to the way the actual mathematics is couched, which is that is the property 'is infinite' but not an object that is 'infinity'.
So those who post uninformed, intellectually bigoted and confused lashing out against set theory would not have the slippery wedge of presenting mathematics as if it is itself absurd. People who are unfamiliar with the mathematics ordinarily don't think of infinitude the way mathematicians do. In everyday life and conversation, and even academically in certain contexts, it would strike as extremely odd to hear that there are "greater and greater infinities". But if it is said instead, "In mathematics, 'finite' and 'infinite' are properties of sets, and there are infinite sets, and there are greater and greater sizes, such as there are more real numbers than natural numbers" then it may strike one as much more sensible and mathematics is not made to look ridiculous as if it has an object that is infinity but that it comes in different sizes.
By the way, Cantor did not work axiomatically. The results of set theory are on much firmer ground now as we work axiomatically now.
I answered that exactly already.
You truly are not in good faith.
You make claims about a subject of which you are ignorant. Then when it is explained to you exactly what your confusion is, you ignore that explanation and instead just go on to make the confused claim again.
So I'll give you the explanation yet again so you can ignore it again:
We do NOT claim that from "after each natural number there is a next number" and "there is no greatest natural number" that we can infer that there is a set of all the natural numbers. Indeed such an inference IS a non sequitur. And every mathematician and logician knows it is a non sequitur. So, we recognize that to have a set with all the natural numbers we need an AXIOM for that, which is NOT an inference.
Also, it is good to study Cantor for historical context, to appreciate his intellectual power, and to gain insights into the concepts. But Cantor has been supplanted for 125 year or whatever by axiomatic set theory. If you are sincere in wanting to fairly critique the mathematics then you would get a book on set theory and read it.
Damn, I knew there was something special about you! :starstruck:
Quoting Philosopher19
A great many of us never go beyond using "unbounded". But we use the symbol for infinity. As for transfinite math, it rarely if ever comes up in classical analysis. But Foundations and Set Theory mathematicians follow the basic axioms and explore what lies beyond. You are in way over your head.
I like your comment on comic relief.
I think I ended yesterday laughing at all this.
It's not always as good. Maybe a winter pastime for some of us.
What?
(1) Whatever jokes you made, you also made the incorrect claim about the size of the set of numbers in the intervals. You still haven't recognized the the thorough explanations why your claim is incorrect.
(2) Yes, theorems:
If C is a cardinal then there is a cardinal greater than C.
If C is a cardinal then there is an infinite set of cardinals greater than C.
If C is an infinite set of infinite cardinals, then there is an infinite set of cardinals greater than C and such that it has infinitely many members that each is greater than every member of C.
There is no set k such that every cardinal is a member of k.
We already know that. It's not something that somehow vindicates your incorrect claim about intervals or anything else you might have said.
(3) Yes, we already know that Turing used a diagonal argument and that the diagonal technique was made prominent by Cantor. Moreover, we know that infinitistic set theory applies to the theory of computability. I don't how in the world you think think any of this some how "backs you up" in terms of any controversy there's been with you.
Of course we prove that there are cardinals of different size. We know the proof well.
That doesn't even the least bit suggest that there is a mathematical object called 'Infinity' that has different sizes.
It is all-seasonal and perennial I assure you. People spouting hyper-opinionated uninformed and confused misinformation about this subject goes on constantly and forever on the Internet.
Still.... especially good (or bad) recently.
As a connoisseur of cranks and sophists, I beg to differ. This thread is run of the mill in that regard. And there are routinely far more risible ignorance and confusion posted.
Depends on what is meant by 'transfinite math'. 'transfinite' is just another word for 'infinite', and, of course, analysis uses infinite sets. Moreover, there are mathematicians who work (and not in obscurity) with higher cardinals vis-a-vis analysis, though that work might not be prominent in the bread and butter mathematics you have in mind.
Here's how I see it for myself, transfinite math = Cardinals above the cardinality of the reals, or, treating infinities as objects. The only place this ever came up for me was a well-known theorem in functional analysis. Even there a slight adjustment in hypotheses removed its necessity.
Of course there are mathematicians who work with higher cardinals in analysis. They are at a higher level then bread & butter math. (there are still lots of questions in the latter, but the former is more attractive nowadays)
But infinite sets are regarded as objects. The set of real numbers is a set theoretic object. Boom, from page 1 we are dealing with an infinite object. The real plane is the Cartesian product of the set of reals with the set of reals. Takes an two objects (or at least one object twice) to make a Cartesian product.
What is the domain of the function f where, for all natural numbers n, we have f(n) = 2*n? f is a function, and every function has a domain, and the domain of f is the set of natural numbers, which is an infinite set.
Even when we say "let n go from 0 to inf", that really is just to say that the domain of the function is the set of natural numbers.
I don't see the point in saying that mathematics such as analysis doesn't use infinite sets, when plainly, at the very outset, to even start in the subject, we see that we are using infinite sets.
Same way we determine a set is infinite without counting it: stipulations.
We cannot determine that a set, S, is infinite by counting the elements (as we would never be able to stop, and this doesnt discern a set that is indefinite from one that is infinite). Instead, we could determine S is infinite either by stipulatione.g., if we are considering the set of all natural numbers, then we thereby know that this set is infinite because there is an infinite amount of them.
Likewise, we cannot determine that S1 is larger than S2 by counting the elements; instead, we come to know it by understanding the stipulations of the sets themselves. If S1 is a set with size 2 elements ad infinitum and S2 is a set with size 1 of elements ad infinitum, then S1 > S2 (and I dont need to count them).
I have explored a topic in classical complex analysis over the years. It is not a popular topic and many of those initially interested have passed away. I have written close to a hundred articles and notes, about a third of which I published before retiring in 2000. After that, publishing was too much a hassle; shorter notes on researchgate.net . None of them use the power set of the set of reals.
You may not know how many topics there are in math. Wikipedia has, I recall, about 26K pages. When I open a page at random I usually am clueless about what I find. ArXiv.org gets over a hundred math research papers a day, listed in various general categories. Even in classical complex analysis, I usually am left behind.
The output of mathematicians is staggering. However, it seems to me there used to be either a category for Set Theory or Foundations in ArXiv.org . It's no longer there. There is one for Logic, and this title caught my eye: Mice with Woodin cardinals from a Reinhardt
I have mentioned before that I am old and outdated. Not a reliable authority for TPF.
Analysis normally does not dwell on set theory. It's there in the background of foundations. And the limit concept arises from it, but when I use limit, as defined using epsilon/delta, I don't go into set theory details. If I say x->1 it is assumed it does so through the reals.
That is circular.
And we don't just stipulate that the set of natural numbers is infinite. We prove it.
Quoting Bob Ross
It's nothing like that.
Right, those working in the various branches don't usually concern themselves with the foundations. So use of infinite sets is ubiquitous without concern for the foundational axiomatization concerning them. Proverbially, infinite sets are the water mathematics swims in. The fish doesn't have to know anything about water, but it still needs that water to swim in.
As I argue, the first day of class when we are told "We have the natural numbers and we have the real numbers and the real number line", boom, we are presented with infinite sets, even if the instructor doesn't happen to mention, "And don't forget, those are infinite sets".
As to age, the mathematics is pretty durable, thus also is the wisdom of those who learn it.
How's it circular? Demonstrate where I am begging the question.
'S is infinite' is equivalent with 'S has infinitely many members'.
Or as you say:
Quoting Bob Ross
'the set of natural numbers is infinite' is equivalent with 'there is an infinite amount of natural numbers'.
Proving that the set of natural numbers is infinite is the same as proving that there are infinitely many natural numbers.
In a textbook in set theory, you would see how a theorem of the form:
S is infinite
is actually proven.
I would say that is one interpretation of "dark energy", but here would be a number of possible interpretations.
Quoting DanCoimbra
What I meant is that "more internal spatial structure" is not consistent with Einsteinian relativity, because that would render a whole lot of predictions about the motions of things as inaccurate. We can posit "dark energy" as the reason why the predictions are inaccurate, but then where is this dark energy, and what is it doing other than making the predictions inaccurate,
.
Quoting Philosopher19
The problem inherent within pragmaticism is that whatever is the purpose at the time (the flavour of the day), the axioms chosen will support that purpose. As time goes by, and needs change, other axioms will be produced to satisfy the evolving needs. At this point, the new and the old are not necessarily consistent, so there may be a degree of contradiction between different logical structures, depending on the purpose which they each serve,
Quoting Philosopher19
I agree that it is appropriate to set a standard of "truth" for mathematical axioms.
By definition, a successor cardinal is strictly greater than its predecessor.
(By the way, aside from successor cardinals, there are cardinals other than 0 and aleph_0 that are not successor cardinals, and they are greater than any previous cardinal.)
Anyway, without even getting into successor cardinals and limit cardinals (cardinals that are not 0 and not successor cardinals), it is easy to prove that for any set, there is a set of greater cardinality.
Mathematics doesn't mention "all existents" or "set of all things".
The heart of your attack on infinitistic mathematics is your own mistaken fabrication of what you think the mathematics is. In other words, you're putting up a huge strawman.
As that was added in edit, I missed it.
Whatever you think of me, or whatever error you think there was in communication, I accurately responded to your posts as they were written.
You claimed that the size of the set of numbers between 1 and 2 is less than the size of the set of numbers between 1 and 3. If that's not what you meant, then it's not my fault. Then you deflected to the fact that the distance between 1 and 2 is less than the distance between 1 and 3, which is true, but it does not bear on the fact that the size of the sets is the same. At the time of posting I saw no post in which you "held yourself accountable" for that error.
And I explained in perfect detail about length, but instead of recognizing that, you incorrectly suggest that I don't understand length and you resort to juvenility such as "too dumb".
Whether described as 'picturing an object' or 'positing that there is such an object' my point is that set theory does not mention, describe or posit any such object, so saying 'Infinity' as a noun as you do is misleading as it does suggest that one should take set theory as suggesting that such an object can be countenanced, considered or pictured, etc.
Quoting Philosopher19
What are your sources? What specific texts in set theory or mathematics do you think have said the things you claim set theory to say?
Quoting Philosopher19
It's the heart of the matter of why your are ignorantly misrepresenting set theory.
You think you've read enough set theory to understand its axiomatic treatment of infinite sets? What specifically have you read, let alone studied sufficiently to competently discuss it?
But all of the above is exactly what I'm saying is contradictory. [b]And my use of infinity which (if I've understood you correctly) you say is not the one that they use in maths, is the reason that I say all of the above is contradictory.
[/b]Quoting TonesInDeepFreeze
So what semantic are mathematicians using when they use the world/label "infinite"?
Quoting TonesInDeepFreeze Quoting TonesInDeepFreeze
Something cannot be both a member of itself and a member of other than itself at the same time. For example, take z to be any set that is not the set of all sets, and take v to be any set. The z of all zs is a member of itself as a z (as in in the z of all zs it is a member of itself). But it is not a member of itself in the v of all vs, precisely because in the v of all vs it is a member of the v of all vs as opposed to a member of itself. If we view the z of all zs as a z, it is a member of itself. If we view the z of all zs as a v, it is a member of the set of all sets. You can't view it as both a member of the z of all zs and a member of the v of all vs at the same time. That will lead to contradictions. In other words, we can't treat two different references as one (as in are we focused on the context of vs or the context of zs?)
Note that the above shows the impossibility of a set that contains all sets that are members of themselves where all equals more than one.
For the fully fleshed out version of this, see my post on Russell's paradox which I posted the link to in this discussion and in the other one.
Quoting TonesInDeepFreeze
If some theory suggests that you can view the z of all zs as both a member of the z of all zs and a member of the v of all vs at the same time, then that theory is contradictory. The z of all zs is either to be treated like a z or a v. If it is to be treated like a z, it is a member of itself. If it is to be treated like a v, it is not a member of itself (precisely because it is a member of the v of all vs)
Quoting TonesInDeepFreeze
I'm not sure what you mean by "So, we recognize that to have a set with all the natural numbers we need an AXIOM for that, which is NOT an inference."
So it is again regarding 'contradiction'.
Go back and read what I wrote about "contradiction".
Quoting Philosopher19
In certain alternative set theories, there are sets that both members of themselves and of other sets.
In ordinary set theory, no set is a member of itself.
By the way, we don't need to use temporal phrases such as "at the same time". Set theory does not mention temporality.
Then the rest of your z's and v's is irrelevant if it is supposed to refute the proofs I gave. Moreover, if you knew anything about this subject or even mathematical discourse you'd see that your prose about it is ungrounded, impenetrable double-talk.
To refute a purported proof, you need to show a step in the proof that is not permitted by the inference rules (which in this case are those of ordinary predicate logic).
And you separately quoted me saying "The axiom schema of separation". What was the point of that? Did you look up what the axiom schema of separation is and you think your remarks relate to it in some way?
Quoting Philosopher19
It is clear. You don't know what it means, because you are virtually completely ignorant of the subject matter.
I'll spell it out even more:
In set theory, to prove there exists a set having a certain property, we must do so from the axioms and rules of inference alone.
In this instance, the property in question is "has as members all the natural numbers"
Without the axiom of infinity, we cannot prove that there is a set with the property "has as members all the natural numbers". But with the axiom of infinity we can prove that there is a set with the property "has as members all the natural numbers".
You argued that from "after each natural number there is a next natural number and there is no greatest natural number" we cannot infer "there is a set of all the natural numbers". And you are correct about that!
So I pointed out that indeed set theory does not make that unjustified inference, but rather, set theory has an axiom from which we CAN infer that there is a set of all the natural numbers. And THAT inference, from the axiom, does NOT use the unjustified inference from "after each natural number there is a next natural number and there is no greatest natural number" to "there is a set of all the natural numbers".
/
You know virtually nothing about set theory. You should present whatever concept of infinity you like, but you shouldn't be presenting it as a refutation of a subject you are ignorant about.
We don't say "using semantic".
Rather, we just state the definitions.
I stated the definitions in my first post in this thread:
https://thephilosophyforum.com/discussion/comment/878326
Mathematicians, like myself, may get a little sloppy about using the word, infinity, at times. For example, for those of us in complex variable theory The point at infinity has a specific reality as the north pole of the Riemann sphere. There is a technical way of saying this.
Please forgive the cliche, but it is especially apt: Above is Dunning-Kruger on steroids.
Great post, thanks. How do you prove then N is different size to P?
Did you give Philosopher19 the finger or is there real math behind the north pole of the riemann sphere? It would be cool if you meant it both ways.
Quoting TonesInDeepFreeze
Evidently, there's no point in continuing this discussion. If you believe your mathematics is free from contradictions or paradoxes, then in my opinion, you are not blameworthy for upholding them or sticking to them (unless of course someone presents a better or more complete thing to you and you reject greater for lesser), but if you see paradoxes and contradictions or incompleteness and you treat them as other than paradoxes/contradictions/incompletions...
I see no paradoxes or contradictions or foundational incompleteness in the beliefs that I uphold (mathematical or otherwise).
Peace
The above point I felt was worth adding to this discussion, but I will probably stop posting here as I don't think there's anything left to add to this discussion.
This isn't really the place to come to get people to agree with you. I think the math boys really did give you a good amount of feedback that would be hard to get anywhere else. So if you want to run something past us we'll tell you what we think and you can react accordingly. Most of what you say really irks a formally trained mathematician.
To me it seems like arguing about mental fantasies but for someone who has studied it there would be something to defend.
It's been one of the more lively threads here...seems to go on all day.
As far as the math profession I do think you should show some respect because the world runs on the math they do and for some things only a few people per million or billion may be able to do it.
Here's actually some advice to all non-mathematicians (from a non-mathematician):
If you really can ask an interesting foundational question that isn't illogical or doesn't lacks basic understanding, you actually won't get an answer... because it really is an interesting foundational question!
Yet if the answer is, please start from reading "Elementary Set Theory" or something similar then yes, you do have faulty reasoning.
We don't. He proved that they are the same size.
As I said, the discussion will go in circles given that you skip answers given you and instead just repeat your refuted claims.
Quoting Philosopher19
It's not my mathematics. I don't have allegiance to it. I find value in it, find wisdom in it, recognize that it axiomatizes reasoning used in the sciences, and enjoy it. But I don't claim that there might not be better approaches - philosophically, intuitively, and practically.
I don't claim to perfect certainty that set theory is consistent. But it seems to me to be an extremely good bet that it is. (1) No contradiction has been found in it under incredibly intense and indefatigable scrutiny for about 125 years. (2) We can see specifically how it was devised to avoid Russell's paradox. (3) The concept of sets as a hierarchy itself suggests an intuitive approach that is consistent.
Again, you use the word 'incompleteness', thus ignoring the information that was given you about incompleteness.
Quoting Philosopher19
You haven't proposed an alternative framework, let alone in axiomatic form. Articulate the principles by which you propose to derive mathematics adequate for the sciences, or, better yet, put it in axioms; then we can put it to the test.
Set theory gets the job done of axiomatizing the mathematics for the sciences. By analogy: Set theory is an airplane that flies. If one thinks it's not a good airplane, then one is welcome to show us a better one.
Which are?
He did it again! He completely skipped recognizing the refutation given him.
From the point of the set N, it looks like it is. But from the point of the set P, it looks like it is only a half set to N. What's going on?
If you look at the posts, I don't think I'm the one that has been showing the disrespect (if I have, it has been in response to disrespect). I wanted a discussion because I felt I had something to offer in response to something that I saw as contradictory. I don't think I entered the discussion closed-minded or dogmatic. And I think I tried to understand the other's point of view.
If someone is an "expert" in the field of something, but that something is evidently paradoxical or foundationally incomplete, it's absurd to treat them like an expert of anything useful. Some people are unreasonable/absurd. They want to hold on to their paradoxical or contradictory theory or belief at the cost of sincerity to Truth/Goodness/Existence/God
If people here witness that their beliefs or theories or axioms lead to no paradoxes or contradictions or foundational incompleteness, then I can't say to them they're misguided or lacking in knowledge.
You have those who recognise/witness that their theories are incomplete and act as such (there is honesty to them), and then you have those who recognise/witness this, but act as though they are the knowledgeable ones whilst all others are ignorant (which to me is the very definition of a "bad guy"). I believe spending hours or years or decades on something that is foundationally corrupt, does not make you an expert in anything other than something that is useless. What good is an expert in multishapism geometry that deals with the study of shapes such as round triangles and circular pentagons?
I don't feel like I have any contradictory or paradoxical theories or beliefs that I need to reconcile. I was trying to address what I saw as contradictory. If it's not contradictory, then it's not contradictory. But if it is contradictory/paradoxical and some are hardcore with regards to holding on to this, what can I say?
The cardinality of N = the cardinality of P iff there is a bijection between N and P.
There is a bijection between N and P.
Therefore, the cardinality of N = the cardinality of P.
Meanwhile, there is no apparent meaning in "from the point of view".
Yes, P is a proper subset of N. Indeed the point is that it is a property of infinite sets that there are bijections between them and certain proper subsets of themselves.
The fallacy is in saying "half" in this context. For infinite sets, there is no division operation such that there is 1/2 the cardinality of an infinite set.
Ok. Let me put it this way. I gave you a refutation with the z example. You started with insults, then you eventually said something like this:
Quoting TonesInDeepFreeze
I decided discussing something with someone who seems to be emotional or biased is a waste of my time so I said I will stop, but I felt the need to add the following to the discussion:
Quoting Philosopher19
This dealt with your temporal phrases response.
You have not yet answered:
Is it logically possible for a set to be both a member of itself and a member of other than itself? If it is a member of other than itself, then it is not a member of itself, is it? And if it is a member of itself, it is not a member of other than itself is it?
And don't say to me something like "some set theories allow for this or that". I'm asking a basic logical question that has a basic and straight forward answer. There is no need to dance around anything. Just deal with the main issue at hand.
All of them are here:
godisallthatmatters.com
Regarding Russell's paradox, sets and infinity:
http://godisallthatmatters.com/2021/05/22/the-solution-to-russells-paradox-and-the-absurdity-of-more-than-one-infinity/
You are too pessimistic. You can have your view and they can have theirs.
You can always declare victory, plant your flag and call it a day. Really, say what you like. I agree there are contradictions and what I brought up about parameters that can be anything your brain can dream up.
Maybe there are real world applications to some of this as has been discussed by those who have actually done it. I assume they use what has proven to work. Math in practice has a precision component, not just theorizing.
I don't care to say you are "disrespectful", but you are irrational and in bad faith when you skip refutations and explanations given you and instead just keep repeating your false and confused claims.
Quoting Philosopher19
You are closed minded to the fact that you are close-minded and dogmatic. And you still won't face that your hyper-opinionating on a subject you know nothing about. If really were the fair minded person you claim to be, then you would get a book and find about the subject rather than posting misinformation and confusions about it.
Quoting Philosopher19
Just for the record, I don't claim to be an expert in anything other than jazz, and even in that field I'm deficient in important ways.
Quoting Philosopher19
There it is again! You say 'contradictory', again ignoring all the explanation given you about that.
Quoting Philosopher19
There is it is again! You say 'incomplete', again ignoring all the explanation given you about that.
Quoting Philosopher19
I don't know anyone who has said that all others are ignorant. You are ignorant on the subject. That doesn't entail that others are ignorant on it. Indeed, there are people who critique classical set theory who are extremely knowledgeable about it. Critiques of set theory are quite fair game and bring profound insights into the subject. But those are knowledgeable, responsible and thoughtful critiques. And better yet, they are critiques that are followed up with actual mathematical alternatives to classical set theory.
Quoting Philosopher19
Nope. Set theory doesn't do that.
I responded to you, you responded me with a refutation, I responded to your refutation with the following:
Quoting Philosopher19
Where is my response? Is it me who ignores you or you who ignores me?
Ok, so you are saying that your beliefs are not incomplete or contradictory in any way. That is not philosophy, that is religion, aka delusion.
Did you read anything from the link I gave you?
I believe my beliefs are not foundationally incomplete or contradictory in any way from a rational/semantical point of view.
I didn't say all others are ignorant. I just said there are people who are like this. I did not specify who.
On some crucial points, you didn't even recognize them, let alone refute them. And when you did attempt to refute points, you failed, as your supposed refutations were false and confused.
Quoting Philosopher19
That's a lie. I started with plain, cold information. And I did that for several posts. Eventually, it became clear that you are immune to rational discussion, and so I factually pointed out that you are confused, ignorant of the subject and in bad faith.
Quoting Philosopher19
As to bias, I have read a pretty good amount of the literature of this field with informed and responsible debates regarding classical mathematics. I am fascinated by and greatly enjoy informed and responsible critiques of classical mathematics. As to emotion, exasperation with cranks is natural.
Quoting Philosopher19
I did answer it. Specifically and exactly.
Quoting Philosopher19
The relative consistency of those theories indicates that it is not contradictory that a set is a member of itself and also a member of other sets.
Quoting Philosopher19
There's nothing terpsichorean about my reply. I gave you an exact refutation. The fact that you are ignorant of the context of set theory and alternative set theories is not my fault.
My response is right where it was when I gave it.
And I also responded to your previous tu quoque, and you ignore that too.
Quoting TonesInDeepFreeze
It is blatantly contradictory for x to be both x and not x. It is blatantly contradictory for a set to be both a member of itself and not a member of itself. Yet you want to persist by saying things like the above. Again, I asked:
Quoting Philosopher19
and I added:
Quoting Philosopher19
It seems that what I added was ignored and what I asked was not answered. Until I see a good enough response, I'm done putting any more time into this. Once again:
It is blatantly contradictory for x to be both x and not x. It is blatantly contradictory for a set to be both a member of itself and not a member of itself.
Who would reject this but the contradictory/unreasonable/irrational/absurd/insincere?
You are very confused. Yes, you didn't say all others are ignorant. And I didn't say that you said that all others are ignorant. Rather, as now you mentions again, you said that some people have regarded all others as ignorant.
You didn't specify anyone in particular. Good. Because there is no one who has even hinted at a suggestion that all others are ignorant. You take the sneaky road of impugning but leaving it open-ended who you are impugning though it is obvious who you mean. And my point stands: You are ignorant on the subject. That doesn't entail that others are ignorant on it.
I skimmed through two pages. It seems to be a collection of semantic games. I am more concerned with what issues you solve with your beliefs. That your beliefs are not contradictory (big claim) is not a selling point for others to adopt it.
It's even contradictory just to say that x is not x.
And set theory does not say there is an x that is not x, nor that there is an x that is x and not x.
You maQuoting Philosopher19
Correct! Indeed that is a crucial point that is used in an important proof I gave you.
Quoting Philosopher19
That's a lie. Stop lying. I never said anything like that.Quoting Philosopher19
That is not "once again". Previously you said that "a set cannot be both a member of itself and a member of other than itself". That is different from "a set cannot be both a member of itself and not a member of itself".
I wondered a while ago whether you did not actually mean "a set cannot be both a member of itself and a member of other than itself" but actually meant " "a set cannot be both a member of itself and not a member of itself". But in my reply I addressed the former in such a way that if you hadn't meant it, then you could revise to what you did mean.
Quoting Philosopher19
Indeed. (Well, except for dialetheists and paraconsistent-ists.)
Relevant: Lawvere's fixed point theorem.
Point at Infinity
:cool:
You are egregiously and flagrantly putting words in my mouth.
I never said cardinalities don't have size.
But I'll say now that cardinalities are sizes.
Two sets are equinumerous iff there is a bijection between them.
The cardinality of a set is the cardinal number with which the set is equinumerous.
'the size of the set' and 'the cardinality of the set' are synonymous.
And we say that two sets have the same cardinality iff they are equinumerous.
And you have it backwards:
The original poster claims that it is contradictory to say that there are different infinite sizes. I have been saying that it is not contradictory to say that there are different infinite sizes. And I have been saying that in set theory it is easy to prove that there are different infinite sizes and indeed that some infinite sets are larger than other infinite sets.
It is amazing that you reversed it completely to characterize me as saying the opposite of what I have been saying.
/
There was no "ass handing" though you like the tough talk sound of that.
/
Quoting Vaskane
I am continually overwhelmed by how much I don't know and could learn. But with you what I have learned is not about mathematics or philosophy.
Quoting Vaskane
I'm happy to be corrected any time I am incorrect.
Good for you. I flamed out at "epimorphism". (i.e., the beginning). And I have actually worked with fixed points in Banach spaces and specifically the complex plane.
Nothing about what you said demonstrated my argument was circular. How was I begging the question?
You wrote:
"if we are considering the set of all natural numbers, then we thereby know that this set is infinite because there is an infinite amount of them."
But:
"there is an infinite [number of] natural numbers" is just another way of saying "[the] set [of natural numbers] is infinite".
So your argument is just that we prove the set of natural numbers is infinite because it is infinite (has infinitely many members).
Proving that a set is infinite is the same as proving that it has infinitely many members.
So it is question begging to assume the set of natural numbers has infinitely many members when that assumption is just another way of saying what you want to prove.
But you can look up actual proofs that the set of natural numbers is infinite.
I disagree. The "math boys" here at the forum tend to respond with 'go read some math texts' to anyone who disagrees with them on fundamental principles. In that case, the issue is not a matter of better learning the mathematical representation of the fundamental principles, and how to apply them mathematically, as a math text will demonstrate, it is a matter of disagreement with those mathematical representations. Therefore the reply of "please start from reading 'Elementary Set Theory' or something similar", is usually just a copout, a refusal to engage with the philosophical matter at hand as if further reading of the mathematics will change a person's mind, who already disagrees with it. That's like telling an atheist to go read some theology, as if this is the way to turn the person around.
Again, the intellectual dishonesty of the crank in action. In this case, blatant strawman by misrepresentation of what his interlocutors have said. And even more egregiously by dint of the fact that this strawman has been pointed out to him many many times.
It's not a matter of disagreement on principles, but rather ignorant and confused misrepresentation of the mathematics that is supposedly being discredited. It is fine and even essential that there be different points of view about foundations including critiques of classical mathematics. But it is pernicious against knowledge and understanding when the attacks on mathematics claim things about the mathematics that are crucially false, when the attacks are premised in an ignorant prejudice that the mathematics works in certain ways that it definitely does not. After the crank's error about this have been explained to him over and over and over and he still persists to spread the disinformation, then the best thing is to recommend that he get a basic textbook to inform himself in the subject that he has spent so much time already cultivating his self-imposed terrible misunderstandings.
Quoting Metaphysician Undercover
It's nothing like that. It's the reverse. It's like telling the zealot denouncing scientific theories to get a textbook in biology.
I don't know anything about microbiology, so I don't spout a bunch of nonsense about. If I did, I should expect someone to kindly tell me to shut up about it and get an introductory text.
You have mentioned, for example, that the limit concept is flawed, although it works well most of the time. But I don't recall your argument beyond that point. A more complete knowledge of space and time and points and continuity? Oh yes, something about the Fourier transform and the Uncertainty principle. What are your suggestions to fix that up? Intuitive mathematics? Remind me where doing something specific makes it better.
Are you working on a change in the fundamentals of math that might calm your concerns? I hope so, no one said math as it stands is perfect.
Quoting Lionino
I believe the solution to Russell's paradox is in here:
http://godisallthatmatters.com/2021/05/22/the-solution-to-russells-paradox-and-the-absurdity-of-more-than-one-infinity/
There are also other things on the website. I think they are appropriately titled with regards to what they try to do or highlight or solve or discuss.
Quoting Lionino
A triangle is triangular (or the angles in a triangle add up to 180 degrees) is not a semantic game. It is use of semantics in a non-contradictory manner. Counting to infinity, or there being no set of all sets is use of semantics in a contradictory manner.
I think the obvious point to start with is divisibility. Generally, mathematics provides that a quantity, any quantity, can be divided in any way. We can call that "infinite divisibility". In reality, there is very clearly many division proposals which simply cannot be done. Because of this fact, that there are real restrictions on divisibility, there is a very big difference between dividing a group of things, and dividing a single object. Each of these two types of division projects has a different type of restrictions or limitations on it.
For example, to divide a group of seven human beings into two equal groups is a project that cannot be done, even though common math would say seven divided by two is three and a half. So we'd have to chop a person in half. But then we'd have eight objects instead of seven, because we'd have have two halves, which are two objects, but unequal to the other six objects. So we have to conclude that the way we divide a group, or quantity of things is seriously restricted.
Further, the way that we quantify something dictates the way that the quantity can be quantized. So if we use weight for example, to measure the volume of a group of grains of sand, we do not count the grains and divide the number of grains evenly, we look at the sand as one thing, with one weight, and divide that weight however we will. But there will still be a issue with precise division, when we get to the point of needing to divide individual grains of sand.
This leads into the problem of dividing single objects. An object is a unit, and this is fundamentally a unity of parts. If there is an object which is not composed of parts, like the ancient atomists proposed for the "atom", this object would be indivisible, and provide the basis for the rules of all division projects. However, such an object has not been found, so the guidelines for dividing a unit must follow the natural restrictions provided by the divisibility of the type of object. Different types require different rules, so mathematics provides for all possibilities (infinite divisibility). What physicists have found, is that the true restrictions to divisibility of all things, are based in mass and wave action, rather than composite "parts".
This means that in order to provide the proper rules or guidelines for the division of units, unities, we need to understand the real nature of space and time. Mass is a feature of temporal extension at a point in space, and waves are a feature of spatial extension at a point in time. Where the common principles of mathematics mislead us is the assumption of "continuity", and this is closely related to the simplistic notion of "infinite divisibility".
Now we have two closely related, but faulty principles of mathematics, infinite divisibility and continuity. They are applied by physicists, and people believe they provide a true representation of reality, when physicists know that the evidence indicates the presence of discrete quanta rather than an infinitely divisible continuity. Therefore our representations of spatial and temporal features need to be completely reworked. To begin with, as I've argued in other threads, representing space with distinct continuous dimensions (Euclidian geometry) is fundamentally flawed. The separations within space indicated by quantum physics, must indicate distinct incommensurable parts. These distinct parts are the parts which may be represented dimensionally, and the parts which cannot be represented that way. However, they must be incorporated together in a way which adequately represents what's real. At the current time, we have a dimensional, continuous line (numberline), with non-dimensional points (real numbers) which may divide the line infinitely, but this is just an unprincipled imaginary concept which in no way represents the real divisibility of space, and it becomes completely inapplicable when physicists approach the real divisibility of space.
Quoting TonesInDeepFreeze
No, I believe it is a crucial point that is used in an important proof I gave you, which to put it in as short a manner as possible is: An item in a subset cannot be both a member of the subset and the set. If it is a member of the subset, it is a member of the subset. If it is a member of the set, it is a member of the set.
Quoting TonesInDeepFreeze
If it's a member of other than itself, this means that it's not a member of itself. So my question to you is, what is the difference?
Quoting Philosopher19
The following must be considered properly:
Quoting Philosopher19
I have never argued against the point that there are different sizes (cardinalities). And I have never said that other people may not also point out that there are different sizes (cardinalities).
Do you mean: If S is a subset of some set T and x is member of S, then x cannot be a member of T ?
That's incorrect. By the definition of 'subset', if S is a subset of T, and x is a member of S, then x is a member T.
Quoting Philosopher19
Every set is a member of certain other sets. For example, every x is a member of {x}.
With the axiom of regularity, no set is a member of itself. So with the conjunction:
x is a member of x and x is a member of other sets
the first conjunct is false, therefore the conjunction is false even though the second conjunct is true.
Without the axiom of regularity, it is not precluded that there are sets that are members of themselves. Therefore, without the axiom regularity, we cannot infer that there is no set that is both a member of itself and a member of other sets too.
/
Or maybe your phrasing is not what you mean. When I take your phrasing literally, as best I can, I take you to be saying that a set cannot be a member of another set and also a member of itself.
But if all you mean is that a set cannot be both a member of itself and not a member of itself, then, of course, we have no disagreement.
Quoting TonesInDeepFreeze
Yes. The list of lists that list themselves is a member of itself in that list alone. Even though it is also a member of the list of all lists, it is not a member of itself in the list of all lists precisely because it is a member of the list of all lists and not the list of all lists that list themselves.
Note that the above shows the need to distinguish between "member of self" and "not member of self". To say no such distinction exists or is possible is to have an incomplete/contradictory theory in my opinion (contradictory because it argues the semantics of "member of self" and "not member of self" do not exist in Existence).
Quoting TonesInDeepFreeze
Quoting TonesInDeepFreeze
I get where you're coming from and I believe I completely get what you're saying. I don't deny the set of all natural numbers encompasses the set of all even numbers. Confusion occurs when one views sets that are not members of themselves (precisely because they are members of the set of all sets and not themselves) as being members of themselves.
To meaningfully talk about "member of self" and "not member of self", a set/context/reference is needed and adherence to it is necessary for the sake of consistency. The best way for me to convey to you what I'm saying in response to what you're saying, is the following:
Quoting Philosopher19
I'm thinking of starting a DIPSHIT PRIZE that we can nominate for and pass around here.
TDF really isn't that bad. He tries hard. You can be in charge.
Edit: To management...Not serious.
Thanks for your thoughtful and intelligent reply.
That's an interesting post. I've seen a few arguments that the success of eternalism in physics, to the extent that many popular physics texts openly suggest "eternalism is what physics says is true," largely flows from similar assumptions in mathematics. That is, it's a similar case of "this is how we think of mathematics, so this is how the world must be."
I am not sure if this is so much a problem with mathematics though as it is with how it gets applied to the sciences and philosophy. It seems to me that infinite divisibility might be worth investigating even if it doesn't accurately reflect "how things are."
That is one thing you say that makes sense and is correct.
I think there is a very close relationship between "mathematics" as the principles, rules etc., and the application of those principles. As Plato said, the people who use the tools ought to have a say in the design of the tool. And in reality they do, because the ones using the tools choose and buy the ones they like, therefore design and production is tailored for the market of application.
So in the case of "infinite" for example, the principles of calculus allow for the representation of an operation which is carried out without a limit. The limit is infinite, which essentially means there is no limit, and the operation proceeds endlessly. This representation proved to be very useful in application.
The issue we can look at, as philosophers, is what exactly is the effect of such an untruthful representation. First, we need to accept the fact that it is untruthful. To allow into any logical "conclusion", that an operation has been carried out without end is a false premise. In reality, the need to carry out the operation endlessly would deny the possibility of a conclusion.
The next step I believe, is to apprehend the level of ignorance which this untruthful representation propagates. There are some very specific problems produced from our conceptions of the continuity of space and time, which were demonstrated by Zeno. The mathematical representation (or more properly misrepresentation) as a premise in calculus, creates the illusion that these problems have been resolved, and so there is denial and ignorance concerning the reality of the problem amongst many people.
Finally, we can see how allowing this untruthful representation actually magnifies the problem rather than resolving it. When the usefulness of the misrepresentation is apprehended and recognized, it, and similar forms are allowed to pervade throughout the logical system (we can call this the propagation of self-deception). This creates the issue pointed to by the op, the need for different types of infinities, infinities of infinities, and the transfinite in general. The issue with the transfinite being, that some applications require the truth, a finite number, while others require the impossible, or false representation of a conclusion drawn from an endless operation, so some applications require a relationship between the two, hence the "transfinite". We can look at it as a bridge between the untruthful, and the truthful, a bridge which enables the self-deception.
I mean that all makes sense, although my understanding was that the question of whether or not space-time is infinitely divisible was an open one. I know there are a lot of physicists who claim that the universe must be computable, in which case it cannot truly require the reals to represent it, and space cannot be truly infinitely divisible. But from what I understand, experiments to support this idea have been wholly inconclusive. There are also folks like Gisin who argue that intuitionist mathematics is the better structure for representing physics, but they are a small minority (albeit seemingly a growing one). But there certainly do seem to be contrary voices who argue that mathematics based on true continuities, infinite points between any two points, works best for predicting empirical results precisely because it does represent the world. That is, perhaps not all elements of the world are infinitely divisible, but some, like space-time, would be.
I know Paul Davies claims to have an experiment that might settle this issue but it's currently impossible to actually perform, involving an insane number of beam splitters and accuracy. Other experiments looking at how light travels from distant stars have made predictions about how it must travel if space conforms to a finitist model, but the data ended up supporting a continuity, although from what I understand, these are no way definitive experiments. And then we can consider that certain discrete limits, like the Landauer Limit, appear in experimentation to turn out not to exist.
So, part of the problem might be that there seems to be informed disagreement on what represents truth here. Although, I do agree that it is problematic when a position becomes the "default" through inertia, despite not having strong evidence for it being the case over the contrary position. I would say reductionism is a strong example of this, where actual empirical support would seem to leave the question undecided, but it remains the default anyhow.
I find the intuitionist view fascinating because it would seem to allow for potential infinites of division, but not actual ones, a sort of near reintroduction of Aristotle.
Honestly, I am having trouble dissecting the arguments used here.
Thoughts, @jgill ?
It appears to me, like the quantum nature of energy demonstrates quite conclusively that the reality of space and time cannot be infinitely divisible. You see, the wave-function represents a continuity, but what it represents is not an observable aspect of reality. Observations indicate discrete occurrences of so-called particles (quanta), with not necessary continuity between the occurrences. If there is a true continuity, it is not represented by the wave-function, which represents possibilities. And, it is not the continuity of space-time, which fails at the quantum level. So it hasn't yet been determined.
To me the best we can do is categorize models of infinity as conceptual mathematical objects.
As such the parameters are arbitrary and their usefulness is in a defined mathematical environment.
Under this categorization scheme, it can be possible that one model can be inconsistent with another and not be false.
Here is my example,
A smaller infinity can reach any finite number that a larger infinity can by freezing the larger infinity and letting the smaller one catch up.
I'm sure there are all kinds of problems with this in the standardized mathematics but in the sense of a conseptual mathematical object it is legitimate. I think I first said it as a bit of a joke but the idea is we can drive the math by abstractions.
We'd have to look at the arguments of people who have said that there is. Who do you have in mind? Naturally, we would look at realists such as Godel. And there are also cranks who at least present as if their own vague, undeveloped, impressionistic and incoherently suggested concept is the true concept, as meanwhile they do explicitly represent that classical mathematics is false.
I'd rather say 'theory' than 'model'.
But then we must ask what we mean by a theory being true or false. In a rigorous sense, a theory is true or false in a model for the language for the theory. A theory may be true in some models and false in others. And of course, if a theory T is inconsistent with a theory S, then there may be models in which T is true but other models in which S is true.
Define 'reach', 'freezing' and 'letting catch up'. Better yet, tell me your primitives and your sequence of definitions from the primitives.
What I've written is about as far as I've gotten on a 'theory'.
I was thinking there might be an application for this in central banking or distributing resources to competing unlimited wants. Maybe the math is out there in some form already. Wouldn't doubt it.
What about two infinity generating machines that spit out consecutive integers at variable speeds endlessly.
Set a dial and one or the other can go faster or slower or stop. If you have a system like that matched to physical systems that have finite limits it might be an interesting model
I'm in over my head but don't infinities have some rubber band like properties that can be set at will.
My interest is mostly going from brain state to doing the math as a basis for a philosophy of mathematics..... Real simple,. Brain; (math processes)
Don't expect everyone to do it perfectly and in learning math or new skills it's always a process of brain programing.
Fight the cranks all you like. Makes things interesting. It's just philosophy here not pure math.
I have addressed that so many times in so many threads. Maybe earlier in this thread too.
When in over one's head, it is recommended to keep one's mouth shut, and head for the shallows. People have drowned in these waters.
I've had college algebra, trig and calculus.
I can also design trusses and figure pressure loss in pipelines. Doesn't that sound exciting.
No, I hate trusses. But hey, more power to civil engineers, though I would rather let the computer handle all those forces in different joints. Don't ask me about hyperstatic structures I don't know.
Well then, when in over your head, retreat to dry land and build a bridge.
I don't really design trusses but in addition to course work I made my own collection of scale model trusses of various designs. I still have them in a folder somewhere. Glue and cardboard.
Russell's Paradox and infinity arguments hold no interest for me. After going round and round with the author on First Causes, I suspect I would learn little from this paper.
I'm interested to know exactly how pressure is lost in pipelines, if there is no leaks. I've heard that in the USA a huge amount of natural gas just goes missing. Where does it go?
Into your posts?
(Ok, but someone had to say it...)
Friction loss but it's way off topic.
Infinity is more of a process of continuing than a quantity?
I don't think the process of continuing forever amounts to anything infinite. I see infinity as the reason for the process of continuing forever as being possible/meaningful.
To me, Infinity and Existence denote the same.
I see Existence as the set of all trees, humans, numbers, existents/cardinalities. I see the set of all existents/cardinalities as Infinite. I'm not sure if I should describe Infinity as a quantity here or not. But I think something like 'the cardinality of absolutely all existents (so that's all numbers, letters, trees, semantics, hypothetical possibilities and so on), amounts to Infinity'. I don't see Existence as incomplete because such a view runs into contradictions, hence the need for Existence to equal Infinite (and possibly the need for Infinity to equal a quantity representative of the cardinality of absolutely all existents).
:roll:
According to the philosophy of intuitionism, a sequence that is said to be "without an end", is only taken to mean a sequence that is without a defined end. This is similar to computer programming, where an infinite loop that is declared in a computer program is only interpreted to imply that the program is to be stopped by the external user rather than internally by the program logic.
So in intuitionism (and computer programming), the difference between a finite sequence and an infinite sequence is taken to be epistemic rather than ontological. From the point of view of the producer of the sequence who gets to control it's eventual termination, the sequence could be said to be "finite", whereas from the consumer's point of view who has no knowledge and control of the sequence's termination, the same sequence could be said to be "infinite", or better, "potentially infinite". Or even better, the word "infinity" can be deprecated and replaced by finer-grained terminology that precisely conveys the information that one has at one's disposal in a given situation, without committing to the idea that the information one has is complete.
Amateur (and even some professional) philosophers demonstrate a profound gullibility, in their face-value interpretation of mathematical symbolism. To believe that infinity means "never ending" in an absolute sense just because an upper bound is omitted from a definition, is like believing that a blank cheque cannot bounce.
1 -> a
2 -> b
3 -> c etc.
If an infinity can be matched in this way it is Aleph Null. If not, it is bigger.
Draw a circle on the X, Y axis with radius pi. All points on the circumference except 4 of them are irrational numbers. No others are rational, even though there is and infinity of both rational and irrational. This is because the irrationals are denser.
Assuming you mean the ordered pairs of real numbers that identify points on the circumference have at least one member, x or y, irrational, what are those four points? x^2+y^2=pi^2.
That was a mistake. I was thinking about another function. No rational points.
"x^2+y^2=pi^2" - x and y will be irrational which is why all end points on they hypotenuse (pi) will be irrational.
Here is something on it https://mathoverflow.net/questions/71305/shortest-irrational-path
Here's an introduction to the exacting infinity and the ultra-exacting infinity.
New Scientist described exacting cardinals as being so large that they contain copies of themselvessort of like a house with many full-scale copies of itself inside. Ultra-exacting copies additionally include mathematical rules on how to create them as if the nested house was also wallpapered with blueprints of itself.
There are "countably many" integers. That doesn't imply they can all be counted, but one can map a counting process to the set of integers. In the real world, that process would never end.
On the other hand, the real numbers can't be counted. There are infinitely many numbers between 1 and 2. In fact, there are infinitely many real numbers between any 2 real numbers. This is the rationale for stipulating that there are "more" real numbers. It's not "more" in the real-world sense of your intuitions; it's "more" in a mapping sense.
Quoting Relativist
Agree with all of the above. But you can't map one infinity to another with one being bigger than another because there isn't more than one. Infinity is one number just as 10 is one number. 10 and infinity do not come in different quantities.
Quoting Relativist
if you map all the numbers from 1 to 2 to all the numbers from 1 to 4 or from any number to any number you would get infinity. Or rather you would get the possibility of an infinite number of numbers. But you'll never successfully map all the numbers from 1 to 2 to all the numbers from 1 to 4 because you cannot count them all. This is because of infinity. Infinitely speaking, there are no more numbers between 1 to 4 than there are between 1 to 2, but finitely speaking, there are more numbers between 1 to 4 than there are between 1 to 2.
Infinity is not a thing that exists. It is a concept, and when it is applied to sets - it can lead to inconsistencies. There are infinitely many integers and infinitely many real numbers, but infinity is not a member of either set. Rather, "infinity" is a property of each of these sets. But is it the same property in both sets?
We can compare sets by defining a mapping between them. There is a 1:1 mapping between the set of even integers and the set of all integers. So although it may seem like there "more" integers than even-integers, that's not the relevant comparison. The comparison that is made is based on abstractly mapping the members of one set to the other. In this example, each integer can be mapped 1:1 to the set of even integers. 1->2, 2->4, 3->6...The mapping applies to all members of both sets; no members are left out.
However, there is no 1:1 mapping between the reals and the integers. Reals map into integers, covering all the integers, but you can't cover all the reals with integers. This is the basis for saying the "size" of the set of reals is greater than the "size" of the set of integers. The formal term for "size" is cardinality: the cardinality of the set of reals is greater that the cardinality of the set of integers. This is the basis for saying there are "more" reals than integers, but this isn't "more" in the everyday sense of the word.
This is where I disagree. I don't believe Cantor's diagonal argument shows anything. Infinity is one cardinality/size, it makes no sense for one infinity to be bigger than another in terms of size.
What's the basis for your claim that it makes no sense?
It makes no sense for one quantity of 10 to be bigger than another quantity of 10. 10 is one quantity. Similarly, it makes no sense for one quantity of infinity to be bigger than another quantity of infinity. Infinity is one quantity.
An interesting question to think about that might help in regards to different sizes of infinity:
What is the length of the circumference of a circle with a radius of infinity?
The circumference would need to be 6.283185307... (Tau) times the size of the infinite radius.
If you can conceive that an infinite radius can form a circle, then it would logically make it necessary that the circumference be at least 6 times the size of the infinite radius. Perhaps this makes sense? If this is inconceivable to you, then disregard the suggestion.
In the everyday use of the term, a "quantity" is always a fixed, real number (e.g. a number of liters, a number of tomatoes, a number of molecules in a mole...). Infinity is not a real number. Your mistake seems to be that you're treating it as one.
Quoting Philosopher19
Quoting Relativist
Forgive me for jumping in, but I think what is needed here is a closer look at what is going on here. I'm not a mathematician, so I hope that you will correct anything I say that is not properly forumulated.
In essence, the problem is that the the normal concepts of number don't apply once one has defined inifite sets. So the mathematical concepts here look very strange unless one looks closely at how they change in this new context.
In one way, this situation is unique. But the concept of number in mathematics has changed several times as mathematics has developed. The ancient Greeks, for example, did not consider that either 0 or 1 was a number; that seems strange to us, but we have got used to the new concept 0 and all the many developments that have happened since the Arabic mathematicians changed everything.
As a start, look more closely at the original question?
Quoting Philosopher19
The definition of size here is the number of members. It is true that the number of members of an infinite set can never be specified, and the set is uncountable in that sense. (But we can confidently assert that the number of members - and hence the size - of an infinite set is larger than any finite set.)
However, the definition of "countable" in the context of infinite sets is that the counting can be started, not that it can be finished.
There's a misinterpretation of "infinity". Inifnity is not a target that can never be reached, but the recognition that counting can never be completed, that it will always be possible to take another step in the series. It is, in my (non-mathematical) book not a number at all.
However, mathematicians work around this, but defining a new kind of number. See Wikipedia - Transfinite Numbers
This does not posit that there is a largest finite number. It is an application of concepts that clearly exist in the case of inifinite sets like [1, 0.5, 0.25, .....] In the case of those sets, there is a number that is smaller than any of the members of the set - 0. It is, paradoxically, not a member of the set. It is called the limit. The analgous numbers in the case of the natural numbers is
One might feel that Cantor's argument does not demonstrate its conclusion. But it does demonstrate that the relationship will persist at each step along the way. A counter-argument would have to show that it will break down at some point, and I don't see how such an argument could be made.
I hope I said enough to show how the apparently impossible conclusion can be established. I'm sure someone will correct anything that I have not formulated properly.
Here's a good definition (from this source) of the term:
A mathematical system consists of:
1) A set or universe, U.[/b]
2) Definitions: sentences that explain the meaning of concepts that relate to the universe. Any term used in describing the universe itself is said to be undefined. All definitions are given in terms of these undefined concepts of objects.
3. Axioms: assertions about the properties of the universe and rules for creating and justifying more assertions. These rules always include the system of logic that we have developed to this point.
4. Theorems: A true proposition derived from the axioms of a mathematical system based on the axioms and derived by the logic.
In my abstract algebra course, I had to learn about a variety of mathematical systems (e.g. groups, rings, fields), that have no relationship to the real world, and to prove theorems about them. As long as the system has the 4 components, it's valid math.
The real number system and integer number system are mathematical systems, and both of these relate directly to the world. They also relate to each other: the integers are a subset of the reals.
Transfinite math is just another mathematical system, and it's one with no direct relationship to the real world. It has an indirect relationship, in that it pertains to the sets of integers and the sets of reals - but that doesn't mean all the concepts of the reals&integers are applicable.
The mistake is to treat all of mathematics as a single system, with a single set of axioms and definitions.
Example: the ordinal numbers (the ordered set of integers) have a "successor function" - an operation that produces the next integer: "+1" There is no successor function with the reals, because there is no "next" real number.
More to the point, the real number axioms don't apply to transfinites. What matters is that there is a universe (the transfinite numbers) and that there are operations that can be performed with them - including a successor function for the transfinite ordinals - which allows treating them as greater than or less than.
It's still true that there is a conceptual relation between the transfinites and the reals and integers, and that was the basis for Cantor defining them. But it needs to be remembered that definitions (like "greater than" "less than" etc) are intra-system.
I'm glad you agree. Nor do I find fault with the definition of a mathematical system. However, there do seem to be some differences of perspective and approach between us; these are not necessarily questions of right and wrong, true or false.
I would not want to say that @Philosopher19 is wrong - just that the argument is based on a different approach to the idea of infinity. Specifically, there is a different understanding of what "countable" means even though there is a common understanding of what "size" means and that infinity means that there is no number that is the number of the members of an infinite set.
There is a constant tension here around the fact that counting cannot be completed and the temptation or desire to think of the infinite as some sort of destination or limit. Compare "the sky's the limit" or "the gold at the end of the rainbow" or even the concept of "transfinite numbers". I don't see that there is a basis here of talking of "correct" or "incorrect" or of mistakes - that requires a shared agreed system, which is not available.
Quoting Relativist
I'm suggesting that, in the face of the concept of infinity, there is more than one way to apply the relevant concepts. If we can choose one way rather than another, we cannot apply correct and incorrect. We need a different kind of argument.
Quoting Relativist
But if definitions like "greateer than" and "less than" are only defined within a system, it follows that they cannot be applied outside it. Isn't that at least close to the OP's conclusion?
Actually, because the reals and integer systems are applicable to the real world (they were developed by analyzing aspects of the real world), the terms "greater than" and "less than" do apply meaningfully.
The transfinite system was not developed directly from real world analysis, but from analysis of implications of sets.
Quoting Ludwig V
Agreed- it results in people treating infinity like a natural, or real, number. Then when non-mathematicians hear of transfinite numbers, it reinforces that false view - because it turns infinities into "numbers" but only in a very specialized sense.
Existence is, from the beginning. It is eternal and infinite. That which exists is eternal. Finite things are events in eternity.
Quoting Relativist
Yes, I put that very badly. What I was getting at was that "largest number" or "smallest number" is not defined, or rather, the possibility is excluded by the definitions of "greater than" or "less than", or, more accurately, by the absence of any definition of "largest" or "smallest".
BTW, I'm actually not entirely happy that "number greater than (smaller than) every other number" is not a definition of "greatest" or "smallest". But I have to accept that in the context of mathematics, the rules allow it.
Quoting Philosopher19
I'm afraid that, although I can see the sense of your conclusion, I do think your argument is mistaken at this point. My reason for accepting your conclusion is that infinity is not a number, so comparisons of size are meaningless.
The concept of "infinity" is a bit like the concept of the horizon. The horizon seems to be located in space, but as you approach it, it recedes; you can never get there. Each step in the number sequnce seems to get you closer to infinity, but it is always infinitely far away. More to the point, any attempt to apply ordinary arithmetic produces nonsense. If infinity was a number, you could add 1 to it and produce a larger one.
What one chooses to make of the transfinite numbers is another question, but, for the purposes of mathematics, I think we have to accept that they work in that context. But even they are not numbers of the same kind as the natural or real numbers.
As I said earlier, your definition of "countable" is different from the one used in mathematics, so your concept of infinity is different from the mathematical concept. It is not a question of right or wrong, but of different ways of thinking.
Well, umm.... in Zermelo-Fraenkel set theory infinity is taken as an axiom. Hence there's no proof for infinity.
Then there's open question of the Continuum Hypothesis and it's status, which tells us that even math / set theory doesn't precisely understand infinity. Even the Cantorian system of a cascade of larger infinities is something that is under debate.
Of course as @jgill mentioned above, mathematicians aren't bothered about all of this as they have their limits (plus you do have non-standard analysis for infinitesimals), so one could describe the situation like mathematicians have outsourced the philosophical problem to set theorists.
I like this. However, category theory - which includes categories of sets - an outgrowth of algebraic topology and what ever else of similar abstraction seem to have gotten into the game.
I was fortunate that the large state university I chose to get my PhD over half a century ago had a perfectly adequate but not elite math faculty, and I was able to do my research in a subject arising from classical complex analysis. Had I been confronted with category theory or a similar abstract topic I probably would have switched to computer science or electrical engineering.
Complex analysis, itself, has apparently moved up the inevitable steps of abstraction to the point that the arXive.org collections of papers on the subject are unreadable to me.
Category theory would be the philosophers companion here, but uh... we haven't been trained in category theory in school or in the university. That is really something lacking!
This has come up before. There are categories in my own subject of complex analysis, but in order to work with them you need a solid background of complex analysis at the beginning grad level. Now set theory can start literally at the bottom and work up. I've mentioned before my intro to the Peano axioms and beginning at 0 and ending (at the end of the course) with exponential functions.
Category theory seems to be more a graduate school offering, whereas set theory can be presented at a much lower level. However, "New Math" of the 1960s and 1970s flopped when this was tried. Feynman was very critical of the effort.
I remember someone saying that basically set theory was first seen as a way to finally solve the problematic nature of analysis.
Quoting jgill
I was a casualty of this "New Math" myself: on first grade they really started with set theory and believe me, for a first grader, it was indeed confusing. The old style with relating numbers to pieces of apples and toy cars with addition and substraction was far more understandable. I only remember how confusing "union of sets", "set substraction" and "intersection" was back then, because the teacher didn't give us any hint that somehow this was related to the old school addition and substraction. I also remember my grandfather and grandmother, both math teachers from my mother's side, having this negative attitude towards the new thing and talking with my parents and my other grandmother, that this is too difficult for a first grader.
Well, when you actually very easily get to "problems" like Hilbert Hotel and can discuss on a Philosophy Forum endlessly the basics of set theory (and the foundations of mathematics), it's no wonder just why "New Math" didn't meet the challenge.
As I studied in the Social Sciences Faculty in the university, I remember this math course what you could basically call a "Getting social science majors up to university-level math, because the school system has failed in this" -course. Or at least the teacher spoke about it so nearly every lecture. One of the best math courses ever that I took, actually. I remember how frustrated the math teacher was every time when some thing in mathematics was "just agreed upon to be so by an international convention" without any actual proof. He could just feel all the young social science majors thinking what on Earth is happening here. I remember just how many of these "agreements" there were in mathematics. Basically there is a huge gap between high school math and then university/graduate level mathematics.
A misuse of the word "size".
So, you disagree with Cantor?
If you want more details on the math, read this post: https://thephilosophyforum.com/discussion/comment/962884