Real numbers and the Stern-Brocot tree

Quoting keystone

Who are you trying to convince here? Philosophers who consider definitions optional?
— jgill

This is a chat forum, not a journal. We should be allowed to spitball here.

That's a strawman. He didn't say that all discourse has to be at the level of a mathematics journal.

Reply to keystone

Here's what you need to provide for your SB proposal:

rigorous definition of 'is an SB_real' (then let SB_R = {x | x is an SB_real})
(and you'll have to provide for the negative real numbers too, though I guess that wouldn't be too hard)

rigorous definition of 'SB_<'

rigorous definition of 'SB_+'

rigorous definition of 'SB_*'

rigorous proof that is a complete ordered field


We also don't yet have a rigorous (not just ostensive) definitions of the SB tree, 'R' and 'L'. But I don't doubt that there are ones, though complicated they probably are, so we could at least provisionally work with the ostensive definitions we know.

Also, you might want to consider taking reals not as paths but as sequences of nodes on paths. Perhaps it's easier to talk about sequences of nodes rather than sequences of edges, or at least it's more familiar.

Reply to TonesInDeepFreeze

Choo choo! All aboard the crazy train!

lol, exactly. But I only suggest it because we already are on a crazy train. When we try to apply formalizations of information quantified as a reduction in uncertainty we get the patently false assertion that receiving a clear message in a code we understand, such as the equation (?1913 • ? ) ÷ 1.934 , makes us just as certain of what the result of that equation is as having received the decimal number. That is, we should instantly know which numbers are greater than the result and which are less than it and the identity of all equations that result in the same number. This is clearly not the case. Digital computers are much quicker, but if you throw a complex enough algorithm at them it would take them trillions of years to complete it, even when the function of program(input) = a number that is quite simple to represent.

It's a broader part of P ? NP. There is really a separate problem with the P ? NP that is not related to specifically reducing complexity classes. It's related to the problem of systematic search when two equivalent terms nonetheless cannot be recognized as such without step-wise transformations, particularly when these can take more computational resources than appear to exist in the visible universe.

The bottom half of the Stanford Encyclopedia of Philosophy article on Philosophy of Information covers this, or you can also look up the closely related "scandal of deduction."

And this has nothing to do with P vs. NP, which is a problem in mathematics that understands that 4 is the same object as 2+2.

I don't think most mathematicians particularly care that much about the philosophy of mathematics. At least that is what I've heard in enough lectures and books on the topic to believe it.

There are several major schools in philosophy of math and some deny that numbers as distinct objects even exist, so I don't get how they can be too offended here. Especially since I will allow that 2+2 = 4 in a sense that the are numerically equivalent, they are just not equivalent vis-á-vis computation.

Right, a description of a number and the number are not the same objects.

Is "the successor integer of 2" a description but 2 + 1 is not? Is "three" not a number but 3 is? A mark on a paper, symbols on a Turing Machine tape and pixels are not numbers. In the mathematics of computation/theoretical computer science this is necessarily true. We don't have to reduce encodings to some one true description, although we do have the shortest description in any one appropriate language, Kolmogorov Complexity.

Wrong. An equation is a certain kind of formula. In an ordinary mathematical theory (such as set theory, which is the ordinary theory for the subject of equinumerosity) there are only countably many formulas. But there are uncountably many real numbers. It's true that the set of equations is not 1-1 with the set of reals, but it's the set of reals that is the greater.

I think you are confusing the set of all computable functions with the set of all equations. The concept of a solution set will be helpful here: https://www.varsitytutors.com/hotmath/hotmath_help/topics/solution-sets

A solution set is the set of all solutions for an equation. For example, "x + 1 = 1 + x" has the reals as its solution set. We are talking about the set of all equations, which is the size of the set of all solution sets for all equations. Thus, the SS of "x + 2 = 2 + x" is also the same size as the reals. We can use complex numbers as well, since "x + ? = ? + x" also = R. We can do this with addition for all the reals, but also do it with multiple addition operations, division, exponents, etc. And of course we have some equations that have only a few or one member in the solution set. The set of all solution sets is not the set of all computable functions, which I'll agree is countable.

So as you can see, there are infinitely more equations than reals, which means that no computer, even an infinite one that allows for infinite operations (which is generally disallowed anyhow), can place encodings and numbers into a 1:1 correspondence. There will always be multiple equations specifying a single number (whereas with computable functions there are not enough functions for all the reals).

Thus, not even an infinite computer can take any one number and treat it like an identity name for all the corresponding equations equal to that number via a preset repertoire (since they cannot be in 1:1 correspondence). This is basically the same thing as saying their are more possible solutions to combinations of numbers than numbers, which is prima facie true, and I imagine it can be shown combinatoricaly as well. Not only this, but in some well known cases the equation -> number relationship is many to many, not many to one.

Now many mathematicians might not care about the scandal of deduction, which is fine, but it's a serious problem created by current definitions nonetheless. Notably, for centuries extremely skilled mathematicians argued that any positive X ÷ 0 = ? and had good arguments for that conclusion. It's not like major shifts don't happen because current established practice ends up resulting in absurdity.

Quoting Count Timothy von Icarus

"And this has nothing to do with P vs. NP, which is a problem in mathematics that understands that 4 is the same object as 2+2."

I don't think most mathematicians particularly care that much about the philosophy of mathematics.

They don't need to care about the philosophy of mathematics to know that 2+2 is 4.

Quoting Count Timothy von Icarus

"Wrong. An equation is a certain kind of formula. In an ordinary mathematical theory (such as set theory, which is the ordinary theory for the subject of equinumerosity) there are only countably many formulas. But there are uncountably many real numbers. It's true that the set of equations is not 1-1 with the set of reals, but it's the set of reals that is the greater."

I think you are confusing the set of all computable functions with the set of all equations.

No, you are confusing what I said and meant with something you want to say or mean.

What I said is exactly correct: (1) There are only countably many formulas. (2) There are uncountably many reals. (3) Therefore, there are more reals than there are formulas.

Quoting Count Timothy von Icarus

We are talking about the set of all equations, which is the size of the set of all solution sets for all equations.

That is clearly incorrect. You know worse than nothing about this subject.

An equation is a formula. There are only countably many formulas. A solution set for an open formula with only one free variable is a subset of the set of reals. A formula may have uncountably many members in its solution set, but still there are only countably many formulas. One more time:

(4) The set of equations is a subset of the set of formulas. There are only countably many formulas, so there are only countably many equations.

(5) A solution set for a formula with only one free variable is a subset of the set of reals. A solution set for a formula may have uncountably many members.

Quoting Count Timothy von Icarus

So as you can see, there are infinitely more equations than reals

No, only as you are deluded. Your delusion is in conflating the cardinality of the set of equations with the cardinalities of solution sets for equations. Again: There are only countably many equations, but some equations have uncountably many solutions.

It might be easy to look at the problem a simpler way.

If we had developed a Platonism of processes instead of objects, we might find nothing weird with saying each unique computation is it's own unique entity. These computations would share a relationship in having a "common output," in the same way numbers can have a "common denominator."

Human biology is adapted to generating a world of discrete objects for itself. Physics increasingly casts doubt on the reality of this perception. Platonism casts a long shadow on mathematics, and there we tend to think of objects as fundemental, not relationships or processes. Hence statements like "? is a rational number, thus the Golden Ratio is a rational number." If we focused on relations instead of objects we would say, "that is ridiculous, you cannot have a ratio of nothing to nothing else as a relationship, ? and all numbers are just useful symbols for representing relationships, not irreducible things in themselves." But instead we think of the ratio was reducible to the numerical object.

Which has obviously worked out pretty well, except for the ludicrous results of assuming any algorithm = x IS x as concerns finite computation instantiated in the world or our knowledge of uncomputable numbers.

We say that an algorithm for finding ? ? ? due to the decision problem, and yet ? + 1 = ? - 1 is true. That is, we can work with the uncomputables in compressed form, and we work with hard to compute numbers effectively in compressed form without identifying their numerical value all the time.

This suggests to me that computation is not simply a statement of identity (else we should jettison the uncomputable numbers).

Reply to TonesInDeepFreeze

Missed this. Mathematics: A Very Short Introduction has a good description in the intro. It's on LibGen, but if you want something that is open access you can check on intuitionism versus Platonism versus formalism, etc. https://plato.stanford.edu/entries/philosophy-mathematics/

Here is a description of some of the problems I mentioned with P ? NP vis-á-vis identity and Kolmogorov Complexity
https://plato.stanford.edu/entries/information/#AnomParaProb

Quoting Count Timothy von Icarus

you can check on intuitionism versus Platonism versus formalism, etc.

I know about intuitionism, platonism and formalism. Meanwhile, you need to learn the most basic mathematics rather than throwing around a bunch of terminology that you don't understand. Clearly, you've never actually studied this subject step by step as the subject requires.

Reply to TonesInDeepFreeze

We're getting our wires crossed. You are talking about all "well formed formulas," which I didn't use as a term for a reason. These are countable under standard set theory because it is assumed that a formula is a finite string, by definition. But that's begging the question on the topic of equations being equivalent to identity. If you want to say most of mathematics generally assumes this, I would agree, I only brought up this tangent because I'm not sure if it should accept this given. Semantics vs syntax.

For example, model theory works with uncountable symbols. For every real there is a 'theorem" in such a system of the form x = x. This is only true where x is a symbol that uniquely identifies each real. If the reals are uncountable then so to are the symbols required to show that x = x for all reals or x + 1 = 1 + x. Set theories formulas are countable, but they are so by definition. A symbolic system with a unique symbol for every real cannot be smaller than the reals, the set of symbols must be, by definition, in one to one correspondence with the reals. And since these symbols can also be combined in an encoding, there are more combinations (infinitely more) than there are reals.

Probably my fault for the language. This is why I started with "encodings," but accidentally slipped into "equations."

Reply to Count Timothy von Icarus

An equation is a formula. An equation is a formula of the form:

T = S
where 'T' and 'S' are terms.

Quoting Count Timothy von Icarus

model theory works with uncountable symbols

You don't know what you're talking about. In ordinary model theory for ordinary first order languages, there are only countably many symbols in the language. That does not contradict that the universe of a model may be uncountable.

Quoting Count Timothy von Icarus

For every real there is the theorem of the form x = x.

Very wrong. You completely misunderstand this subject.

Every theorem is a sentence and every sentence is a formlula. There are only countably many theorems. There is not a different theorem for each real. Rather, the open formula 'x=x' is satisfied by uncountably many reals (i.e., the solution set of 'x=x' is uncountable). You are making the same mistake you started with, which is conflating solution sets with formulas. AGAIN:

There are only countably many formulas. But some formulas have uncountable solution sets.

Quoting Count Timothy von Icarus

A symbolic system with a unique symbol for every real cannot be smaller than the reals, no?

Correct, if there is a unique symbol for each real, then the set of symbols is uncountable.

But in ordinary first order languages, there is NOT a unique symbol for each real. We don't ordinarily work with uncountable sets of symbols, because doing so would defeat the intent that the set of symbols, the set of expressions, the set of formulas, the set of sentences, the set of axioms and the set of proofs are all decidable sets.

Get a good book on mathematical logic and study it step by step. Right now, you're very much ill-informed on the subject.

Reply to TonesInDeepFreeze

Correct, if there is a unique symbol for each real, then the set of symbols is uncountable.

Thank you. That's all I was saying. Now will you allow that, given an alphabet where every real has a unique symbol, those symbols could be used in arithmetic, such that, for example, the symbol for 1 added to itself is equivalent to 2?

Everything else you responded to was you jumping over yourself to demonstrate knowledge about terminology and irrelevant. I mentioned all possible encodings of the [I]type[/I] "any real = itself,"etc. (i.e. one such string for every last real), which I thought was apparent given the context. This set has nothing to do with well formed formulas re: standard set theory. You jumped to the formula x = x; that's not what I was referring to. If I meant "x + 1 = 1 + x" in terms of the variables I would not have bothered listing out the variations using integers or mentioned solution sets as an analogy to illustrate the point. "x +1 = 1 + x," is not one such an encoding because x is a variable, not a real. Since the context was encodings' equivalence with the thing they encode, in an abstract sense, I thought this was obvious.

I'm talking about the informational encoding of any object such that a system can recognize that encoding X uniquely specifies Y without any symbolic manipulation having to be performed. As I mentioned originally, this is in the context of communications theory. My point is that, even if we imagine an infinite computer with an infinite alphabet, it still must use step wise transformations to relate symbols to each other.

Which yes, is a point far adjacent to mathematics proper, but this isn't a math forum lol.

You don't know what you're talking about. In ordinary model theory for ordinary first order languages, there are only countably many symbols in the language. That does not contradict that the universe of a model may be uncountable.

Yes, because the statement "model theory can be used to examine infinite symbolic alphabets," is equivalent to my saying "ordinary model theory for ordinary first order languages uses infinite alphabets." Notice how you had to throw in a bunch of specifiers into that sentence so you could show how wrong I am and how smart you are? Have you heard of the principal of charity?

Reply to Count Timothy von Icarus

As I wrote, we don't ordinarily work with languages with uncountably many symbols. It's not even clear what "use" would mean with a language of uncountably many symbols: If there are uncountably many symbols, then there is no decision procedure to decide what the symbols of the language even are, so it's not clear what it would mean to "use" symbols when we can't even know what is or is not a symbol for the language. Languages with uncountably many symbols are "theoretical" in the sense that they are not used for working mathematics but are instead used for theoretical investigations about languages.

Quoting Count Timothy von Icarus

Everything else you responded to was you jumping over yourself to demonstrate knowledge about terminology and irrelevant.

Your attempted mind reading is incorrect and presumptuous. You don't know that my purpose was to show off my knowledge and not instead to correct and explain your error. I have given you, gratis, information and ample explanations. Doing so does not warrant your snide and incorrect "jumping over yourself to demonstrate knowledge".

And what I wrote is exactly relevant in response to what you wrote. If you meant something different from what you actually wrote, then it is not my fault to have taken what you wrote as you actually wrote it.

Quoting Count Timothy von Icarus

you jumped to the formula x = x;

YOU wrote and discussed that formula, and I replied regarding it.

Quoting Count Timothy von Icarus

I'm talking about the informational encoding of any object such that a system can recognize that encoding X uniquely specifies Y without any symbolic manipulation having to be performed. As I mentioned originally, this is in the context of communications theory.

It's not my fault then that you also jumbled together whatever you mean by the above with continued and incorrect claims about equations and solution sets, then also to post an egregious falsehood about model theory, and a number of other misconceptions about mathematics.

"informational encoding"
"system"
"can recognize"
"uniquely specifies"
"without any symbolic manipulation having to be performed"

Would you please say where I can see a glossary of that terminology as you are using it?

Meanwhile, among other points, I hope that at least you understand that in ordinary mathematics '=' means identity, which is to say, for any terms 'T' and 'S',

T = S

means that 'T' and 'S' name the same object, which is to say that T and S are the same object.

Reply to TonesInDeepFreeze

I took the tone from posts starting with:

"All aboard the crazy train,"

"No, only as you are deluded. "

"Wrong."

Generally in field with multiple subfields where the same term can refer to multiple concepts, it's common to ask if there might be a communication problem, not call someone an idiot. And I'll readily admit I misused the term sequence earlier, which was pointed out to me in a helpful way.

Second, that assumption also came from the fact that I tried expressing the point at length and you only responded to small fractions of each post, only where there appeared to be an in for calling me deluded or uniformed.

Anyhow, thanks anyways, as it's always good to see what the least charitable take would be on an idea so as to better polish up the description. Although, even if I know "less than nothing," about mathematics, I think I know enough about conversation to know that when someone starts an interchange with calling you deluded, or responds to a point about how some philosophers of mathematics don't think numbers exist outside formal systems, games we set up, with "they don't need to care about the philosophy of mathematics to know that 2+2 is 4," they aren't particularly interested in a discussion.

Let's flag this right away:

Quoting Count Timothy von Icarus

responds to a point about how some philosophers of mathematics don't think numbers exist outside formal systems, games we set up, with "they don't need to care about the philosophy of mathematics to know that 2+2 is 4," they aren't particularly interested in a discussion.

Here is what I posted:

Quoting TonesInDeepFreeze

I don't think most mathematicians particularly care that much about the philosophy of mathematics.
— Count Timothy von Icarus

They don't need to care about the philosophy of mathematics to know that 2+2 is 4.

YOU said that most mathematicians don't "particularly care that much about the philosophy of mathematics." And I agree with that. But I replied that they still know that 2+2 is 4. That does not imply that I don't care about the philosophy of mathematics or discussion about it.

Quoting Count Timothy von Icarus

I took the tone from posts starting with:

"All aboard the crazy train,"

"No, only as you are deluded. "

"Wrong."

That tone does not imply that my purpose is to show off my knowledge.

Quoting Count Timothy von Icarus

Generally in field with multiple subfields where the same term can refer to multiple things, it's common to ask if there might be a communication problem, not call someone an idiot.

(1) It's not a matter of terms having different meanings. We are not comparing definitions of 'equation', 'solution set', and 'model'. Rather, you made flat out incorrect claims about them.

(2) I didn't comment on your intelligence.

Quoting Count Timothy von Icarus

deluded or uniformed.

You are deluded and uniformed regarding the mathematics and mathematical logic claims that you made and that I mentioned.

Quoting Count Timothy von Icarus

when someone starts an interchange with calling you deluded

And I didn't start that way. You can see the first post to see.

Quoting Count Timothy von Icarus

you only responded to small fractions of each post

Your posts include a wide mess of undefined terminology thrown around. I replied specifically to parts where I could best offer exact corrections and explanations. Instead of recognizing those corrections now, you spread a lot of smokescreen as above.

/

Quoting Count Timothy von Icarus

"Correct, if there is a unique symbol for each real, then the set of symbols is uncountable."

Thank you. That's all I was saying.

No, that's not all you said:

Quoting Count Timothy von Icarus

For every real there is a 'theorem" in such a system of the form x = x.

And I explained why that is incorrect.

So your correct claim 'If there is a unique symbol for each real, then the language is uncountable' was used for a non sequitur that there is a theorem for each real.

Quoting keystone

If we continue down the tree with this alternating pattern RLRLRLRLRLRL... we approach the Golden Ratio.

Is there anything wrong with completing this tree and saying that the infinite digit RL is the Golden Ratio?

Constructively speaking, there's nothing wrong with your identification of real numbers with "infinite" paths, i.e. the non-wellfounded sets known as "streams", provided such paths are finitely describable. For a computable real is equivalent to a circularly defined equation that can be lazily evaluated for any desired number of iterations to yield a finite prefix. In your case, that would be an impredicatively defined binary stream such as S, defined as the fixed point condition

S = 1 x ~S

where _x_ is the cartesian product and ~ is logical NOT (i.e. S is the liar sentence).

To faclitate the identification of streams with cauchy convergent sequences, S can be considered equivalent to other streams for which it shares a bisimulation with respect to some filter for deciding how streams should be compared. The stern-brocot tree can also be interpreted as a game-tree, such that a computable real number is identified with a "winning strategy" for converging towards an opponent's position who attempts to diverge from the player's path to some epsilon quantity.

Surreal Numbers also share a similar binary- tree construction, and their fabled ability to embed the real-numbers might be recalled. But this rests upon the assumption that transfinite induction is valid, which isn't constructively permissible due to it's reliance on the axiom of choice. Your indicated idea of using fixed-points to define real numbers, although not original is more promising.

I believe the non-standard identification of real-numbers with streams and more generally co-algebras, was originally due to Peter Aczel in the eighties, who became famous for inventing/popularising non-wellfounded set theory. For an alternative approach to non-standard analysis that is constructive and sticks to well-founded sets by merely augmenting them with additional axioms to denote terms at the fixed points, see Martin Lof's notes under "The Mathematics of infinity"

Reply to TonesInDeepFreeze

Meanwhile, among other points, I hope that at least you understand that in ordinary mathematics '=' means identity, which is to say, for any terms 'T' and 'S',

T = S

means that 'T' and 'S' name the same object, which is to say that T and S are the same object.

I said as much at the outset.

Here is the problem, if computation is not reversible. If 4 + 4 = 8 and 10 - 2 = 8, what does that mean for the instantiation of the abstraction?

Having 4 $20 bills and being given 4 more is not the same thing as having 10 ($200) and giving away 2 ($40). Having 5 apples and picking two more isn't the same as having 9 and throwing two in the fire.

This entails that computation is not instantiated in the world at all, that adding two apples to seven is the same thing as taking two from nine, a tough argument to make, or claiming that the relationship described in 4 + 4 is acomputational, that it exists without reference to computation. That is, numbers exist as real abstract objects but computation is just a human language for describing their relationships.

That's a fine way of looking at it, maybe one of the more popular. My problem is that it seems hard to explain why we can recognize identity sometimes but not others, and formal systems for describing information do not account for this. So there appears to be a problem with the formalizations or the ontology in this respect.

Quoting Count Timothy von Icarus

I said as much at the outset.

You said the contrary at the outset and some time afterwards also.

Quoting Count Timothy von Icarus

If 4 + 4 = 8 and 10 - 2 = 8, what does that mean for the instantiation of the abstraction?

I don't know what you mean by "the instantiation of the abstraction".

In any case, 4+4 = 8 and 10-2 = 8. 4+4 is 8 and 10-2 is 8.

Quoting Count Timothy von Icarus

Having 4 $20 bills and being given 4 more is not the same thing as having 10 ($200) and giving away 2 ($40). Having 5 apples and picking two more isn't the same as having 9 and throwing two in the fire.

So what? Numbers are not bills or apples. Not only are you deluded and uninformed about mathematics, you're also jejune in your arguments about it.

Quoting Count Timothy von Icarus

That is, numbers exist as real abstract objects but computation is just a human language for describing their relationships.

I don't have a special opinion to state about that. Except that there is also an abstraction of computation in the theory of computability.

/

Meanwhile, a number of misconceptions by you that I explained in my first post:

https://thephilosophyforum.com/discussion/comment/803226

Quoting Count Timothy von Icarus

If 4 shares an identity with 2+2, 3+1, 5+ -1, 8/2, etc. then the P?NP problem doesn't make sense

But P vs NP does make sense, so from your conditional we would have to infer that 4 is not identical with 2+2. (By the way, I have no idea why you think that 4 being +2 implies that P vs NP does not make sense. I'm not asking though, since I'm not inviting you to spew yet more jumbles of undefined terminology and vague premises.)

Quoting TonesInDeepFreeze

"informational encoding"
"system"
"can recognize"
"uniquely specifies"
"without any symbolic manipulation having to be performed"

Would you please say where I can see a glossary of that terminology as you are using it?

I'm still interested. If you are earnest about communicating, then the least you could do is provide a resource for the definitions of your terminology.

Reply to TonesInDeepFreeze

You could look at the link I shared about it from SEP.

Instantiation =

The word "instantiate" is related to "instance". If someone says, "Name some things that are red." you could answer, "For instance, roses are red, apples are red, blood is red." In other words, roses, apples and blood are instances of the property red. In other words, roses, apples, and blood instantiate the property red.

That's all "instantiate" means. An object x instantiates a property p if p(x). That is, x instantiates p if x has the property p, if x exhibits p, if x is an instance of p. All are ways of saying the same thing (with possibly some subtle metaphysical distinctions).

So, yes a property can be instantiated by another property. The property "is a color property" is instantiated by the property red.

Five apples is an instantiation of the number five, etc. e.g., Plato's Theory of Forms.

Informational encoding: the signal in Shannon-Weaver information if you're familiar with that.

System: from physics, either physical or in an abstract toy universe. By this I just meant any computer we can envisage. I didn't want to get into a specific definition because I want to consider all possible computers.

Can recognize: not a signal sent in ambiguous code, one signal cannot refer to two+ different outcomes for a random variable.

Without symbolic manipulation having to be performed: I should have said "without multiple computational steps (quintuples)," to be more precise. One step would be "see symbol x, print symbol y on that section of the tape (doesn't matter which way to move the tape after)." As opposed to how computation generally has to be performed, utilizing multiple spaces on the tape in a stepwise fashion.

The only way to avoid having to use multiple spaces on a tape for at least some computations, even with an infinite symbol system for every real number, would be to have a unique symbol for every arithmetic combination of those symbols.

Quoting TonesInDeepFreeze

In the formalist interpretation of mathematics, where "an entity is what it does,"
— Count Timothy von Icarus

Where can I read that that is a formalist interpretation?

Still interested. I know what 'formalism' is in the philosophy of mathematics. But I don't know of formalism claiming "and entity is what it does".

Reply to Count Timothy von Icarus

I'll look at that link.

/

I know what 'instantiate' means. I just don't know what you mean by "the abstraction". Which abstraction? And I don't know what you mean by "what does it mean"? Is there a particular implication from 2+2 = 4 that you are wondering about?

There are computations of arithmetic. And of course they can be conveyed as Turing machine computations. I don't know what makes problematic the theorem '2+2 = 4' or the interpretation of '=' as standing for the identity relation. I don't see why you think it's a problem that the steps are single and stepwise. Any routine, of course, is reducible to steps.

Reply to TonesInDeepFreeze


Ah, my mistake. This is supposed to say "If 4 + 4 = 8 and 10 - 2 = 8, what does that mean for the instantiation of the computation?
"

I.e., how is computation instantiated in the world? (E.g., https://plato.stanford.edu/entries/computation-physicalsystems/).

If you tell a Turning Machine to add 2 to 2, that's different than subtracting 2 from 6, right? But they have the same answer, 4. However, if all arithmetic expressions that = 4 are identical with it, why does this seem so unintuitive and why do we think dividing 16 by 4 is different from adding 2 and 2 when it comes to computation. We tend to think of computation as being the thing a Turing Machine does to produce outputs, not the outputs themselves.



Yeah, that was very unclear with the wrong word in there.

You are talking about two different subjects together: Mathematics and mathematical logic and, as I take your word for it, information theory.

I have no opinions about what you say about information theory. But I have corrected you on a number of mathematical points.

Here's one:

Quoting Count Timothy von Icarus

I think you are confusing the set of all computable functions with the set of all equations.

That came from your incorrect notion that the set of equations is uncountable. So I am not conflating the set of equations with the set of computable functions. The set of equations is a set of syntactic objects, viz. a set of certain kinds of sequences of symbols. The set of all computable functions is also countable, but I do not conflate those two sets. I'm talking about basic mathematical notions as formalized in mathematical logic. And in context of ordinary mathematical languages in which the set of symbols is countable.

I'd very much like to know whether you understand now that the set of all equations is not a countable set.

Quoting Count Timothy von Icarus

If you tell a Turning Machine to add 2 to 2, that's different than subtracting 2 from 6, right?

Of course.

Quoting Count Timothy von Icarus

However, if all arithmetic expressions that = 4 are identical with it

Right there, you're committing the error of not distinguishing the name from the object. The expressions are not the number.

'4' is not 4 and '2+2' is not 2+2. But 4 is 4 and 2+2 is 2+2 and 4 is 2+2.

Reply to TonesInDeepFreeze

Yeah, I thought I understood the miscommunication there and I did not. You're correct re: well formed formulas. I had always thought formula = well formed formulas and equation could be defined more broadly as "any two equivalent expressions," such that an equation would allow for things that a formula would not, e.g. having infinite length or an infinite symbolic alphabet. Also that formulas have variables whereas an expression needs none. What would be the term for all statements following the form "1 + n = n + 1", but actually using the real numbers, not the variable?

Basically, if the reals are the solution set of all the values that can be put into 1 + n = n + 1, what do we call the set of all the "things" (I said equations before) that the values are solutions to?

The set that has 1 + ? = ? + 1, etc. as its members?

Reply to Count Timothy von Icarus

Thank you for that.

Different authors of textbooks in mathematical logic define these terms somewhat differently. I go with Enderton, and this is for first order logic in particular:

We have a countable set of symbols. Each symbol is of only one of these kinds:

quantifier
sentential connective
variable
n-place operation symbol, for some n >= 0
n-place predicate symbol, for some n >= 0

(also left and right parentheses, but I depart on that point, as officially I formulate with only Polish notation so that parentheses are not needed, though informally I use infix notation for some terms and formulas with parentheses.)

(I like that, in classical logic, a first order language needs only one quantifier (the other quantifier can be defined from the first) and only one sentential connective (either the Nicod dagger for "neither nor" or the Sheffer stroke for "not both", as from either one of those, all other connectives can be defined).)

the set of expressions is the set of finite sequences of symbols. (A bit odd to call them all 'expressions' since some of them are just gibberish and have no interpretations, but so it goes.)

the set of terms is defined inductively.

the set of closed terms is the set of terms that have no variables.

the set of well-formed formulas is defined inductively (and, commonly, we just say 'formula' with the same meaning).

the set of sentences is the set of formulas that have no free variables.

an equation is a formula of the form:

T = S, where 'T' and 'S' are terms.

/

it is not the case that all formulas have variables. of course there are formulas without variables, for example:

0 = 0

is a formula with no variables.

Quoting Count Timothy von Icarus

"1 + n = n + 1", but actually using the real numbers, not the variable?

If 'n' is replaced by a closed term, then the result is a sentence, in particular a sentence that happens to be an equation with no free variables.

Thanks for that.

I apologize for this whole digression anyhow because I had the realization that the thought I had that kicked this off is irrelevant to the thing I'm actually interested in, the ways in which the "process" by which 2 and 2 are added together, the computation, is different from just the output of the process. Total blind alley.

It's shockingly hard to find a discussion of computation that isn't just "computation is the processes that lambda calculus, Turing Machines, etc. can define." You can find a lot of articles on "what are numbers," or "what is entailment," but I've had trouble saiting my interests on this front. You'd think that all the interest in physics re: pancomputationalism would have sparked more philosophical interest in the topic? IDK, maybe I'm just looking with the wrong terms.

Of course, I agree that the computation is not the same as the result.

/

Also, back to an earlier juncture, it is decidedly not the case that I want to show off my knowledge. I don't claim to have very much knowledge about mathematics and mathematical logic. I have a real good firm grasp of some basics, but beyond those mere basics, I falter. I have forgotten so much of the mathematics (especially aside from mathematical logic and set theory) that I studied that I am not even competent when the subject gets very far. So the notion of me wanting to impress anyone is ridiculous. But I also am enthusiastic and like to share what I do know, and it bothers me when I see clearly incorrect or confused claims posted, so I do take some solace in posting corrections and explanations.

/

Anyway, I am interested in the idea of SB used for defining the reals, as another poster has proposed, but I'd like to see that notion developed beyond mere handwaving.

Quoting jgill

He may have devised the continued fraction expansion of the equation

That's certainly a beautiful definition of the golden ratio.

Quoting TonesInDeepFreeze

I'm not talking about that tree in that context. I was talking about the three competing definitions of 'is a real number' and how easy or difficult it is to define the operations for real numbers based on those definitions.

Accepted. I suppose I changed the topic a little, hoping to show how arithmetic with real decimals (which troublesomely starts from the right of the string) is less manageable than arithmetic with real SB strings (which conveniently starts from the left of the string).

Quoting TonesInDeepFreeze

so if 'is a real number' would be defined as just one particular Cauchy sequence of rationals, then which of the infinitely many should it be?

The SB tree might offer something here since it appears that each real number has a single path which can correspond to a sequence of rationals [as you hinted].

Quoting TonesInDeepFreeze

'is a path on the left side of the SB tree' as a fourth competing definiens? It would be of 'is a real number between 0 and 1 inclusive'.

Can you rephrase this? I'm not sure what you're asking.

Quoting TonesInDeepFreeze

And are you sure that every irrational number is one of the denumerable paths? And that the sequence of nodes of every denumerable path converges to an irrational number?

My intuition says yes, but I wouldn't know how to prove it. I think it's clear that every decimal number can be captured by a SB string (of L's and R's) but that is no proof.

Quoting TonesInDeepFreeze

Aren't there denumerable paths that stay constant on a single rational number?

Every path (whether finite or infinite) leads to a different number. Finite paths lead to rational numbers. Infinite paths lead to irrational numbers. Or I think you'd be more comfortable saying that infinite paths that don't end in R or L lead to irrational numbers. I think of the limit of the tree as the real number line as depicted here:

https://imgur.com/vWBO6U9

But as we agreed, there is no bottom of the tree. This limit exists somewhat like an ever distant mirage.

Quoting TonesInDeepFreeze

But 2=1.9. If your method entails that that is not the case, then I doubt that your method actually provides a complete ordered field.

I agree that RLR (1.9) converges to 2 (R). So in the conventional sense of equality they are equal. However there is no row of the tree where the two paths intersect. If equality corresponds to intersecting paths, then in this stricter sense of equality I argue that they are not equal. I believe there is value in both uses of equality. The former is useful for practical purposes (e.g. working with calculus) while the later is useful for philosophical purposes (e.g. calculus, which is inseparably tied to real numbers, is the mathematics of the journeys (paths), not of the destinations (nodes)). If we move our focus away from arriving at a destination (which is a mirage after all) then we lose our attachment to actual infinity. Potential infinity will suffice.

Quoting TonesInDeepFreeze

That's a strawman. He didn't say that all discourse has to be at the level of a mathematics journal.

Fair enough. I think I got defensive because I interpreted it as 'get educated and then talk' when his comment may have simply been something like 'if you want to take this to the next level you need to formalize your ideas'.

Quoting TonesInDeepFreeze

Here's what you need to provide for your SB proposal: rigorous definition...We also don't yet have a rigorous (not just ostensive) definitions of the SB tree, 'R' and 'L'. But I don't doubt that there are ones, though complicated they probably are, so we could at least provisionally work with the ostensive definitions we know. Also, you might want to consider taking reals not as paths but as sequences of nodes on paths. Perhaps it's easier to talk about sequences of nodes rather than sequences of edges, or at least it's more familiar.

I agree that for this to be taken seriously I must present a rigorous definition. However, at this time I'm not equipped to do it. I like the idea of paths/journeys because it corresponds to a process. But yes, I like the idea of describing a path/journey by the intermediate stops along the way (i.e. sequence of nodes).

Quoting TonesInDeepFreeze

Anyway, I am interested in the idea of SB used for defining the reals, as another poster has proposed, but I'd like to see that notion developed beyond mere handwaving.

Cool. Yeah, I would too.

EDIT: When I mentioned a rational path intersecting an irrational path, I meant that rational dashed line never intersects with an irrational solid line, similar to image below.

https://upload.wikimedia.org/wikipedia/commons/3/37/SternBrocotTree.svg

Reply to sime
It's clear you're providing useful comments but you've mentioned multiple topics/terms which I'm not familiar with. I've got some homework to do! Thanks.

Quoting keystone

arithmetic with real SB strings

I can only take your word for it that you've satisfactorily worked out that arithmetic. Don't forget that you have to manage not just finite sequences but infinite ones too.

Quoting keystone

it appears that each real number has a single path which can correspond to a sequence of rationals

If the details truly work out, then, yes, that is a nice feature.

Quoting keystone

'is a path on the left side of the SB tree' as a fourth competing definiens? It would be of 'is a real number between 0 and 1 inclusive'.
— TonesInDeepFreeze

Can you rephrase this? I'm not sure what you're asking.

Nevermind it; I was not on the right track there.

Quoting keystone

Potential infinity will suffice.

The tree itself is infinite. And every real is an infinite path.

It seems you're back to your old tricks again. If you don't want infinite sets, then state your axioms in which you derive mathematics without infinite sets. We went over all this 'potential infinity' business a while ago. To save my valuable time, rather than go full circle yet again with you, I'd do best to recommend that you or anyone can read those threads.

Quoting keystone

Every path (whether finite or infinite) leads to a different number.

Right. I was distracted by the dashed lines in the Wikipedia illustration. I recognize now that they're just for place keeping.

Quoting keystone

an ever distant mirage

For example, the square root of 2 does not remind me of a mirage. It is not problematic that it is the limit of a sequence of rationals but is not one of the entries in that sequence. But some people just can't grok the idea of the entries of a sequence getting arbitrarily close to a point but that point is not itself an entry in the sequence. But, alas, this brings us back again to the threads from a few months ago. We've been through it already.

Reply to TonesInDeepFreeze

I did come across the term I meant to use if reference to solution sets. It's the replacement set, so for a formula like 1 + x = x + 1 this would be the variables replaced with each real number.

Now if the solution set is the real numbers, does that mean the replacement set is the same size? And if so, what do we call the members of the replacement set if not equations? Expressions? Or, as I thought might be the case from your post, can we say that the replacement set is actually smaller than the same formula's solution set?

Here is why I thought the replacement set might smaller, but tell me if I'm wrong:

The things in the replacement set seem to be equations, 1+2 = 2+1 is an equation, at least as defined as two expressions with a equals sign (which maybe is a definition lacking rigour?). However, if equations are necessarily finite, how could we have one for every real number? We would need an uncountable number of such equations, one for each real, which seems to violate the logic you described.

You'd either need an infinite string for the equation to put the real number in, since you can't do it with a finite number of digits, or a unique symbol for each real, something like ?. But then you would need an uncountable number of symbols, which also violates the logic. This would mean that there is not a well formed equation for every solution of 1 + x = x + 1, the old "some truths aren't expressible in a system."

I looked around for answers but it's hard to find something specific and then some online sources aren't vetted and conflict. I figured the answer has to be either that the members of the replacement set have a different name than "equation" or that the replacement set is counterintuitively smaller than the corresponding solution set in cases where the solution set it the real numbers.

But could you have a set of "equations" from a system that does allow an infinite alphabet? These wouldn't be valid equations under set theory, but they would be a set in the way we can have a set of mathematical models, or a set of library books. I guess "a set of statements from a language in an infinite symbol language."

Quoting TonesInDeepFreeze

I can only take your word for it that you've satisfactorily worked out that arithmetic. Don't forget that you have to manage not just finite sequences but infinite ones too.

Aside from never being able to complete the computation involving infinite strings, there is a scenario where the algorithm may appear to hang. This is where it would continue to absorb more input digits without generating more output digits. A decimal analogy might be when subtracting 0.9 from 1.0 where (depending on your approach) you might get stuck in the step of continually looking ahead for a non-zero digit to borrow from.

Quoting TonesInDeepFreeze

For example, the square root of 2 does not remind me of a mirage. It is not problematic that it is the limit of a sequence of rationals but is not one of the entries in that sequence. But some people just can't grok the idea of the entries of a sequence getting arbitrarily close to a point but that point is not itself an entry in the sequence.

From the Stern-Brocot perspective, what is a point if not a node on the tree? Sqrt(2) does not converge towards any node on the tree. However, it appears to converge to a node that exists at 'row infinity'. Of course, there is no 'row infinity' which is why I relate it to a mirage. There is an inconsistency in claiming that both (1) the real number line exists and (2) 'row infinity' does not exist. You can't have it both ways. The real number line is an incredibly useful mirage.

Reply to Count Timothy von Icarus

The crux of this is that there uncountably many reals but only denumerably many names.

But it's difficult to reply point by point to your post, because it's too tangled and knotted up. It would be much better to just start from the beginning. That would be to move step by step through a textbook treatment of this subject in mathematical logic.

It's like when the cables among a lot of electronic components are so tangled and knotted that you can't tell what is connected to what, so you have to just unplug everything and then reattach all the cables in a methodical way.

But I'll address a few points anyway, reiterating some of what I've already said:

First, just to be clear: 'countable' doesn't meant 'finite'. Rather, 'S is countable' means 'either S is finite or S is 1-1 with the set of natural numbers'. And then 'S is denumerable' means 'S is 1-1 with the set of natural numbers'. So there are finite countable sets and infinite countable sets. And a denumerable set is an infinite countable set. And, 'S is uncountable' means 'S is not countable'. Lastly, 'uncountable' doesn't just mean 'infinite'. Yes, if S is uncountable, then S is infinite, but also S is not 1-1 with the set of natural numbers.

"PLUGGING IN"

Because we are grappling with the notion that there are uncountably many real numbers but only denumerably many names, we need to be more exact in what we mean by 'plug in'. This gets pedantic, but it's necessary:

An equation is a syntactical object. So when we substitute a constant symbol for a variable, we are not "plugging in " a number. Rather we are plugging in one symbol (a constant symbol) for another symbol (a variable).

The constant symbol STANDS FOR a real number, but it is not itself a real number.

Recognizing that fact helps to dispel bafflement about the fact that there are uncountably many reals in the solution set but only denumerably many substitutions we can make for the variable.

BOTTOM LINE

There are uncountably many real numbers.

So the solution set for

x+1 = 1+x
(where '+' is defined as the addition operation on the set of real numbers)

is an uncountable set.

But there are only denumerably many names, so there are uncountably many unnamed real numbers, so there are uncountably unnamed real numbers in that solution set.

And, since there real numbers that don't have names, there are no names for those real numbers to plug into the equation.

FURTHER EXPLANTION:

SYNTAX

Ordinary mathematical languages have a denumerable (countably infinite) set of symbols.

The syntactical objects are the terms ("names") and formulas ("statements").*

Every equation is a formula.

Every term and formulas is a finite sequence of symbols.

With only denumerably many symbols, there are only denumerably many finite sequences of symbols. So there are only denumerably many terms and denumberably many formulas.

So there are only denumerably many names and denumerably many equations.

The terms with no free variables are the closed terms.

Every closed term can be abbreviated with a constant symbol per a definition for that constant symbol. For example, we can provide a formulation that is a definition for the constant symbol 'pi':**

Ax(x = pi <-> Ecmd(c is a circle & m is the circumference of c & d is the diameter of c & x = c/d))

And there are only denumerably many constant symbols.

The formulas with no free variables are the sentences.

A theory is a set of sentences closed under deduction.

By 'theorem of a theory' we mean a sentence that is a member of the theory.

Usually, with a theory we also mention an axiomatization of that theory. So a theorem is a sentence provable from those axioms.

In our usual mathematical theories (i.e., any of the usual extensions of Z set theory), we have this formulation that is a theorem:

{x | x is a real number} is uncountable

/

SEMANTICS

For a given mathematical language, we provide the "meaning" for the terms and formulas through the method of models. A model is a function from the set of symbols:

To the universal quantifier, the model assigns an non-empty set, which we call the 'universe for the model' or 'the domain of discourse' for the model.

To each n-place operation symbol, the model assigns an n-place function on the universe.

(a constant is just a 0-place operation symbol)

To each n-place predicate symbol, the model assigns an n-place relation that is a subset of the universe.

(a sentence letter is just a 0-place predicate symbol)

The model does not assign anything for the variables. It shouldn't, because variables are not supposed to have a fixed designation. But we can make a separate assignment for the variables and then we have a model plus an assignment for the variables.

Then, for closed terms, the model assigns members of the universe, inductively per the assignment for the operation symbols.

And, for sentences, the model assigns a truth value, inductively per the assignments for the operation symbols and the predicate symbols.

For open terms, the model plus an assignment for the variables inductively assigns a member of the universe.

For open formulas, the model plus an assignment for the variables inductively assigns a truth value.

A model M is a model of theory T if and only if every theorem of T is true per model M.

Now, per a given model, a mathematical object is a member of the universe of that model.

Each real number is a mathematical object.

Our theory says has the theorem:

{x | x is a real number} is uncountable

Now, for any model of our theory that is also a model that "correctly captures"*** the "intended meaning" of 'uncountable', the subset of the universe that is mapped to from the predicate symbol 'R' (for "is a real number") is indeed uncountable.

/

SOLUTION SETS

For example, the solution set for the equation

x+1 = 1+x
(where '+' is defined as an operation on the set of real numbers)

is

{x | x is a real number} = R

and we have the theorem:

R is uncountable

And, looking at it semantically, if we have model in mind that "correctly captures", then 'R' maps to the set of real numbers (or an isomorphic variant), thus the model maps R to an uncountable set.

In general, for a formula P with free variables x1....xn, the solution set is:

{ | P}

where n=1, we may drop the tuple notation and just say:

{x | P}

/

REPLACEMENT SET

As far as I can tell, that is a notion used in beginning informal high school algebra or instruction at that level. I don't know of an actual serious mathematical definition in this context. For the purpose of this discussion, I recommend just forgetting about "replacement sets". It is not needed for any explanatory purpose and only clutters an otherwise rigorous exposition of this topic.

/

* To be more accurate, only terms with no free variables are names, and only formulas with no free variables are statements.

** Throughout, I use some English in the formulations to facilitate exposition. In principle, these formulations would be just symbol sequences of the formal language.

*** To get avoid certain cases provided by Lowenheim-Skolem.

Quoting keystone

never being able to complete the computation involving infinite strings

Exactly.

Meanwhile, with the other common definitions, we do define addition and multiplication of real numbers and that is not blocked by the fact that computations do not accept infinite sequences as inputs.

Quoting keystone

what is a point if not a node on the tree?

Unless instructed otherwise, I would take 'point' and 'node' as synonymous in this context.

Quoting keystone

Sqrt(2) does not converge towards any node the tree.

If we are redefining 'is a real number' as 'is a path in S-B' (I would prefer 'is a sequence of nodes on a path in S-B'), then of course such a sequence does not converge to sqrt(2), since sqrt(2) is a sequence and not a node.

Quoting keystone

it appears to converge to a node that exists at 'row infinity'

Right, it doesn't converge to any node on the tree. But with the definition

a real number is a path on the S-B tree

convergence is no longer relevant in this context, since a real number is path and not a node to which a sequence converges.

And about "row infinity":

Quoting keystone

Of course, there is no 'row infinity' which is why I relate it to a mirage.

Exactly, you don't have a mathematical basis, so you resort to merely figurative, undefined, subjective language. One should not mind figurative language used to convey intuitions about mathematics, to help us get a "mind's eye" grasp of certain concepts. But when the figurative language ends up not backed by actual mathematics then it is fitting to respond, "Okay, that might be interesting, so get back to us when you've worked out the math."

Quoting keystone

There is an inconsistency in claiming that both (1) the real number line exists and (2) 'row infinity' does not exist.

No there is not. You're lying. Inconsistency is the derivability of the conjunction of a statement and its negation. You can't show any such derivation.

The real line is constructed in our mathematical theory. And also, in our mathematical theory, the S-B tree does not have a "row infinity". If you claim that is inconsistent, then PROVE it. And if you can't, then you should desist from lying about it.

Your speciousness and intellectual dishonesty here is similar to the previous threads with you.

And it's even WORSE in this thread, because in the other threads, the discussion was about thought experiments, which are informal analogies about mathematics, while in this thread, we are talking about an exact mathematical object.

Reply to keystone

Keeping track of where the discussion stands:

I wrote:

Quoting TonesInDeepFreeze

It is not problematic that it is the limit of a sequence of rationals but is not one of the entries in that sequence.

That still stands, notwithstanding your reply (which ends with a lie).

Quoting TonesInDeepFreeze

Exactly. Meanwhile, with the other common definitions, we do define addition and multiplication of real numbers and that is not blocked by the fact that computations do not accept infinite sequences as inputs.

All I said was that the computation when the inputs were infinite strings wouldn't complete. The SB algorithm can certainly accept infinite strings as inputs though, it would just absorb the digits little by little and emit output digits little by little, never actually completing the output. In other words, I wasn't presenting a limitation of the algorithm, I was conveying my disbelief in supertasks. I would relate this to computing the digits of pi. The pi generating algorithm works, it would just never actually output all digits of pi.

Quoting TonesInDeepFreeze

sqrt(2) is a real number. A real number doesn't converge. A sequence converges.

As we've informally agreed, irrationals (e.g. sqrt(2)) are unending paths on the SB tree. In an informal sense, it is reasonable to say that paths converge. The paths LR and RL both converge to the node 1. When I mentioned sqrt(2) not converging to a point, I was referring to the path sqrt(2) not converging to any node (in that sense). Would you rather I use another term than converge?

The difference betweenQuoting TonesInDeepFreeze

And it's even WORSE in this thread, because in the other threads, the discussion was about thought experiments, which are informal analogies about mathematics, while in this thread, we are talking about an exact mathematical object.

The difference here is that we have a mathematical object (SB tree) which is very simple but yet still (it appears to) sufficiently captures to topic of concern. But unlike with our other thread, you cannot defend your position by making your arguments more and more complex. I, on the other hand, am defending my position by making my augments more and more simple. I suppose one way for you to end this would be by explaining why the SB tree (and my argument) oversimplifies the issue. If you don't like my description of the real number line as it relates to the SB tree, can you propose a better description using the SB tree?

Quoting TonesInDeepFreeze

For example, the square root of 2 does not remind me of a mirage. It is not problematic that it is the limit of a sequence of rationals but is not one of the entries in that sequence.

It does not bother me that there are (unending) paths on the SB tree having no destination node. I also think there's value in saying that such a path is approaching a 'virtual' node. Calculus demonstrates the utility of such a belief. But I wouldn't accept that virtual nodes and actual nodes are the same thing. I think when we say rationals and irrationals are both numbers, we are essentially doing that.

Quoting keystone

the computation

There are two separate matters:

(1) The definitions of the operations.

For addition, this is of the form:

x=y = z <-> P
where P is a formula in which only x, y and z occur free and such that we have the theorem:

AxyE!zP

If you propose the tree as a basis for the real numbers, then you have to provide such a definition.

Please tell me whether you understand that. Actually, don't bother, because I already know you know nothing about mathematical definitions.

(2) Computations. That's not what I have been talking about.

Quoting keystone

As we've informally agreed, irrationals (e.g. sqrt(2)) are unending paths on the SB tree.

I don't take it as merely an informal proposal. If it's going to hold up, then it will be formalized.

Quoting TonesInDeepFreeze

sqrt(2) is a real number. A real number doesn't converge. A sequence converges.

I had edited that to:

Quoting TonesInDeepFreeze

If we are redefining 'is a real number' as 'is a path in S-B' (I would prefer 'is a sequence of nodes on a path in S-B'), then of course such a sequence does not converge to sqrt(2), since sqrt(2) is a sequence and not a node.

Quoting keystone

unlike with our other thread, you cannot defend your position by making your arguments more and more complex

Not only are you lying about there being an inconsistency, but you're moving on to mischaracterize what I posted in other threads. And indeed you even more obnoxious, because what I did in the other threads was to generously give you increasingly detailed explanations of things you're ignorant about. As you continued not to grasp the basic mathematics, I generously explained it to you in yet more detail. That is not me resorting to defending my arguments by making them more complex. You are a real piece of work.

Quoting keystone

I, on the other hand, am defending my position by making my augments more and more simple.

Yes, it's a simple lie that the existence of the real line is inconsistent with the existence of the S-B tree.

Again, you have evaded the point:

Since you claim there is an inconsistency, then PROVE it.

Quoting keystone

I suppose one way for you to end this would be by explaining why the SB tree (and my argument) oversimplifies the issue.

What? Why should I do that? I don't claim that the S-B tree "oversimplifies" anything. Now, you're also resorting to strawman.

Quoting keystone

If you don't like my description of the real number line as it relates to the SB tree

What I don't like, because it's a lie, is your claim that the existence of the real line contradicts the existence of the S-B tree.

Quoting keystone

can you propose a better description using the SB tree?

What? I have no problem with saying that set of real numbers is the set of paths in the S-B tree, with whatever other finer qualifications need to be given to make it all work out rigorously. And then defining an ordering to define the continuum. I said explicitly that I can imagine it all working out.

But your lie is saying that there is contradiction between the existence of the continuum and the existence of the S-B tree.

Quoting keystone

For example, the square root of 2 does not remind me of a mirage. It is not problematic that it is the limit of a sequence of rationals but is not one of the entries in that sequence.
— TonesInDeepFreeze

It does not bother me that there are (unending) paths on the SB tree having no destination node. I also think there's value in saying that such a path is approaching a 'virtual' node.

(1) You are evading my point. I'll say it again (whether regarding ordinary analyis or an S-B proposal):

It is not problematic that a real is the limit of a sequence of rationals but is not one of the entries in that sequence.

(2) "virtual node". More undefined figurative language. The last refuge of the crank.

The mathematical axioms prove the existence of the S-B tree.

The mathematical axioms also prove the existence of a complete ordered field with the carrier set being the set of Dedekind cuts.

The mathematical axioms also prove the existence of a complete ordered field with the carrier set being the set of equivalence classes of Cauchy sequences of rationals.

Those two complete ordered fields are isomorphic.

We can also try to figure out showing that we have a complete ordered field with the carrier set as the set of paths in the S-B tree. And, since all complete ordered fields are isomorphic with one another, the one based on the S-B tree would be isomorphic with the others too.

The set of paths in the S-B tree is not part of the S-B tree. But that doesn't contradict anything else in the mathematical theory from the axioms. The S-B tree exists, and lots of things other than the S-B also exist. The set {p | p is a path in the S-B tree} is one of them.

I don't know, but maybe you are thinking that there is ONLY the S-B tree and anything other than the S-B tree doesn't exist (except as a "mirage" or whatever crank dodge concept you invoke)?

If that is the case, then dump that thought. The S-B tree exists; the Dedekind cut real numbers exist: the Cauchy sequence real numbers exist; and the set of paths in the S-B tree exists (and possibly it too as a basis for a complete ordered field).

Quoting TonesInDeepFreeze

If you propose the tree as a basis for the real numbers, then you have to provide such a definition.

Fair enough. The author of the paper proved the algorithm to work specifically for the rationals (not reals). I imagine the proof is much more difficult (impossible?) to extend to reals. You're right, I don't have the knowledge to produce a formal definition, let alone prove it. My take was based on my experience coding and utilizing the alorithm.

Quoting TonesInDeepFreeze

of course such a sequence does not converge to sqrt(2), since sqrt(2) is a sequence and not a node.

(RL) converges to the node corresponding to 1. RL (phi) does not converge to any node.

Quoting TonesInDeepFreeze

what I did in the other threads was to generously give you increasingly detailed explanations

You brought the conversation to a level of complexity/formality that I wasn't comfortable with so the conversation ended. I don't believe it, but if that level of complexity/formality is required to discuss infinity then so be it.

Quoting TonesInDeepFreeze

What I don't like, because it's a lie, is your claim that the existence of the real line contradicts the existence of the S-B tree.

Quoting TonesInDeepFreeze

You are evading my point.

Let's table this for a moment as I better understand your interpretation of real numbers on the SB tree.

Quoting TonesInDeepFreeze

I have no problem with saying that set of real numbers is the set of paths in the S-B tree

Do you think the real number RL (phi) is the path itself or the limit of the path?

Quoting TonesInDeepFreeze

And, since all complete ordered fields are isomorphic with one another, the one based on the S-B tree would be isomorphic with the others too.

I agree with this. The reason why I prefer the S-B tree view is that it's more understandable to amateurs like myself. What I want to reiterate is that I'm not trying to abandon the reals or the existence of continua like the real line. Instead, I'm trying to understand what the real line is from the perspective of the S-B tree.

Quoting TonesInDeepFreeze

The set of paths in the S-B tree is not part of the S-B tree.

I understand that there are no sets in the tree, but the paths do exist in the tree, right? The tree is composed of both the paths and the nodes, right?

Quoting keystone

What I don't like, because it's a lie, is your claim that the existence of the real line contradicts the existence of the S-B tree.
— TonesInDeepFreeze
You are evading my point.
— TonesInDeepFreeze

Let's table this for a moment as I better understand your interpretation of real numbers on the SB tree.

I'm not tabling it for anything. It is flat out incorrect that

Quoting keystone

There is an inconsistency in claiming that both (1) the real number line exists and (2) 'row infinity' does not exist.

The S-B tree is clear enough by the ostensive definition we have of it. (Eventually though we should have a rigorous definition.)

And we agree that there is no row of the S-B tree that is not indexed by a natural number. (I.e., we agree that there is no "row infinity" of the S-B tree.)

And I understand the notion of the irrational reals being the denumerable paths of the S-B tree.

So we don't have to wait to see what my interpretation is.

These are consistent with one another:

The S-B tree exists.

The set of finite paths of the S-B tree exists and there is a 1-1 correspondence between the set of finite paths and the set of rational numbers. So the proposal is to take the rational real numbers as the finite paths.

The set of denumerable paths of the S-B tree exists. And there is a 1-1 correspondence between the set of denumerable paths and the set of irrational real numbers. So the proposal is to take the irrational real numbers as the denumerable paths.

There is no row that is not indexed by a natural number. (I.e. there is no "row infinity".) Therefore, no irrational real number has a path with a final node.

Now, you claim there is an inconsistency there. So PROVE that there is an inconsistency there. Otherwise, you are making the claim utterly without basis; you are fabricating, which is to say you are lying.

/

Quoting keystone

Do you think the real number RL (phi) is the path itself or the limit of the path?

I'm accepting whatever coherent proposal YOU are making. You want to make phi the path. I've said that that is fine with me.

Just to be clear on what we're talking about:

'graph' is defined different ways. I use the definition by which a graph is a certain kind of triple. It follows from the definition that:

Every graph is a triple . V is the set of vertices (aka 'the nodes'), E is the set of edges, and f is a function on E that assigns to each edge either an unordered pair or ordered pair of vertices.

Watch out now, here comes something quite pedantic:

=
< f> =
{{} { f}} =
{{{{V} {V E}}} {{{V} {V E}} f}}

So the only two things that are literally in the graph (i.e., are members of the graph) are {{{V} {V E}}} and {{{V} {V E}} f}.

I mention that to explain why I personally don't like to say "the paths are in the tree" but rather "the paths are of the tree".

But it would be unduly pedantic indeed to disallow that conversationally we use the word "in" more loosely (i.e., not just for membership), and that most mathematicians don't say "of" but say things like:

"The nodes are in the graph", "the edges are in the graph", the "paths are in the graph" and "the rows" in the graph".

So I'll go along with that usage.

Now the bullet points:

Every path is a sequence of edges.

A path is NOT a sequence of nodes.

Every tree is a graph.

The S-B tree has both finite and denumerable paths.

As I understand the proposal, the finite paths are the rational real numbers, and the infinite paths are the irrational real numbers. (I suggest that it might be better to make the rational reals the finite sequences of nodes and the irrational numbers the denumerable sequences of nodes. I.e. "eliminate the middleman".)

Quoting keystone

of course such a sequence does not converge to sqrt(2), since sqrt(2) is a sequence and not a node.
— TonesInDeepFreeze

(RL) converges to the node corresponding to 1.

Actually 1/1 is the one exception. 1/1 is not represented by a path but by a node, since there is no path leading to the node 1/1.

Quoting keystone

RL (phi) does not converge to any node.

At least for right now, I won't quibble with the notion of a path converging to a node. But I don't see what your reply has to to do with my point that sqrt(2) is a sequence not a node, except that you give phi as another example.

Quoting keystone

You brought the conversation to a level of complexity/formality that I wasn't comfortable

And that does not entail that I did that as something similar to a "No true Scotsman" ploy (as I surmise you were suggesting).

Quoting keystone

The reason why I prefer the S-B tree view is that it's more understandable to amateurs like myself.

There are attractions in using the S-B tree:

Every fraction is in lowest terms. (But we will still have to deal with justifying, e.g., 2/4 = 1/2. There is no node 2/4, so there is no path to 2/4, so 2/4 is not a rational number. So what is it? This is why the standard approach takes rationals to be equivalence classes.)

It's intuitive that the rational reals would be the finite paths and the irrational reals the denumerable paths. And that also means that, e.g., 1/3 doesn't need to be mentioned as represented by a denumerable sequence.

Real numbers can be sequences and not equivalence classes of sequences.

Quoting keystone

I'm trying to understand what the real line is from the perspective of the S-B tree.

The continuum is where R is the set of real numbers and S is the standard 'less than' ordering on the set of real numbers. There are some details to work out in taking the set of real numbers to be the set of paths (but tossing in 1 also, as 1/1 has no associated path), but it seems to be a plausible idea. But then you have to define the 'less than' ordering.

But you are also trying to impugn the standard theory, which you have objected to for being infinitistic. But the S-B approach is no less infinitistic. Then, irrationally, and even defeating your own purpose ("cut your nose to spite your face") of using the S-B tree, you say there is an inconsistency. And, aside from S-B, it's fine to propose a finitistic or constructive alternative or computable analysis or non-standard analysis or even a theory with a paraconsistent logic. But that doesn't permit propounding the lie that the standard theory is inconsistent or inconsistent with the S-B tree (indeed, the S-B tree itself is within the standard theory).

Thanks for providing a detailed and thoughtful response.

Quoting TonesInDeepFreeze

I'm accepting whatever coherent proposal YOU are making.

Let me re-share this image of the S-B tree (https://imgur.com/vWBO6U9) because it shows how I see the connection between the S-B tree and the real number line. The nodes are projected down and at the limit we have nodes for all real numbers forming the continuous real number line in totality. Granted, the limit is not a part of the tree, after all, there are no nodes for irrational numbers in the tree. However, the limit can still be a property of the tree just as an irrational number can be a property of a Cauchy sequence. So from the perspective of the Stern-Brocot tree, the real line is like a mirage. It's not actually a part of the tree, but it's a valid property of the tree nonetheless. With this view, the objects of concern are the nodes (not the paths). But since some paths are not actually linked with nodes, I proposed we say that they converge to virtual nodes. Or in other words, the real number line is composed of points and virtual points.

While it may be convenient to think in terms of paths, there is no analogous pictorial connection between paths and the real number line. I also think paths are more like the Cauchy sequences themselves while the nodes are more like the limits of Cauchy sequences.

Quoting TonesInDeepFreeze

Now, you claim there is an inconsistency there. So PROVE that there is an inconsistency there. Otherwise, you are making the claim utterly without basis; you are fabricating, which is to say you are lying.

With my view now outlined above, I am arguing that it is inconsistent to say that the real number line is composed entirely of actual points. That would only be possible if there was a row infinity in the tree, which we agree that there is not. Instead, the real number line can only exist at the limit and must be composed of both points and virtual points.



Quoting TonesInDeepFreeze

I mention that to explain why I personally don't like to say "the paths are in the tree" but rather "the paths are of the tree".

Might we also be allowed to say that the limit is of the tree?

Quoting TonesInDeepFreeze

(I suggest that it might be better to make the rational reals the finite sequences of nodes and the irrational numbers the infinite sequences of nodes

Can we say that the rational/irrational numbers are the limits of their corresponding paths.

Quoting TonesInDeepFreeze

And that does not entail that I did that as something similar to a "No true Scotsman" ploy (as I surmise you were suggesting).

I followed my statement with "I don't believe it, but if that level of complexity/formality is required to discuss infinity then so be it." In other words, you made the conversation too complex for me. Whether it was too complex for the topic is not for me to decide.

Quoting TonesInDeepFreeze

so there is no path to 2/4, so 2/4 is not a rational number. So what is it?

I suppose you can use the S-B tree to compute 2 divided by 4 and you'd get the rational 1/2. But I see your point.

Quoting TonesInDeepFreeze

But you are also trying to impugn the standard theory, which you have objected to for being infinitistic. But the S-B approach is no less infinitistic.

One can write a finite (but complete) computer program to create the entire S-B tree. My point is simply that that program cannot actually be executed to completion. As such, the object of study should not be the complete output of the program (which cannot be generated) but instead the program itself whose execution is potentially infinite.

So while I don't believe in 'the set of all natural number', I don't mind a computer program that prints all natural numbers. And when 'we' work with infinite sets, 'we' don't actually work with the infinite sets themselves, 'we' work with the 'algorithms' used to generate them. If my view were to be proven true, standard theory wouldn't fall apart. It would just be a philosophical shift from actually infinite objects to potentially infinite algorithms/programs. For example, in set theory, it could be as small as moving from saying 'there exists a set' to 'construct a set'.

We'll go with our continuing lead story:

Quoting keystone

Now, you claim there is an inconsistency there. So PROVE that there is an inconsistency there. Otherwise, you are making the claim utterly without basis; you are fabricating, which is to say you are lying.
— TonesInDeepFreeze

With my view now outlined above, I am arguing that it is inconsistent to say that the real number line is composed entirely of actual points.

That's yet more dishonest obfuscation by you.

For about the 1000th time: Inconsistency is having both a statement and its negation as theorems.

"actual" is not a terminology of the theory.

So there is no "inconsistency" regarding it.

You need to stop using the word 'inconsistency' with your own private meanings. Inconsistency is an exact rigorous notion in mathematics. When you appropriate the term to use it with your own private and undefined meaning, you disservice honest and coherent discussion.

"Just remember it's not a lie if you believe it".

Look up 'lie' in Merriam-Webster.

Quoting keystone

he nodes are projected down and at the limit we have nodes for all real numbers forming the continuous real number line in totality.

You mangle terminology. Choose a definition and stick with it.

First you said the reals are the paths. Now you say they are nodes.

And you use "limit" in an undefined way.

Your insouciance in not making definitions and sticking to them invites confusion and is annoying.

Quoting keystone

Can we say that the rational/irrational numbers are the limits of their corresponding paths.

(1) What is a limit of a path? I can see what the limit of a sequence of nodes is, but you would need to define what you mean by a limit of a sequence of edges.

(2) You said you wanted the reals to be the paths. Now you want to switch to something else. Let us know when you've reached a stable decision.

(3) Your notion doesn't even make sense, as follows: You are trying to define 'is a real number' per the S-B tree. But you need also to prove the existence of the objects that meet that definition. You haven't proven the existence of whatever the limits are supposed to be.

Look, you have the existence of the paths. And you have the existence of the sequences of nodes. But you don't have the existence of whatever you think are going to serve as "the limits". If you want to have the things that are going to serve as the limits, then you need to prove they exist. And not "mirages"'. Or give a mathematical definition of 'mirage'.

Quoting keystone

One can write a finite (but complete) computer program to create the entire S-B tree.

To be clear, since you write ambiguously "create the entire tree". Yes, there is a program such that, for any n, the program will generate up to and including the nth row and stop. But there is no program that generates all rows and stops. I take it that you agree.

Quoting keystone

One can write a finite (but complete) computer program to create the entire S-B tree.

A program can generate any arbitrary finite part down the tree, but it can't create "the entire" tree.

Again, the tree is an infinite object. And if you consider it to be the limit of the finite stages, then it still has to exist to BE that limit!

Meanwhile, you seem now to regard the reals as limits of successive finite sequences of finite paths. I don't see a problem in that. But, again, it entails that those limits are THEMSEVLES objects - existing. Indeed, they are the denumerable paths. You can't say that the denumerable paths are limits but that they don't exist.

The notion of limits in this context is infinitistic. The limits of sucessive finite sequences of finite paths are the denumerable paths. If you want them to be our real numbers, then they exist and it's just jejune to say [paraphrase], "They don't "actually" exist but they exist as "mirages"". That's just undefined verbiage, nice for poetry but it's not mathematics. Please don't insult intelligence that way.

Look at sqrt(2) in standard mathematics. It is the limit of a sequence of rationals, but it has to exist ITSELF to BE there to be the limit. We can't coherently say, "There are just the finite approximations and the sqrt(2) is the mirage (something that does not exist but only seems to exist) at the end of those approximations." No, for the sqrt(2) to be a real number, it has to exist.

Now, you can say that there are only successive finite sequences of rationals getting closer to one another progressively in succession. You could cook up a theory in which it works that way. But then you cannot say that there is a limit that is the sqrt(2).

Quoting keystone

As such, the object of study should not be the complete output of the program (which cannot be generated) but instead the program itself whose execution is potentially infinite.

Ah, so you don't want the S-B tree after all. You want instead a program that generates successive finite number of rows.

But does the tree exist for you or not? Please don't answer with yet more wiffle waffle undefined terminology. Please just say whether it exists or not.

I understand the notion of the entire tree as a "limit" of the finite sequences of accumulations of rows. But a limit in mathematics is itself an existing object. It's not a "mirage". You can't have it both ways: You can't say BOTH (1) there is not an entire tree but only finite approximations and (2) the entire tree is the limit of those approximations. To be the limit of a sequence is to already exist to be the limit. Sequences don't converge on something that doesn't itself exist.

If the paths don't exist in their entirety, then those real numbers that are supposed to be those paths don't exist, or whatever you think are supposed to be the "limit" nodes don't exist.

But let's say the object of study is the program itself. Okay, but then pray tell how do you extract from that study of the program real analysis for the mathematics for the sciences? I think there are rigorous theories (I guess they can be made fully rigorous?) that do that kind of thing. But you don't provide a clue how you would do it.

Actually, this is a huge bait and switch by you. You said that the real numbers are to be the paths in the tree. But now you don't want to have the existence of those paths, so you switch to saying "study the program". I was game for talking about the initial proposal, but now you've switched to something undefined to the point of nebulousness.

Quoting keystone

I don't mind a computer program that prints all natural numbers.

To be clear, you mean a program that at any stage generates up to the nth natural number for some n and always goes to the next stage and never halts. That is called a 'recursive enumeration'.

Quoting keystone

'we' don't actually work with the infinite sets themselves, 'we' work with the 'algorithms' used to generate them

There is mathematics formulated along those lines.

But that doesn't entail that there is inconsistency in the standard approach!

Do your thing, whatever it is, but you should lay off spreading disinformation that there is inconsistency in infinitistic mathematics.

SB is a fairly deep mathematical subject, as seen here.

Reply to jgill

It is a fascinating subject. I wish I were up to speed to absorb and appreciate that paper.

Reply to TonesInDeepFreeze

Thanks for the detailed explanation. That makes sense, I was thinking the resolution was somewhat along those lines.

Reply to TonesInDeepFreeze

Wiki puts SB in the category of number theory. Perhaps it's analytic number theory since limits appear. Apart from the simple continued fraction expansion it's beyond me. :cool:

I want to characterize all paths on the S-B tree by their destination. Infinite paths do not actually have destinations but neither does 1/x have a value at x=infinity. After all, x never is infinity and there is no row infinity on the S-B tree. But there is value in saying that 1/x approaches 0 as x approaches infinity. Similarly, there is value in saying that the path RL approaches 1 as we descend the tree and approach 'row infinity'. In both of these examples, infinity is a useful fiction. But it's important to acknowledge that it is a fiction, however you want to phrase it.

Now, given that paths (as described by infinite sequences of rational numbers) are analogous to Cauchy sequences, it doesn't seem like a big jump to say that the limit of a SB path is analogous to the limit of a Cauchy sequence, or in other words, that the limit of infinite SB paths are nodes corresponding to real numbers. But these nodes only exist at row infinity (which is a fiction) and so they too are useful fictions. Of course, it's obvious that they're fictions since, once again, infinite paths do not end at any node (in a similar way that Cauchy sequences do not end at any rational number).

As we agreed earlier, nodes are analogous to points. The issue then is that in the S-B tree, we only have nodes corresponding to the rational numbers. The nodes corresponding to the real numbers are fictional. Comparatively, with the real number line, we do not distinguish between the rational and irrational points on the real number line. They are of the same essence. This disagreement between the S-B tree and the real number line is what I'm trying to highlight.

Quoting TonesInDeepFreeze

You need to stop using the word 'inconsistency' with your own private meanings.

Granted, the branches of the S-B tree never actually intersect, but if we're trying to illustrate the real line as the limiting row of the tree, one approach we might be able to use is projective geometry. With this approach, the parallel lines of 0/1 and 1/0 (and all the branches in between) meet at a single node at infinity. So in a sense, maybe one can say that at infinity (if it were not a fiction) all of the real numbers would be equal. Or analogously, going back to the 1/x example, at x = infinity we have a singularity. Perhaps all numbers being equal is more in line with a standard notion of inconsistency. However, this is all 'handwavy'. For example, what if 0/1 and 1/0 were instead diverging lines? Anyway, until I can support arguments like this with something more concrete, I'll refrain from using the term inconsistency.



Quoting TonesInDeepFreeze

"Just remember it's not a lie if you believe it".
Look up 'lie' in Merriam-Webster.

Lie - to make an untrue statement with intent to deceive.
Perhaps cranks are deluded, but our intentions are pure.

Quoting TonesInDeepFreeze

Choose a definition and stick with it.
First you said the reals are the paths. Now you say they are nodes.

I think it is a virtue to be able to adjust one's view upon new evidence. I wanted to treat the real numbers as unending journeys along a path but you were inclined to treat them as the paths themselves. While I went along with this for a while I realized that it doesn't align with my views regarding infinity. As such, I have adjusted my view to something that seems more agreeable to both of us - limits. We can both speak of limits while in our heads, you are imagining an actually infinite object (path) and I am imaging a potentially infinite process (journey).

Quoting TonesInDeepFreeze

Your insouciance in not making definitions and sticking to them invites confusion and is annoying.

Sorry, I can see how this can frustrate someone.

Quoting TonesInDeepFreeze

If you want to have the things that are going to serve as the limits, then you need to prove they exist.

In the 1/x analogy above, does one have to prove that infinity is a 'thing' to talk about x approaching it?

Quoting TonesInDeepFreeze

To be clear, since you write ambiguously "create the entire tree". Yes, there is a program such that, for any n, the program will generate up to and including the nth row and stop. But there is no program that generates all rows and stops. I take it that you agree.

There is a difference between writing a program and executing it. I can write a program that generates all rows and stops. However, I cannot execute it. The program is no less of a program just because the output doesn't exist. The output is a fiction. The execution of the program is a potentially infinite process, the output of the program is an actually infinite object.

Quoting TonesInDeepFreeze

But, again, it entails that those limits are THEMSEVLES objects - existing.

Analogously, the output of the program doesn't have to exist for us to use it to describe the program and its execution. The [fictitious] output is used to describe the program/execution (not the other way around).

Quoting TonesInDeepFreeze

But does the tree exist for you or not? Please don't answer with yet more wiffle waffle undefined terminology. Please just say whether it exists or not.

FOR ME - The program to create the tree exists, and it can be written with finite characters. But the program cannot be executed to completion (it doesn't halt - i.e. it is potentially infinite) so the output of the program does not exist. There is no actually infinite object corresponding to the S-B tree. Nobody has ever seen the actually infinite object with their minds eye. What we see with our minds eye is the program.

Quoting TonesInDeepFreeze

But let's say the object of study is the program itself. Okay, but then pray tell how do you extract from that study of the program real analysis for the mathematics for the sciences?

I believe that reformulating calculus using limits did just that. In my view, calculus doesn't need any reformulation. Consider that I can write a program to perform Newton's method but if I'm hunting for an irrational solution it will never halt. The Cauchy sequence of intermediate rational 'approximations' describes the program/execution, not the solution. There is no solution.

Quoting TonesInDeepFreeze

Actually, this is a huge bait and switch by you. You said that the real numbers are to be the paths in the tree. But now you don't want to have the existence of those paths, so you switch to saying "study the program". I was game for talking about the initial proposal, but now you've switched to something undefined to the point of nebulousness.

This is not a bait and switch. Infinity is inseparably tied with calculus. If one denies infinite objects one has to then accept infinite processes otherwise calculus doesn't get off the ground. And the simplest way to talk about processes is to treat them as programs being executed. It is unreasonable for you to disallow programs from the discussion, especially given that computer science is so closely tied with mathematics.

EDIT: You're not disallowing programs from the discussion so my last paragraph should be retracted. Instead, perhaps I should have acknowledged that my ideas and method of communication are evolving. However, I wouldn't call this a bait and switch since I'm not trying to trick you.

Reply to keystone

Four posts to follow. Untangling your confusions and lies.

Unfortunately, it's likely that you'll reply with even more confusions and lies.

Reply to keystone

For reference, these are the approaches that have been discussed here:

STANDARD

(1) Reals are Dedekind cuts.

Advantages:

* Easy to visualize.

* 'less than', addition and multiplication have been defined.

Disadvantage:

* Unintuitive that a real is the entire set of all the rationals less than the real.

(2) Reals are equivalence classes of Cauchy sequences of rationals.

Advantages:

* Intuitive that sequences rather than entire "less than sets" are involved.

* 'less than', addition and multipication are handily defined.

Disadvantage:

* Unintuitive that a real is an entire set of equivalent Cauchy sequences of rationals.

(3) Reals are sequences of rationals.

I have heard about this, but don't know how it works as a definition or how 'less than', addition and multiplication are defined. Also, I don't know why we couldn't take reals in (0 1) to be the ordinary binary(or decimal, whatever) expansion (excluding sequences of all 1s after some index, or all 9s after some index, whatever). But since that is not one of the two standard methods - (1) or (2) - I can only guess that there's a complication in this method.

Advantage:

* Intuitive that a real is a particular sequence.

S-B TREE

(4) Reals are paths.

This seems plausible to me.

Advantage:

* Intuitive that rational reals are a finite path and irrational reals are a denumerable path.

Disadvantages:

* What about fractions not in lowest terms? I guess division would be defined in a special way. And I think we might end up with having to deploy equivalence classes of integers, which is what we do anyway in the standard definition of the rationals.

* Need to define the negative reals.

* Need to define 'less than', addition and multipication.

* Sequences of nodes are slightly more intutive (familiar) than sequences of edges.

(5) Reals are sequences of nodes.

This seems plausible to me and better than (4).

Advantage:

* Intuitive that rational reals are finite sequences and irrational reals are denumerable sequences.

* Sequences of nodes are slightly more intutive (familiar) than sequences of edges.

Disadvantages:

* What about fractions not in lowest terms? I guess that division would be defined in a special way. And I think we might end up with having to deploy equivalence classes of integers, which is what we do anyway in the standard definition of the rationals.

* Need to define the negative reals.

* Need to define 'less than', addition and multipication.

(6) Reals are limits of paths.

This is NONSENSE. "limit of path" is not defined. If paths converge to something that is a real number, then those things they converge to must already be defined and proven to exist.

No Advantage, because it is NONSENSE.

Disadvantage:

* It isNONSENSE.

(7) Reals are programs.

I think there is mathematics for this, but I don't know enough about the details - either the definition of 'is a real' or the definition of 'less than', addition and multiplication.

Advantage:

* Perhaps it's more friendly to finitism, but I don't know that infinite objects can be actually dispensed with.

Disadvantages:

* Unintuitive that numbers are programs.

* Might be very complicated to formalize.

* I guess it would only account for the computable real numbers. So it might be complicated to formalize calculus.

Reply to keystone

FALSE TELLERS

Quoting keystone

"Just remember it's not a lie if you believe it".
Look up 'lie' in Merriam-Webster.
— TonesInDeepFreeze

Lie - to make an untrue statement with intent to deceive.
Perhaps cranks are deluded, but our intentions are pure.

Now you are even lying about lying. I mentioned a dictionary definition, and instead of reporting back the whole definition, you truncated it to just the sense you like while leaving off the senses that dispute you:

The two other senses you left off:

": an untrue or inaccurate statement that may or may not be believed true by the speaker or writer

: something that misleads or deceives"

It is not uncommon for people to believe their own lies. People convince themselves of untrue claims, and over time they entrench themselves deeper and deeper into the falsehoods. And when they are confronted with refutations of those falsehoods, they respond by entrenching themselves even deeper. They intentionally ignore all clear evidence and irrefragable arguments contrary to their false beliefs so that they can continue to propagate the falsehoods. That's not just lying; it is systematic lying. And when a person makes a ludicrous claim but has no basis for it, even if they irrationally believe it, we may still call it a lie. "There are transmitters in vaccines that let the government know where you are and who you are with at all times". Even if a person believes it, it's a lie, not a mere mistake.

And you know who epitomizes all of that? Cranks.

It is wonderful for you to have just provided such an ironic example where you deceptively left off the part of the definition of the word 'lie' itself!

And if you were to reply that you didn't mean to deceive but that you forgot to read the rest of the definition or whatever, then you're still lying about the definition, because failing even a modicum of diligence to get the facts right but nevertheless spouting egregious falsehoods is another form of lying.

But anyone can make a mistake, right? Sure, of course, people get things wrong and sometimes because they were not thorough in checking the facts or whatever, and that doesn't ordinarily deserve to be called 'lying'. But when it is a pattern, over and over and over, such as with cranks, then it deserves to be said that it is lying.

So, no, cranks do not have "pure" intentions.

Quoting keystone

I think it is a virtue to be able to adjust one's view upon new evidence.

Of course. But what is annoying with you is that you don't clearly say that you've changed your plan.

First you talked of reals as paths. Then you talked of reals as "limits" of paths (though that is not defined). Then you said actually there are only programs. But the way you casually slip around among those plans makes it difficult to follow and know really at each juncture what it is you actually claim. If you are to keep changing, then it would be helpful and just the least of consideration to say something like "Previously my approach was X. But now I see that X doesn't work. So from now on my approach is Y". Otherwise, it's an undo strain to follow your continual swerves.

Quoting keystone

This is not a bait and switch.

First it is was paths, then suddenly and briefly nodes, then paths again briefly, then programs.

Quoting keystone

It is unreasonable for you to disallow programs from the discussion, especially given that computer science is so closely tied with mathematics

As you added in an edit, I don't. I don't disallow computable analysis or any other rigorous mathematics. But I also don't disallow myself from commenting in full when you're spouting bull.

Quoting keystone

I should have acknowledged that my ideas and method of communication are evolving. However, I wouldn't call this a bait and switch since I'm not trying to trick you.

Regarding moving from paths to nodes to programs, it's not a question of tricking. Rather, it's that you are so confused and sloppy that you shift from one half-baked idea to another instead of organizing a coherent proposal or even point of view. That careless shifting adds up to misdirection even if not intended. And there is your continual disinformation about mathematics, some of it so egregiously produced as to be lies.

And there is this whopper lie:

Quoting keystone

I wanted to treat the real numbers as unending journeys along a path but you were inclined to treat them as the paths themselves.

I'll address that in another post.

"INCONSISTENT"

Quoting keystone

You need to stop using the word 'inconsistency' with your own private meanings.
— TonesInDeepFreeze

Granted, the branches of the S-B tree never actually intersect [...]

That and the rest of your paragraph have nothing to do with my point, which is (AGAIN) that 'inconsistency' has an exact mathematical definition. A theory is not inconsistent because it doesn't comport with your hand waving! A theory is inconsistent if and only if it there is an S and its negation that are both theorems of the theory.

Quoting keystone

until I can support arguments like this with something more concrete, I'll refrain from using the term inconsistency.

Not just "something more concrete". You need to prove that there is a sentence such that both it and its negation are theorems of the mathematics in which both the S-B tree and the continuum exist. You can't fudge it. It's flat out the case that NOTHING else justifies claiming inconsistency.

PHILOSOPHY

Quoting keystone

There is no actually infinite object corresponding to the S-B tree. Nobody has ever seen the actually infinite object with their minds eye. What we see with our minds eye is the program.

Oh no no no, you don't get away with that.

First of all, it's ludicrous to say "we" see a program when considering the S-B tree. Who all are the people that constitute the "we" who say the tree doesn't exist but there is a program held in their mind?

Of course, EVERY mathematical object in this context is abstract. No one has seen the number 0 nor any other number. No one has ever seen a line or plane or space. No one has ever seen Euclidean space or any other space, finite or infinite.

To say what I mean by saying "exists", I merely need to say that there is such and such an existence theorem. Of course, one may have philosophical views that certain things exist independently of formalizations. But all that is required at minimum are theorems. So when I say "exist", without opining on philosophical notions, I mean at least there is an existence theorem.

Now, it is true that, at least in principle, one can picture all the members of a finite set together but not of an infinite set. Moreover, aside from capability in principle, there are finite sets that humans can't picture - take any sufficiently large finite set, not even just abstractions, but concretes. It doesn't even have to be very large. Try closing your eyes and picture seventeen similar looking but different objects as distinct and all at once.

More importantly, one can also understand the meaning of a predicate such as 'is a natural number' and then take the set of natural numbers to be the abstract mathematical object that "stands for" that property.

Anyway, this is a diversion from the mathematical subject we've been talking about. The subject is not philosophical senses of 'exist'. That's a fine subject, but nothing in it alters that when one says "the S-B tree and the continuum exist" one may reduce that to the fact that the mathematical axioms prove that statement. So, for the record, notwithstanding any other philsophical bents, when I say "X exists" I mean that, at least, the axioms prove the theorem "X exists".

Reply to keystone

DID SOMEBODY SAY 'PATHS'?

Quoting keystone

I wanted to treat the real numbers as unending journeys along a path but you were inclined to treat them as the paths themselves.

You are SUCH a liar.

To be clear, my point here is not to claim that one should be discouraged from revising one's position or proposal, but rather that you're lying that it was my idea, prior to yours, to take the reals as paths.

From the very first post in this thread:

Quoting keystone

A number on the tree can be described by the right/left turns needed to get there from the top.
For example:
R = 2/1
RL = 3/2
RLR = 5/3

If we continue down the tree with this alternating pattern RLRLRLRLRLRL... we approach the Golden Ratio.

Is there anything wrong with completing this tree and saying that the infinite digit RL[...] is the Golden Ratio?

You were asking, obviously basically rhetorically, about an irrational number being a denumerable* sequence. First you mentioned a sense in which a denumerable sequence of rationals (each rational is a node coded itself as a binary finite sequence of Rs and Ls) approaches a real, but then you asked about just taking the denumerable sequence itself to be that real. There was no mention of "journey". As you wrote it, you were suggesting taking reals to be the denumerable sequences themselves.

* Denumerability is indicated by '...', which is ordinary informal notation for a denumerable sequence.

The notion of reals being infinite sequences of S-B branching came from YOU not me.

Quoting keystone

might irrationals be all the infinite strings which do not end in R_repeated or L_repeated?

Right there, you suggested reals as infinite strings (i.e. denumerable sequences). Literally, "infinite strings" (not "journeys") is what you said.

The notion of reals being infinite sequences came from YOU not me.

https://thephilosophyforum.com/discussion/comment/803226

There I mistakenly rejected your idea. I mentioned the standard approach, which is fine to do, but also I misconstrued that you meant that an irrational could be a node. Ironically though, even though at that juncture you weren't suggesting irrationals as nodes, later you advocated for irrationals as "mirage"-like limits. In the link above, I mistakenly took you to mean nodes, and I unnecessarily explained that there are no such nodes. But later I also explained that there are no "mirage"-like limits either. Or if the real is to be taken as a limit of such a sequence, then the object that is that limit must have been proven already to exist.

The notion of reals being infinite sequences came from YOU not me.

Quoting keystone

Why can't we say that (non-repeating) infinite decimals are journeys that are described by unending processes (e.g. limits) and not 'destinations' (numbers)?

"journey" and "destination" had not been given mathematical definitions by you - you had not stated a difference between a path and a "journey" or a difference between a node and a "destination". The crux of your point could only be understood to be that a real is an infinite string as that is what you literally suggested in the previous quote.

Quoting keystone

While irrationals do not correspond to any node in your tree, they do describe a paths on that tree (from the top all the way down), no?

There it is, right there: Associating reals with paths. Now, explicitly 'paths'.

The notion of reals being paths came from YOU not me. You are lying when you say it is the reverse.

Quoting TonesInDeepFreeze

No, the paths are not real numbers. First, a path is a sequence of edges, not a sequence of nodes. Second, a sequence of nodes is not a real number. Rather the limit of the sequence is a real number.

There, I was still adhering to the standard approach, in which it turns out that every real is a limit of a sequence of rationals. So clearly I was not pressing you to say that reals are paths, which you had just done yourself anyway. I hadn't yet realized the plausibility of the notion, and I was mistakenly RESISTING that notion, not pressing you to adopt it.

You have it backwards: It was not I who first suggested reals as paths; it was YOU who first suggested it. You're lying when you say it is the reverse.

Quoting keystone

If RL[...] looks like the golden ratio and it behaves like the golden ratio, why do you not say that it is the golden ratio?

There it is again, right there: You were arguing that reals may be taken to BE paths.

The notion of reals being paths came from YOU not me. You are lying when you say it is the reverse.

Quoting keystone

I think we both agree that RL[...] corresponds to a specific path on the Stern-Brocot tree, not a node. If the algorithm treats RL[...] as the golden ratio, then it seems reasonable to say that the golden ratio (and all real numbers) are paths on the Stern-Brocot tree.

There it is again, right there in plain quotes and explicitly 'paths': "(all the real numbers) are PATHS" [emphasis added]

The notion of reals being paths came from YOU not me. You are lying when you say it is the reverse.

Quoting TonesInDeepFreeze

I take it that by 'RL', you mean the particular denumerable path.

So by that time, based on all you posted, I said I take it that you mean the reals are paths.

I didn't first suggest that idea. YOU did. You are lying that it was initially my idea and not yours.

Quoting TonesInDeepFreeze

CORRECTION:

I initially misconstrued you. I was not reading carefully enough. I made the point that no irrational is a node on the tree. That is true, but not relevant, since your point (which I failed to read correctly) is that irrationals may be certain paths (not nodes).

There I even corrected myself to align with YOU that reals are paths, as I had earlier misconstrued you to be taking reals to be nodes.

I didn't first suggest that idea. YOU did. I was even mistakenly arguing against the plausibility of it. You are lying that it was initially my idea and not yours.

Quoting keystone

The SB tree might offer something here since it appears that each real number has a single path which can correspond to a sequence of rationals [as you hinted].

Again, paths.

The notion of reals being paths came from YOU not me. You are lying when you say it is the reverse.

Quoting keystone

I think it's clear that every decimal number can be captured by a SB string (of L's and R's) but that is no proof.

Again, paths.

The notion of reals being paths came from YOU not me. You are lying when you say it is the reverse.

Quoting keystone

Every path (whether finite or infinite) leads to a different number. Finite paths lead to rational numbers. Infinite paths lead to irrational numbers. Or I think you'd be more comfortable saying that infinite paths that don't end in R or L lead to irrational numbers. I think of the limit of the tree as the real number line as depicted here:

There you slipped to undefined "leads to". And "limit of the tree", while there is no such thing defined in this discussion.

Quoting keystone

what is a point if not a node on the tree? Sqrt(2) does not converge towards any node on the tree. However, it appears to converge to a node that exists at 'row infinity'. Of course, there is no 'row infinity' which is why I relate it to a mirage.

Right there you switched from saying a real is a path to saying a real is a node at "row infinity" which is a "mirage".

First, as I've explained in spades, that's nonsense.

Second, this is where it would have helped if you had clearly said, "I am abandoning reals as paths and I am switching to reals as mirage nodes."

Quoting TonesInDeepFreeze

If we are redefining 'is a real number' as 'is a path in S-B' (I would prefer 'is a sequence of nodes on a path in S-B')

Again, I was following YOUR view that the reals are paths, as it was not clear to me that you had actually abandoned that proposal.

Quoting keystone

I think it's clear that every decimal number can be captured by a SB string (of L's and R's) but that is no proof.

'string' again. If not a paths, then what other sequences? Later, I gave reasons why sequences of nodes would be better, but, if I recall correctly, you ignored that, so I kept on with paths. So, by that time you were suggesting two proposals at the same time:

reals are paths
reals are "mirage" nodes that the paths converge to

The first notion is plausible.
The second notion is nonsense.

In any case a path is not a "mirage" node, so it is incoherent to suggest that reals are both paths and "mirage" nodes.

Quoting keystone

Every path (whether finite or infinite) leads to a different number. Finite paths lead to rational numbers. Infinite paths lead to irrational numbers. Or I think you'd be more comfortable saying that infinite paths that don't end in R or L lead to irrational numbers. I think of the limit of the tree as the real number line as depicted here:

No, I am UNcomfortable with that. I don't use the word 'leads'. It is not mathematically defined here.

And you need to mathematically define what you mean by the continuum being a "limit" of the tree, and prove that it is.

Quoting keystone

As we've informally agreed, irrationals (e.g. sqrt(2)) are unending paths on the SB tree.

Putting aside in what sense this is formal or not, there you again explicitly said that reals are paths. LITERALLY "irrationals are unending paths".

Quoting keystone

With this view, the objects of concern are the nodes (not the paths).

So it was paths, then "mirage" nodes, then paths again, then nodes again.

Quoting keystone

Can we say that the rational/irrational numbers are the limits of their corresponding paths.

"limit" is UNDEFINED by you in this context. I guess you're still talking about your nonsense "mirages".

Quoting keystone

As such, the object of study should not be the complete output of the program (which cannot be generated) but instead the program itself whose execution is potentially infinite.

And there you moved from reals being limits (after they were previously paths) to reals being programs.

So you started by taking reals as paths, and I eventually went along, then very recently you quickly and briefly switched to nodes. Then paths again, and now they're not paths or nodes but programs. Then you lie by saying it was my notion, not yours, that reals are paths.

You are such a piece of work.

Reply to keystone

VARIOUS

Quoting keystone

Infinite paths do not actually have destinations but neither does 1/x have a value at x=infinity.

There are two different notations:

(1)

lim 1/x [x = 1 to inf] = 0

There 'inf' does not stand for an object named 'inf'. Rather, it is an informal placeholder as we may replace it with a formulation that does not invoke 'inf':

Let f = { | x is a natural number greater than 0}.

lim f = 0

No mention of 'inf' there. Saying "to inf" is merely a figure of speech and does not imply that there is an object named "inf".

(2)

An extended system where there are the objects -inf and +inf.

Note that those may be any objects other than real numbers. They don't have to actually be infinite sets.

Ordinarily the stipulative (and I stress 'stipulative') definition of the division operation yields:

1/inf = 0

Quoting keystone

But there is value in saying that 1/x approaches 0 as x approaches infinity.

We don't just "say" it. We prove it.

Quoting keystone

there is value in saying that the path R L[...] approaches 1 as we descend the tree and approach 'row infinity'.

(1) You've cited two different versions of the S-B tree. Please choose one and stick with it.

In one version, 1/1 is the initial node and it is not obtained by any path. In the other version, 0/1 and 1/0 are the initial nodes, and 1/1 is the sole rational that is obtained twice by two different path, viz. R from 0/1 and L from 1/0.

In any case, 1/1 is not reached by by any finite or denumerable sequence R L[...].

And again, please choose one of the two versions and stick with it.

(2) Please stop saying "row infinity". There IS NO row infinity. We PROVE that there is no row infinity. It is not coherent mathematics to keep saying "row infinity" no matter that you put single scare quotes around it or call it a "mirage" or "fiction".

Quoting keystone

In both of these examples, infinity is a useful fiction.

Wrong. In the case of standard mathematics, we PROVE the existence of whatever objects we use. But in your case, you just hand wave that somehow there are "fictions" not defined even as abstract mathematical objects that explain your arbitrary claims.

Quoting keystone

however you want to phrase [the notion of "row infinity"].

I don't phrase it in any way. Because I don't have the intellectual mathematical dishonesty of trying to get by with nebulous undefined hand waving terminology.

Quoting keystone

paths (as described by infinite sequences of rational numbers)

Just to be clear, I still don't know whether you understand that paths are not sequences of nodes.

Quoting keystone

it doesn't seem like a big jump to say that the limit of a SB path is analogous to the limit of a Cauchy sequence, or in other words, that the limit of infinite SB paths are nodes corresponding to real numbers.

For crying out loud, I gave you DETAILED explanation why that is incoherent.

One more time: With Cauchy sequences, there IS an object that is the limit. But with your bull, there is no object (except you preposterously and egregiously hand wave that these objects are "mirages" or "fictions").

I gave the example of sqrt(2), but you SKIPPED.

AGAIN, you can't just take a sequence and say that there is a limit; it's not enough to say that the terms of the sequence get closer to each the next - you have to PROVE that there is an object such that the terms get arbitrarily close to it.

When you just say that there is a "limit" of the sequence of paths (or now it's nodes, you keep switching). You have to PROVE that there is such a limit.

In the rational numbers, there is no limit to the sequence 1, 1.4, 1.41, 1.414, 1.4142 ... But we prove that in the reals there is a limit, viz. the least upper bound of the range of the sequence. And that least upper bound is the square root of 2.

That is, we prove that, in the reals, every bounded set has a least upper bound. {x | x^2 < 2} is bounded. So it has a least upper bound. Then we prove that the least upper bound of {x | x^2 < 2} squared is 2, thus the least upper bound of {x | x^2 < 2} = sqrt(2) . And the sequence of rationals that are approximations converges to the limit, which is sqrt(2).

You can't just say, "The terms of the sequence get closer, so PRESTO POOF OF MAGIC, there's this mystical, fictional "mirage" thing that I say is the limit!" You have to PROVE that there is such a limit.

The bottom line being that if your magical mystical mirages are merely fictions, not proven mathematical objects, then you don't have any math about them!

I don't know how I can make this any clearer for you.

Quoting keystone

it doesn't seem like a big jump to say that the limit of a SB path is analogous to the limit of a Cauchy sequence

It is a HUGE jump, because 'limit of a Cauchy sequence' has an exact mathematical definition, while 'limit of a path' in this context is undefined and your notions about it are INCOHERENT. See above

Quoting keystone

infinite paths do not end at any node (in a similar way that Cauchy sequences do not end at any rational number

There is a similarity only if we ignore, as you do, the decisive difference that the Cauchy sequences DO converge to objects that we DO prove to exist, as we don't rely on magic wand waving as you do.

And you need to drop the word "end". The Cauchy sequences do not end at a limit. The sequences do not have an end. Rather they CONVERGE. Every time you say things like "end" you only entrench in your mind a basic misconception and then add it to the landfill of misinformation on the Internet.

Quoting keystone

The issue then is that in the S-B tree, we only have nodes corresponding to the rational numbers.

First you said the rationals are the finite paths. In that case each node is the terminal node on a finite path. So a rational is a finite path and has its corresponding terminal node.

Quoting keystone

The nodes corresponding to the real numbers are fictional.

Your "fictional" is meaningless. Anyone can say, "Here are my objects that are solutions. They're fictional objects that I made up. I like them. So there ... my math! By the way, I have an answer to Goldbach's conjecture! There is an even number that is not the sum of two primes. It's a fictional number I made up. I call it 'the Emerald number'. Behold my math!" It's bull.

Quoting keystone

Comparatively, with the real number line, we do not distinguish between the rational and irrational points on the real number line. They are of the same essence.

Oh, yes, "essence". Yeah right, everyone knows what an essence is in mathematics.

Meanwhile, I'll inform you:

A rigorous formulation is that rational numbers are equivalence classes of integers. Then real numbers are equivalence classes of Cauchy sequences of rational numbers. So, strictly speaking, the set of rationals is disjoint from the set of reals. But there is an embedding of the rationals in the reals. So, in another sense of 'rationals', we take the rationals to be the image of that embedding.

Why is it important to correct you on this? Because garbargy undefined rubrics like 'essence', 'mirage', etc. only invite fallacious inferences based not on math but on mere suggestibility.

Quoting keystone

This disagreement between the S-B tree and the real number line is what I'm trying to highlight.

You're only highlighting your ignorance, confusion and dishonesty.

I've explained more than once that there is NO "disagreement" between the S-B tree and the continuum. They are both proven to exist from the mathematical axioms without inconsistency.

Indeed, if we countenance a proposal to take real numbers as the paths in the S-B tree, then presumably we get a number system isomorphic with the standard treatment. There's not a quandary about this.

Quoting keystone

I can write a program that generates all rows and stops

No, you can't.

Quoting keystone

However, I cannot execute it. The program is no less of a program just because the output doesn't exist.

If someone says I "wrote a program to do X but it doesn't do X", I wouldn't know what that would mean.

Ordinarily, by "program that does X" we mean that it does X.

Quoting keystone

The [fictitious] output is used to describe the program/execution (not the other way around).

There you go again with your "fictitious" stuff. It's bull.

A program is not specified by an outcome. A program is specified by instructions, and those instructions entail outcomes upon inputs. One such algorithm provides for printing natural numbers increasing in size and never stopping. But there is no algorithm that provides for printing every one of the natural numbers and then stopping. That is, there is a program such that for any natural number, the output will write that natural number; but there is no program that will finish writing all the natural numbers. That is, the set of natural numbers is computably enumerable but it is not finitely enumerable.

PS. As I mentioned, in my first few posts I mistook your initial proposal. In those posts I have now added edits in brackets, and marked as 'Edit', to note my mistake.

@TonesInDeepFreeze

As always, I appreciate your detailed responses.

I believe in what I'm saying but IF it's untrue then you're right - according to Merriam-Webster, my statements would be considered lies. I don't want to argue with you about my intentions. I was wrong to claim that you came up with the path definition. I will not claim something is 'inconsistent' unless I can prove that both a statement and its negations are theorems of the mathematics of concern. Sorry. Overall, I've been sloppy with my communication. I intend to be more clear. I ask that we set aside my earlier comments in this thread and I'll start afresh to better capture my position as I respond to your last set of messages. I'll need some time to digest all of your comments and formulate my response.

Reply to keystone

Probably, the idea of reals as programs has been written about. I wonder whether it's covered in the subject of computable analysis.

And it seems interesting to apply the notion of programs to the S-B tree. I can see the basic idea: Each program would generate sequences of Rs and Ls. A real number would be a program. For a rational real, execution of the program would halt. For an irrational real, execution of the program would not halt.

But we have to keep in mind that this only accounts for the computable reals. So we'd have to explain how to formalize calculus with only the computable reals. I think that has been written about, though I don't know enough about it.

Consider the follow Python function:

def endless_loop()

	while True:

	    print("Looping indefinitely...")

	Return 1

Do you consider this a valid function? It certainly obeys Python syntax rules and can be interpreted by a computer program. Does it make sense to say that it is designed to return 1 even though it never actually could? I see in your recent post that you can see the basic idea of reals being programs so perhaps these questions don't need answering.

THE EXTENDED S-B TREE

Moving forward, let's use this extended version of the S-B tree so that all real numbers are covered.

• At every row we have a blue line corresponding to the real number line.
• The white nodes indicate that the blue lines corresponds to the open interval (-inf,+inf).
• One can travel from the 0/1 node to any rational node in the tree by a unique sequence of left and right turns as described by S-B strings. Some examples of S-B strings are RL (1/2) and LL (-2/1).
• There are two ways to interpret infinite journeys: (1) as completed processes, whereby they correspond to the entire path travelled and (2) as incomplete unending processes. I prefer the latter but until it becomes an issue, let's use journey ambiguously without saying whether it is completed or not.
• Row infinity does not exist. From now on, when I say 'approaching row infinity' it will be an informal placeholder corresponding to an unending journey down the tree.
• I have an issue with infinite sets but I see value in the concept of equivalence classes. Instead of discussing what an equivalence class might mean in the absence of infinite sets, please grant me slack to use this concept loosely. While each node only contains the reduced form of a rational number, it actually correspond to an equivalence class of integer pairs.
• As we progress down the tree, nodes are placed directly on top of their corresponding point on the real number line. For example, the node corresponding to 2/3 lies directly on top of the point corresponding to 2/3. Although the nodes are illustrated as large circles, let's imagine that they are actually points. So while the blue line gets hidden from the illustration as the nodes become more numerous, let's imagine that the real number line is still present in its entirety at every row.

Quoting TonesInDeepFreeze

Who all are the people that constitute the "we" who say the tree doesn't exist but there is a program held in their mind?

When I imagine the Stern-Brocot tree, I visualize the first few rows of the tree and then internally mutter 'and so on as governed by the rules for constructing it'. That 'and so on' is not a formal program, but it refers to an algorithm which we can fit into our finite brains and be comprehended.

Quoting TonesInDeepFreeze

AGAIN, you can't just take a sequence and say that there is a limit; it's not enough to say that the terms of the sequence get closer to each the next - you have to PROVE that there is an object such that the terms get arbitrarily close to it.

With the real number line now embedded in the tree, hopefully it is reasonable to say that as we approach row infinity, the journey corresponding to RRL converges to the point corresponding to the golden ratio.

Quoting TonesInDeepFreeze

So when I say "exist", without opining on philosophical notions, I mean at least there is an existence theorem.

Given this, I'll stop using 'exists' since my view of existence extends beyond mathematics. I know you don't like it when I make up new terms, but hopefully you'll give me slack here as I'm trying not to hijack 'exists'. Instead, I'll use 'actualized' for what I mean. In my view, an object is actualized if it is present in the memory of a 'computer'. I am actualized because I am present in the 'computer of the universe'. I'm thinking of a purple cow so that purple cow is actualized because it is present in the 'computer of my mind'. This statement is actualized as I type because it is present in the memory of my laptop. When a memory of an object is flushed, it becomes 'potentialized'.

Quoting TonesInDeepFreeze

But we have to keep in mind that this covers only the computable reals. So we'd have to explain how to formalize calculus with only the computable reals.

The infinite decimal corresponding to pi is forever potentialized.
Leibniz's formula for pi/4 is currently actualized because it is present in my mind.
If I understand correctly, non-computable reals cannot be present in the memory of any computer so they are forever potentialized.

Let's say we color the necessarily potentialized points on the real number line red and all possible actualized points green. Given that the computable reals are countable, they have measure 0 so the real number line is entirely red. But we still have the line and can perform calculus as usual. I don't understand why we would need to adjust anything in our formalization of calculus.

@TonesInDeepFreeze

I just realized I incorrectly described the black lines in the S-B tree image. They do not correspond to paths from 0/1 to all the rational numbers. They link any node to its two parent nodes. Doh! It was a while ago when I made that figure...

You still systematically resort to sophistry. But I will spend some of my finite supply of patience anyway, though I shouldn't.

First, do you agree that any program you can write can in principle be stated as a Turing machine?

Quoting keystone

At every row we have a blue line corresponding to the real number line.

At every row you have a finite set of rational numbers ordered by 'less than'. That's not the real number line.

You say "corresponding to the real number line". There you go again, invoking an infinitistic object (the real number line) that you claim doesn't exist. That's a cheat. You want to develop calculus without infinite sets. So you can't then smuggle in infinite sets. You can't say that something corresponds to something else that you yourself claim does not exist! Please, don't continue to insult my intelligence.

Quoting keystone

Row infinity does not exist. From now on, when I say 'approaching row infinity' it will be an informal placeholder corresponding to an unending journey down the tree.

(1) I am wary that you'll smuggle in the existence of a "row infinity" anyway. It would be more clear to just stay away from the threat of such sophistical pits and instead just use defined terminology.

(2) "unending journey". Back to undefined terminology again.

(3) There is no "tree" in this context (in the sense of all the tree).

(4) So you are taking "approaching row infinity" as informal for "unending journey down the tree", so an informal rubric to stand for another informal rubric. As they say in show business, the jokes write themselves.

Quoting keystone

I have an issue with infinite sets but I see value in the concept of equivalence classes. Instead of discussing what an equivalence class might mean in the absence of infinite sets, please grant me slack to use this concept loosely. While each node only contains the reduced form of a rational number, it actually correspond to an equivalence class of integer pairs.

In other words, "I don't accept that there are infinite sets, but I'm going use infinite sets anyway".

No, I don't grant you that "slack". The very point is that you propose to do calculus without infinite sets. If you're going to use infinite sets anyway, the all bets are off. You're wasting our time.

Quoting keystone

As we progress down the tree

No, in your context, there is no tree. There is only a program. That program never outputs what you are calling "the tree". Even as the program runs with no upper bound of steps, it never outputs what you are calling "the tree". Only, at any given step, the program outputs a finite number of rows. Please stop trying to have your cake and eat it too.

Quoting keystone

let's imagine that the real number line is still present in its entirety at every row.

In other words, let's imagine a contradiction. Let's imagine that the real line does not exist but that the real line exists.

Quoting keystone

Who all are the people that constitute the "we" who say the tree doesn't exist but there is a program held in their mind?
— TonesInDeepFreeze

When I imagine the Stern-Brocot tree, I visualize the first few rows of the tree and then internally mutter 'and so on as governed by the rules for constructing it'. That 'and so on' is not a formal program, but it refers to an algorithm which we can fit into our finite brains and be comprehended.

(1) You believe the tree does not exist. Yet, earlier in your post, you refer to "the tree", so one must think you are referring to something that does exist.

You need to state some axioms, then definitions and proofs. Otherwise, you're doomed to perpetually post nonsense.

(2) I asked who are the "we". Your reply is that it's you.

Quoting keystone

With the real number line now embedded in the tree, hopefully it is reasonable to say that as we approach row infinity, the journey corresponding to RRL converges to the point corresponding to the golden ratio.

What an obnoxious mess that is.

(1) There is no real line in this context. There's only a program that does not output the real line but only successive finite proper subsets of rationals.

(2) "The tree" (as in the full tree) does not exist in this context. Only, at any given stage of output, finitely many rows.

(3) The real line is not "embedded" in the tree in this context. Moreover, 'embedded' has a mathematical definition. But you haven't defined your own sense.

And even in standard mathematics, one would have to say in what way the real line is "embedded" in the three. The real line is the set of points of the form where x is a real number, along with the less than ordering. The S-B tree (or your variant) do not have a row that is the real line. And no row has irrational nodes anyway! But, of course, we do prove the existence and define 'the real line' irrespective of the S-B tree anyway.

You propose a program, to avoid committing to infinite objects. Fine. But then you invoke infinite objects anyway. That is stupid and childish.

(4) You said you use 'row infinity' in this sense

"when I say 'approaching row infinity' it will be an informal placeholder corresponding to an unending journey down the tree."

So let's plug it in and see what we get:

as we unending[ly] journey down the tree, the journey corresponding to R RL[...] converges to the point corresponding to the golden ratio.

Aside from, 'journey' undefined, and there being no full tree, and convergence in this context not being defined:

There IS NO point in the tree corresponding to Phi. We've been through this already!

Now we are back to where we were near the beginning of this thread!

Your idea of reals (the computable reals, I think) being programs might be okay. But your latest post does not contribute a coherent extension of the idea nor much hint of making it rigorous. Your latest arguments sputter at the level of nonsense from the start.

Quoting keystone

I'll use 'actualized' for what I mean. In my view, an object is actualized if it is present in the memory of a 'computer'.

That's too vague for me to take as mathematics. Let me know if you ever get around to moving past word poetic metaphors and instead present some actual mathematics.

Quoting keystone

Let's say we color the potentialized points on the real number line red and the actualized points green.

Better yet, howzabout you give mathematical definitions for 'potentialized' and 'actualized".

Quoting keystone

I don't understand why we would need to adjust anything in our formalization of calculus.

Because the standard formalization uses infinite sets. Meanwhile, you don't have formalization.

Get a book on mathematics, why don't you?

Reply to keystone

Among all the nonsense you posted, this is a good item to especially highlight:

Quoting keystone

as we approach row infinity, the journey corresponding to RRL converges to the point corresponding to the golden ratio.

Now we are back to where we were near the beginning of this thread!

Aside from, 'journey' undefined, and there being no full tree, and convergence in this context not being defined: There IS NO point in the tree corresponding to Phi.

After I've posted so much to explain to you why your notes are nonsense, you regress back to your original fallacy!

You are a consummate crank: You cop out with undefined terminology. You assert contradictions but seem not to realize it. And most of all, the very essence of a crank: You SKIP refutations that have been given you and just go back to reasserting what has been refuted as if it was not refuted, not even RECOGNIZING the arguments that refute you, thus putting the conversation in a circle.

A little knowledge (from Wikipedia) can be a dangerous thing. :roll:

Quoting TonesInDeepFreeze

There you go again, invoking an infinitistic object (the real number line) that you claim doesn't exist.

So that I can respond it will be helpful if you clarify your position on programs by answering these questions I asked in my earlier message to you.

Quoting keystone

Consider the follow Python function:

def endless_loop()

	while True:

	    print("Looping indefinitely...")

	Return 1

Do you consider this a valid function? It certainly obeys Python syntax rules and can be interpreted by a computer program. Does it make sense to say that it is designed to return 1 even though it never actually could?

Quoting keystone

Do you consider this a valid function?

I like this better:

(1) Print "Hello". Go to (2).
(2) Go to (1).
(3) Print "Goodbye". Halt.


That's an algorithm. The execution of it successively prints "Hello", and it never prints "Goodbye", and it never halts.

Algorithms may have an instruction that is never executed because executions always loop ahead of the instruction.

That's different from saying that an algorithm executes infinitely but then finally provides an output after executing infinitely.

An algorithm outputs successive finite approximations of Phi. But it never outputs the infinite expansion of Phi. That would be a supertask, while you disavow that there are supertasks! You are again trying to have your cake and eat it too. Grow up. And maybe have the courtesy not to keep looping back to an already refuted starting point in this discussion, which is to say, to not exercise your right as a crank to SKIP whatever refutes you.

Algorithm to emulate conversation between keystone and TonesInDeepFreeze:

(1) Print "keystone says: Phi is a node in the tree." Go to (2).
(2) Print "TonesInDeepFreeze says: Phi is not a node in the tree, because first, in your proposal there is not an infinite tree, and second, even in standard mathematics, there are no nodes in that tree that are irrational numbers, and third, there is no "row infinity" and so no "row infinity" that has nodes for Phi or any other irrational number, and fourth, paths don't "converge" to points the way you incoherently imagine they do." Go to (1).
(3) Print "keystone says: I see now. I should learn some mathematics." Halt.

Instruction (3) is in the program, but never executed.

You said reals are programs. Fine. As far as I can tell, those are computable reals. Fine. So go ahead and write a template for programs that are reals. Then define 'less than', addition and multiplication of programs. Then show how to do calculus with this. But, meanwhile, looping back to junctures that have already been refuted is stupid and obnoxious.

Quoting jgill

A little knowledge (from Wikipedia) can be a dangerous thing.

Oh, you said it so well.

Quoting keystone

the journey corresponding to RRL converges to the point corresponding to the golden ratio.

That is egregious.

First you said reals are paths. And that is plausible.

Then you said reals are not paths but they're nodes. That is absurd since there are no nodes for irrationals.

Then you said reals are nodes that are in "row infinity". That is absurd since there is no "row infinity".

Then you said reals are programs. That is plausible, and I mentioned that there might be something like that in the mathematical literature.

Then you said you'd clean up your act and reply with something more clear.

Then you come back only to circle back again to saying reals are (or "correspond to", whatever that means) nodes in "row infinity".

Like I said, bait and switch. But with you, it's bait and switch and switchbait and switch again back to one of the earlier switches. In another thread I learned a term "Motte and Bailey fallacy", for when someone can't defend their position so they switch to an easier position to defend but not acknowledging that they've switched. But you actually switch back to the earlier impossible to defend position!

Bait: reals are paths
Switch: reals are nodes in "row infinity" as those nodes are converged on
SwitchBait: reals are programs
Switch: reals are nodes in "row infinity" as those nodes are converged on

So you SKIPPED all the explanation I gave you why the notion of "row infinity" is nonsense. And you SKIPPED extensive explanation why it is question begging to say that such nodes are the limits (per convergence) of sequences: For x to be a limit of a sequence, x has to exist to even be a limit. It is nonsense to say that a sequence converges to a point Phi when you haven't proven that there IS such a point. I've told you about that over and over, but you blithely and dishonestly SKIP it.

Then the piece de resistance:

Quoting keystone

I'll use 'actualized' for what I mean. In my view, an object is actualized if it is present in the memory of a 'computer'. I am actualized because I am present in the 'computer of the universe'. I'm thinking of a purple cow so that purple cow is actualized because it is present in the 'computer of my mind'. This statement is actualized as I type because it is present in the memory of my laptop. When a memory of an object is flushed, it becomes 'potentialized'.

That is not progress when the subject is a mathematical account of reals as programs.

Let's contrast with mathematics:

We have axioms and definitional axioms.

We provide an algorithm by which to determine whether a given string is or is not an axiom. We provide an algorithm to determine whether a given sequence of strings is or is not a proof from the axioms.

We provide an algorithm to determine whether a given formula is or is not a definitional axiom.

By asserting 'there exists an x such that blah blah about x' we at least mean that we have a proof of that formalized as a theorem.

We define 'is a natural number'.

We prove that there exist natural numbers.

We prove that there exists {n | n is a natural number}.

We prove that there exists an equivalence relation on the natural numbers to form the set of integers as a set of certain equivalence classes of ordered pairs of natural numbers.

We prove that there exists an equivalence relation on the integers to form the set of rationals as a set of certain equivalence classes on ordered pairs of integers.

We define 'is a Cauchy sequence of rationals'.

We prove there exists an equivalence relation on the Cauchy sequences of rationals to form the set of reals as a set of certain equivalence classes on the Cauchy sequences of rationals.

We define 'less than', addition and multiplication on the the reals.

We define 'is a complete ordered field'.

We prove that is a complete ordered field.

We prove that all complete ordered fields are isomorphic with one another.

We prove the theorems and provide definitions used in real analysis (including calculus and mathematics for the sciences).

Mic drop.

But, sincerely, I do like your new version of the tree, accounting for the negative rational numbers. Also, in context of standard math, I like the elegance of Phi generated by such a simple instruction (I'm taking your word for it that that sequence of rationals does converge to Phi, and I'm assuming we can prove that).

Quoting TonesInDeepFreeze

Do you consider this a valid function?
— keystone

I haven't the foggiest.

def endless_loop()

	while True:

	    print("Looping indefinitely...")

	Return 1

I do plan to respond to all of your comments (it may take some time for me to collect my thoughts) but please allow me to dwell on this point some more as your response will help me compose a response to your earlier comments. In case you 'haven't the foggiest' simply because you're not familiar with Python, what this function is designed to do is loop indefinitely (printing the string 'Looping indefinitely...' with each iteration) and at the end return (i.e. output) 1.

This function follows all of the required Python syntax and so it can be interpreted and run. It is a valid program.

What I want is your view on whether it makes any sense to say what this function returns. I want to distinguish between the design of the code and the execution. The code can never be executed to completion so it can never actually return 1. However, as described above the code is designed to return 1.

I would phrase this by saying that the output of the function is potentially 1, but it is never actually 1.

What do you think?

I'm not looking forward to a response in which I would read you compounding your confusions due to ignorance of the basic mathematics and dodges to fancifulness instead while SKIPPING the decisive refutations I provide you.

/

"I haven't the foggiest" - TonesInDeepFreeze

I revised the above response, since actually I did know what you were driving at. My revised response:

Quoting TonesInDeepFreeze

(1) Print "Hello". Go to (2).
(2) Go to (1).
(3) Print "Goodbye". Halt.

That's an algorithm. The execution of it successively prints "Hello", and it never prints "Goodbye", and it never halts.

Algorithms may have an instruction that is never executed [...]

/

Quoting keystone

What I want is your view on whether it makes any sense to say what this function returns.

I'm not interested in evaluating computer code. I do know what you have in mind. It's more readable (does not require that anyone know the particulars of a particular computer language) the way I've described it.

The algorithm never halts. It prints "Hello" over and over again. It never prints "Goodbye".

Quoting keystone

as described above the code is designed to return 1.

"Designed to" is not defined. Without a definition, a reasonable sense would be psychological: what is the intention of the programmer, which is mental. That's not mathematics.

So, not to yet again detour through such subjectivity, we merely observe an objective and mathematical fact: There is an instruction in the algorithm that is never executed.

Quoting keystone

I would phrase this by saying that the output of the function is potentially 1, but it is never actually 1.

Definitions:

T is a potential output per an algorithm G if and only there is an instruction in G to print T but execution of G does not print T.

T is an actual output per an algorithm G if and only execution of G prints T.

We don't need to say 'actual output' as we can just say 'output'.

But now we're YET AGAIN back to the same mistake you keep making:

For Phi to be a potential output, Phi has to first exist. Your confused fanciful musings claim Phi is a potential output, but you don't first prove existence of Phi to have Phi as a potential output.

But you revert to an analogy with '1' as a potential output.

In context of your finitist proposal, the number 1 (or "Goodbye") already exists to be a potential outupt, but Phi (or an infinite string of "Peach Desk Happiness Sink Rain Courage [...]") does not already exist. Your analogy is clearly inapt.

As in other threads, you just keep rephrasing your most basic misconception. You shift from one imaginary picture to another in which, somehow, your own mentation allows impossible things. The imaginary pictures are not mathematics, no matter that you're able to shift from one to another after another.

The basic misconception in this thread is the same as in the other threads, such as with Thompson's lamp. In Thompson's lamp, it's an impossible situation. There is no final state in the infinite sequence of states, and the sequence does not converge to a limit state. Here, there is no final row in the tree, but there is a sequence that converges to the limit Phi, but we must FIRST have the existence of Phi for it to be the limit. And with "Hello Goodbye", successive finite strings of "Hello"s are printed, but never an infinite string of them, and "Goodbye", though it exists, is never printed. "Goodbye" is NOT like a limit there.

Reply to keystone

I asked you a question:

Do you agree that in principle any algorithm can be stated as a Turing machine?

Please answer that.

The set theoretic formulation of Turing machines is the lingua franca of computability. We don't need to evaluate particular computer code in particular computer languages that discussants and readers of the thread might not know. Moreso, for simple algorithms, we can convey the ideas in plain English such as:

Quoting TonesInDeepFreeze

(1) Print "Hello". Go to (2).
(2) Go to (1).
(3) Print "Goodbye". Halt.

But if you do need a real formalization, then I suggest Turning machines, which is common ground in computability.

And I would like your answer whether or not you understand this:

"[...] the limit of infinite SB paths are nodes corresponding to real numbers."
— keystone

In your proposal, you don't prove that there IS such limit. [Saying it is "potential" but not "actual" is just more hand waving by you; or, in the sense we can mathematically define 'potential' and 'actual', they still don't obviate that you have not proved that there is such a limit.]

In contrast with your mental pictures, in mathematics with Cauchy sequences, there we DO prove that there is an object that is the limit.

You can't just hand wave to a say that there is a limit; it's not enough to say that the terms of the sequence get closer to each the next - you have to PROVE that there is an object such that the terms get arbitrarily close to it.

Consider sqrt(2). In the rational numbers, there is no limit to the sequence 1, 1.4, 1.41, 1.414, 1.4142 ... But we prove that in the reals there is a limit, viz. the least upper bound of the range of the sequence. And that least upper bound is the square root of 2.

That is, we prove that, in the reals, every bounded set has a least upper bound. {x | x^2 < 2} is bounded. So it has a least upper bound. Then we prove that the least upper bound of {x | x^2 < 2} squared is 2, thus the least upper bound of {x | x^2 < 2} = sqrt(2) . And the sequence of rationals that are approximations converges to the limit, which is sqrt(2).

You can't just say, "The terms of the sequence get closer, so PRESTO POOF OF MAGIC, there's this mystical, fictional "mirage" thing that I say is the limit!" You have to PROVE that there is such a limit.

Reply to keystone

You are asking basic questions that concern the topic of "Denotational Semantics", which use partially ordered sets (more specifically, Scott Domains) to denote partial states of evaluation with respect to the computation of a term such as a number. Terms of any type are represented as having a totally undefined value prior to evaluation, a partially defined value during the course of evaluation, and in the case of finite terms that can be fully evaluated, a totally defined value after evaluation known as a "normal form".

In denotational semantics, the Type corresponding to 'Computable real numbers' refers to the set of fix-point equations that if iteratively applied on a given rational number, generates a sequence of prefixes that are Cauchy convergent. To obtain an extensional value for a term of 'computable real number type' requires iteratively evaluating the term and then terminating the iterative evaluation abruptly after an arbitrary number of finite iterations, to produce a finite prefix representing a rational number that is very misleadingly said , to "approximate" the real number concerned (it is misleading since we are comparing apples, namely fix-point equations that are defined intensionally in terms of equations and that refer to types, to oranges that are observable states of computation that refer to terms.

A question remains as to who gets to decide when to terminate the iterative evaluation : the interpreter/compiler, or the user of the program? In programming languages with strict semantics, their respective interpreters and compilers always evaluate the term of every type to the fullest extent possible, meaning that real numbers cannot exist as types in such languages, since their terms have no "normal form" and would cause programs to loop endlessly if evaluated. In such languages, real number constants tend to be denoted by rational numbers with a priori fixed values decided at compile time.

By contrast, in a language with lazy semantics such as Haskell, terms can be used and passed around in partially evaluated form. This means that real numbers can exist in the sense of partially-evaluated "infinite lists" consisting of an evaluated prefix and an unevaluated tail. These lazy languages allow runtime conditions to decide what rational value is used in place of a term of real-number type, which is allowed to vary during the course of computation and which corresponds more closely to the notion of "potential infinity".

I'll take your word for it that R RL[...] yield successively more accurate approximations of Phi.

Algorithm [named 'Chasing mirage-Phi']

(1) X = R. Go to (2).
(2) X = XR. Go to (3).
(3) X = XL. Go to (2).
(4) X = mirage-Phi. [whatever that is!]

In keystoneland is mirage-Phi an infinite sequence RRLRLRL... ? No, because in keystoneland, there are no infinite sequences.

In keystoneland, is mirage-Phi a "potential" output? It would be, if it were something that exists in keystoneland.

Lesson: Of course an algorithm may have an instruction that is never executed. But if the instruction mentions an object as an output, then that object has to exist, whether as an output or a "potential" output.

Please, no more or your handwaving! You don't have a coherent proposal. If you are sincere, then state axioms, rules of inference, definitions, and prove the theorems that justify a logical construction.

@TonesInDeepFreeze

You raise some good criticisms. I have reformulated my proposal to address them.

The object that I want to talk about moving forward is not the actually infinite S-B tree (which you're right, I don't believe in), but instead the blue line as illustrated below.



Don't think of this figure as depicting an infinite collection of blue lines, but instead as a single line, and each row describes a finite set of cuts which can be performed on the line. Although I'm not talking about the S-B tree, the cuts are closely linked with the S-B tree, which I'll call the S-B algorithm moving forward to avoid any confusion on my philosophy).

This line is not the real number line (which is composed of uncountably infinite points). On the very top row we have segment [] as described by the interval (-inf,+inf). There are no points on this line. I recognize that intervals across continua are typically associated with the infinite set of points between them so I'm not using the term 'interval' as it is typically used. To avoid any confusion, moving forward, in this context I will use pseudo-interval instead of interval.

In row 0, the line is cut into two segments:
(1) segment L corresponding to the pseudo-interval (-inf,0)
(2) segment R corresponding to the pseudo-interval (0,+inf)

With each cut, a point is introduced on the line corresponding to a rational number. This point is not an actual object, but instead only demarks the space between adjacent line segments.

With each successive row, the line is cut more and more times.

If asked, what number is a solution to 'phi^2-phi-1=0?' I would say that I cannot provide a number. But what I can provide is the pseudo-interval within which it lies at any row. For example,

Row 0: R
Row 1: RR
Row 2: RRL
Row 3: RRLR
Row 4: RRLRL
Row 5: RRLRLR
Etc.

We can proceed arbitrarily far down the tree, continually shrinking the pseudo-interval within which the solution lies. However, since there is no bottom of the tree, perfect precision cannot be achieved (i.e. the pseudo-interval can never be reduced to a single number, and the line cannot be reduced to a point). The solution to this problem is therefore not a number, but instead a vanishingly small pseudo-interval.

One might say that the sequence of pseudo-intervals describing phi is (R,RR,RRL,RRLR,RRLR,RRLRL,RRLRLR,…) but this sequence is infinite and so I do not believe it exists. What exists is the potentially infinite algorithm to generate successive cuts to narrow the pseudo-interval. In this case, the algorithm can be succinctly captured by the string: RRL. Similarly, the algorithms corresponding to 1 are RLR and RRL. Infinite S-B strings are algorithms. Reals are algorithms.

The claim that there are only countably many algorithms/programs does not imply that this view has gaps in the line. The line is there in full from the start. It simply means that it is impossible for any computer to hone in on some sections of the line.

It is my suspicion that this view is so closely tied with Cauchy sequences that maybe few new proofs are needed. Perhaps only a proof to shown that Cauchy sequences are actually talking about this. All that's really changed is the philosophy - but even that hasn't changed as much as one might think. For example, when explaining integrals, we use ever shrinking rectangles - this is not unlike the ever shrinking pseudo-intervals used here.

P.S. I've moved on from talking about the output of an unending program, but I do agree that any program you can write can in principle be stated as a Turing machine.

Quoting sime

By contrast, in a language with lazy semantics such as Haskell, terms can be used and passed around in partially evaluated form. This means that real numbers can exist in the sense of partially-evaluated "infinite lists" consisting of an evaluated prefix and an unevaluated tail. These lazy languages allow runtime conditions to decide what rational value is used in place of a term of real-number type, which is allowed to vary during the course of computation and which corresponds more closely to the notion of "potential infinity".

Actually, the algorithm used to perform arithmetic on the Stern-Brocot tree was originally written in Haskell (I rewrote it in Python) and it does just this!

https://www.sciencedirect.com/science/article/pii/S1570866706000311#:~:text=The%20Stern%E2%80%93Brocot%20tree%20is,rope%20between%20zero%20and%20infinity'.

Quoting keystone

The solution to this problem is therefore not a number, but instead a vanishingly small pseudo-interval.

You STILL don't get it. You just keep putting new clothes on an old pig. Every time, you reformulate but you retain the essential fallacy.

There is no single "vanishingly small pseudo-interval". There are only successively smaller pseudo-intervals on successive rows.

This is still essentially the same subject, as you still (after I asked twice) SKIPPED answering my question*, though I can see for myself that the correct answer is 'no'.

* https://thephilosophyforum.com/discussion/comment/806060

I really should not continue to reply when you so obnoxiously continue to apply the same fallacy clothed differently each time though I have explained it over and over and over and even asked you whether you understand, yet you don't reply even to that question itself.

But I'll look under the hood yet again:

As I understand, there is just one line, and then each row is a different set of pseudo-intervals for that line.

I don't see a problem with that. Except that I don't see that it improves the more simple approach:

There is an algorithm that successively outputs the S-B rows. And there are algorithms to output successive S-B finite paths, each path ending in a node. We take the comp-reals to be those algorithms, calling them real-ithms.

Then, define '<', '+' and '*' on real-ithms, and you'd be on our way to something.

In other words, for arbitrary real-ithms G and R:

G < R <-> [fill in definiens]

G+R = [fill in definiens]

G*R = [fill in definiens]

But then this:

Quoting keystone

The solution to this problem is therefore not a number, but instead a vanishingly small pseudo-interval.

No, throw that out. There is no such thing. There are only successive rows with smaller and smaller pseudo-intervals.

Your "vanishingly small pseudo-interval" is just a variation on the "mirage last row" fallacy.

Quoting keystone

The claim that there are only countably many algorithms/programs does not imply that this view has gaps in the line.

"Gaps". What I actually say: There are only countably many comp-reals, so they don't provide a complete ordered field, thus there is no isomorphism with the continuum. But the standard calculus depends on the completeness of the reals. So if you claim your proposal can do mathematics for the sciences, then you have to show how it does that. And by 'show' I don't ostensive examples, handwaving or picture stories. I mean axioms and theorems - formulas. And formulas then used to solve problems of science.

Quoting keystone

The line is there in full from the start.

You are SUCH a self-contradicting liar. You said:

Quoting keystone

There are no points on this line.

[bold ORIGINAL]

You seem to have no compunctions about insulting intelligence.

Quoting keystone

All that's really changed is the philosophy

Changed from standard analysis? No, your proposal is radically different from standard analysis, from the start: Standard analysis uses infinite sets; you disclaim infinite sets. Standard analysis has a continuum; you don't. Standard analysis has uncountably many reals; with you, it's not clear how many comp-reals there are, since there are denumerably many real-ithms but you disclaim that there are denumerable sets (but maybe you could argue that there is not a set of all real-ithms but instead a program that itself generates real-ithms).

Quoting keystone

I've moved on from talking about the output of an unending program

Right, you dressed it up differently again, now as "a vanishingly small pseudo-interval". Same pig, new dress.

In sum:

Your 'pseudo-intervals' are okay, up to, but not including, a 'vanishingly small pseudo-interval'.

But it's unnecessarily complicated when instead all we need is:

There is an algorithm that successively outputs the S-B rows. And there are algorithms to output successive S-B finite paths, each path ending in a node. We take the comp-reals to be those algorithms, calling them real-ithms.

Then, define '<', '+' and '*' on real-ithms, and you'd be on our way to something.

In other words, for arbitrary real-ithms G and R:

G < R <-> [fill in definiens]

G+R = [fill in definiens]

G*R = [fill in definiens]

/

Your move to this yet next iteration of you proposal (same pig, new dress) only serves to DISTRACT from the fact that AGAIN you want to eat your cake and have it too. You want only finite objects, but you also want real numbers, and you wanted them to be at the "mirage last row", or the last output of after an infinite loop, but now courtesy of a "vanishingly small pseudo-interval" while those DON'T EXIST with just your finite-only approach.

Here's an explanation of this process converging to the Golden Ratio. Leaving out the details, we begin with the

cf = [math]1+\frac{1}{1+\frac{1}{1+\frac{1}{1+\cdots }}}[/math], corresponding to the pattern of movement down the SB chart. This cf can be generated by iterating the function

[math]f(x)=\frac{1}{1+x},\text{ }{{f}^{(2)}}(x)=f(f(x))=\frac{1}{1+f(x)}=\frac{1}{1+\frac{1}{1+x}},\text{ etc}\text{.}[/math]

Setting [math]f(x)=x\text{ }\Rightarrow \text{ }x=\alpha ,\beta [/math], where the first of these fixed points

is an attractor: [math]f(x)-\alpha =K(x-\alpha ),\text{ }\left| K \right|<1[/math], [math]K=K(x)[/math], for most values of x.


And [math]{{f}^{(n)}}(x)-\alpha ={{K}^{n}}(x-\alpha )\to 0[/math] as [math]n\to \infty [/math]

Now, the value of the cf is [math]1+\alpha[/math] = Golden Ratio



This curiosity has a relation to arguments about endless cause/effect chains and first causes. Viewing the cf as going back in time, there is no end to the process, but stopping at any value of n , the x we have chosen at random, say, is a first cause.

Reply to jgill

What does 'the GR' stand for there?

Reply to TonesInDeepFreeze Fixed. :cool:

I've left out details to give an outline.

Reply to jgill

I shoulda gleaned it meant the golden ratio. I was distracted by fact that coincidentally I used 'G' and 'R' for something different.

Quoting TonesInDeepFreeze

I really should not continue to reply when you so obnoxiously continue to apply the same fallacy, clothed differently, each time though I have explained it over and over and over and even asked you whether you understand, yet you don't reply even to that question itself.

I appreciate you continuing this discussion. I'm getting plenty of value out of this dialogue. I don't think your infinite loop programming example of me not listening was a fair representation as I believe I am learning (perhaps not fast enough).

Quoting TonesInDeepFreeze

You STILL don't get it. You just keep putting new clothes on an old pig. Every time, you reformulate but you retain the essential fallacy.

There is no single "vanishingly small pseudo-interval". There are only successively smaller pseudo-intervals on successive rows.

You're right. I shouldn't have used 'vanishingly small pseudo-interval'. That makes it sound like I'm talking about an object at the bottom of the tree. The only object is the line. There is no bottom of the tree.

Let me rephrase a paragraph from my earlier post here:
If asked, what number is a solution to 'phi^2-phi-1=0?' I would say that I cannot provide a number. But what I can provide is an algorithm (which inputs integer, ROW, and outputs pseudo-interval, PI) such that for any positive rational number, EPS, there is an input ROW which outputs a PI, whose potential cuts (rational number, m) would all yield |m^2-m-1| < EPS. This algorithm can be described as RRL and I call it the golden ratio.

Quoting TonesInDeepFreeze

Except that I don't see that it improves the more simple approach

The idea behind this proposal is that the fundamental object is the line, not the point. This difference is significant.

Quoting TonesInDeepFreeze

But it is the standard calculus depends on the completeness of the reals.

But doesn't that mean that standard calculus depends on there being no gaps on the line? That's what this suggests:
https://en.wikipedia.org/wiki/Completeness_of_the_real_numbers

Quoting TonesInDeepFreeze

There are no points on this line.
— keystone
[bold ORIGINAL]

You seem to have no compunctions about insulting intelligence.

Infinite sets are so deeply embedded in your thinking that you're not even willing to imagine the possibility that points are not fundamental. Flip your thinking upside down just for a moment. In my proposal I start with a line and points get added as we progress down the rows. I'm starting with an unmarked 'stick', not a perfect precision 'ruler'. If I'm starting with a 'perfect precision ruler' on the top row then please tell me the coordinate of even a single specific point on that line [].

Quoting TonesInDeepFreeze

Changed from standard analysis? No, your proposal is radically different from standard analysis, from the start: Standard analysis uses infinite sets, you disclaim infinite sets. Standard analysis has a continuum; you don't. Standard analysis has uncountably many reals. With you, it's not clear how many comp-reals there are, since there are denumerably many real-ithms but you disclaim that there are denumerable sets (but maybe you could argue that there is not a set of all real-ithms but instead a program that itself generates real-ithms).

Are my proposed algorithms that different from Cauchy sequences?
Is my proposed line that different from the real number line?
Is my proposed line not continuous?
Standard analysis achieves length by having uncountably many points. Is length not also achieved by having pseudo-intervals?
Of course I'm proposing something different, but we must remember that calculus came before set theory. I'm only proposing a different foundational underpinning. If you don't think that's philosophy then sure.

Quoting TonesInDeepFreeze

Then, define '<', '+' and '*' on real-ithms, and you'd be on our way to something.

In other words, for arbitrary real-ithms G and R:
G < R <-> [fill in definiens]
G+R = [fill in definiens]
G*R = [fill in definiens]

This is sound advice. I need to research this. I appreciate you advising me on next steps.

Quoting TonesInDeepFreeze

You want only finite objects, but you also want real numbers

I don't want all real numbers, only the computable ones when they're needed. And to calculate the area between y=1, y=0, x=0 and x=1, I don't need to there to exist infinite points. As long as I know that there are no gaps across the lines then I can calculate the area just fine. Would anyone claim to explicitly make use of infinite points in their calculation?

Quoting keystone

I'm getting plenty of value out of this dialogue.

I've supplied some pretty good posts.

Quoting keystone

I don't think your infinite loop programming example of me not listening was a fair representation

The infinitude of it was a joke to express that it feels interminable. More than fair in that way.

Quoting keystone

The idea behind this proposal is that the fundamental object is the line, not the point.

The simple version is programs. Bringing in a concept of an initial object that is determined solely by -inf and +inf and then pseudo-intervals is extraneous. Everything you need is captured by:

"There is an algorithm that successively outputs the S-B rows. And there are algorithms to output successive S-B finite paths, each path ending in a node. We take the comp-reals to be those algorithms, calling them real-ithms."

Quoting keystone

This algorithm can be described as R RL[...] and I call it phi.

Exactly. The algorithm is comp-Phi. That's what I said. And the algorithm only needs to print successive rows - not worrying about cuts and pseudo-intervals. My guess is that you wanted to gussy it all up so that it seduces us to think that in a prettier getup there's some kind of vanishing point horizon mirage-limit. But at least now (hopefully, sincerely and stably*) you've backed off from claiming that.

* Let's set the stopwatch to see how long before you relapse and try to smuggle it in dressed up differently again.

Quoting keystone

But doesn't that mean that standard calculus depends on there being no gaps on the line?

What? I just said, "gaps" is not defined by you. The mathematics meanwhile is that there is a continuum. A continuum requires that the ordering is complete, meaning that for every bounded set, there is a least upper bound.

Quoting keystone

Infinite sets are so deeply embedded in your thinking that you're not even willing to imagine the possibility that points are not fundamental.

You don't get to incorrectly say what I'm willing to imagine. I don't even have notion of "points are fundamental", let alone that I won't imagine that it's not the case.

I'm happy to see any mathematics or formal logic - no matter how different from set theory and standard logic - as long as there are axioms, rules of inference, and definitions such that we can rigorously verify whether something is an axiom, an application of an inference rule or a definition. Or any algorithm, hopefully that can be conveyed as a Turing machine.

Please don't make pronouncements about what I am willing to imagine.

Quoting keystone

In my proposal

I understand your proposal. I am just saying that you don't need all those gooey toppings. Plain vanilla does the job just as well.

Quoting keystone

Are my proposed algorithms that different from Cauchy sequences?

Indeed they are! I EXPLAINED this. I don't understand what you don't understand in my explanation.

(1) An algorithm is finite. A Cauchy sequence is denumerable. And an equivalence class of Cauchy sequences has the uncountable cardinality of the set of equivalence classes of Cauchy sequences.*

* I think that sentence is right.

(2) There are only denumerably many algorithms, but uncountably many equivalence classes of Cauchy sequences.

(3) Cauchy sequences have a limit. But if we somehow defined the limit of an algorithm, then that would be infinitistic (unless some actual rigorous workaround could be formulated).

Why don't you already know this?

Quoting keystone

Is my proposed line that different from the real number line?

Your use of 'line' is only a figure of speech. It's not a line. It has nothing on it; it's a placeholder only - as YOU said. It's not a line in the sense of geometry or analytic geometry.

Already, we have confusion because you use a word in an utterly personal way and it gets conflated with the actual mathematical sense. So you should call it 'the k-line' so that it doesn't get mixed up again with 'line' in the mathematical sense.

And at no output does the cutting remotely resemble the continuum. First, at every output, there are only finitely many cuts and thus only finitely many rationals described. Second, there are no irrationals described. That is VERY different from the continuum that has both rational reals and irrational reals and altogether not just finitely many, but uncountably many, and proving a continuum.

And even if we took the UNION of the rows (which you can't do, because that is infinitistic) we still would have only the denumerable set of rationals described and none of the uncountably many irrationals.

Quoting keystone

Is my proposed line not continuous?

What? Are you TROLLIING me? Your questions are so ignorant and stupid that I can't help but suspect that you are.

Your "line", the k-line, has NOTHING on it, as YOU said. So 'continuous' is not even applicable.

And there is no infinite set of cuts on the k-line that comes after all the rows. You just now admitted that.

Quoting keystone

Standard analysis achieves length by having uncountably many points.

Wrong. You don't know ANYTHING about this. You don't even know high school mathematics.

Length is the absolute value of a difference. Even without irrationals, we have length with just rationals. Uncountability is not required to define length. Sheesh!

Quoting keystone

Is length not also achieved by having pseudo-intervals?

I don't know. You can do the arithmetic to see whether differences restricted to only those between rationals all on a row work out as desired.

But even if it worked out, calculus needs more than just lengths. It seems you don't know what calculus is. Do you?

Quoting keystone

we must remember that calculus came before set theory

So what? It used infinitisitic methods. Set theory provided axioms to make those methods rigorously derived from axioms.

Quoting keystone

I'm only proposing a different foundational underpinning. If you don't think that's philosophy then sure.

I respect and encourage philosophical frameworks for various notions of finitism. But your own mathematical proposal inspired by your particular finitism is incoherent. You are have a massive mental block that doesn't allow you to understand the basic illogic in your thinking. You keep wanting to have both only finite objects but also objects that exist only as provided as an end of an infinite process, while refusing in different forms to recognize that there is no such end hence no such objects.

/

I haven't tried to formalize your latest idea, but the rough sketch I come up with is this:

There is a primitive object, called 'the k-line'.

There are two more primitive objects, called '-inf' and '+inf. They are ordered so that -inf is less than +inf.

There are two more primitive objects: R and L.

The k-line but also associated with -inf and +inf is the base row.

There is an algorithm, call it the 'k-S-B algorithm', that generates rows, starting with the base row, then to the next row that is row 0, ad infinitum. The k-S-B algorithm recursively exhausts all "turn decisions" of R and L.

The k-S-B algorithm also associates each row with a set of fractions and an ordering on them, and the fractions are grouped in "cuts" which provide "pseudo-intervals".

So, a row is the k-line, with associated fractions, along with associated cuts and associated pseudo-intervals.

A real-ithm is an algorithm that executes non-terminatingly and each successive output is a finite sequence of Rs and Ls depending on a sequence of "turn decisions".

A k-real is a real-ithm.

NOTE:

We don't need the k-line. It is extraneous to capturing the information we want. We can just say a row is the set of cuts.

We don't need cuts. They are extraneous to capturing the information we want. We can just mention the fractions and their ordering.

I think the reason you want all that is to give the illusion that it amounts to a kind of pseudo-"continuum". But it doesn't. Essentially it's a big red herring. Toss out the red herring and simplify as I showed you, which is basically what you proposed yesterday.

NOTE:

There is no final row.

No real-ithm outputs a denumerable sequence.

Only computable reals are described.

A continuum cannot be described (as I explained a few posts ago and again here).

Reply to keystone

I've responded to all your main points and nearly all your secondary points. And I answered the exact questions you asked me.

Meanwhile, I've asked you three times now whether you understand this post:

https://thephilosophyforum.com/discussion/comment/806060

But you still say not a word about it.

What's up with that?

@TonesInDeepFreeze

To continue to clean up some of the language:

• A point in the context of my view is a k-point
• A line in the context of my view is a k-line
• An interval in the context of my view is a k-interval
• An algorithm which takes as input any natural number (corresponding to the row) and always outputs a k-interval (in finite time) is a k-algorithm
• A function which takes as input rational number(s) and outputs rational number(s) is a k-function
• A string of L's and R's (and underlines to represent repeats) is a k-string
• A k-line which has no gaps is k-continuous.
• ..and so on.

Quoting TonesInDeepFreeze

Meanwhile, I've asked you three times now whether you understand this post:
https://thephilosophyforum.com/discussion/comment/806060
But you still say not a word about it.

Sorry, I thought I was answering this question indirectly but let me be more clear. The successive outputs of a k-algorithm do not converge to any object. Ever. The S-B algorithm does not terminate (or to someone who believes in actual infinity - there is no bottom of the S-B tree).

Quoting TonesInDeepFreeze

Bringing in a concept of an initial object that is determined solely by -inf and +inf and then pseudo-intervals is extraneous.

The problem is that we've only been talking about numbers so far. The k-line becomes important when exploring higher dimensions. Consider the following k-functions:

• y=3x^2+2
• y=0
• x=1
• x=0

These k-functions can be illustrated as k-lines as depicted in the k-graph below. Please do not concern yourself with the shape of these k-lines. All properties of these k-lines are invariant under any continuous deformation.



In this figure there are only four k-points: (0,0), (0,2), (1,5), and (1,0). Notice how these k-points emerge when the k-lines cut each other.

The question is: What is the area between these k-lines?

One can estimate the area by introducing more k-functions and using rectangles.


So the area is approximately (5)(0.5)+(2.75)(0.5) = 3.875.
I can continue adding more k-functions and summing the areas of smaller and smaller rectangles but this process will never be exhausted. I cannot provide an exact answer of the area between the four k-lines.

However, because the above k-lines are k-continuous, I can using integration and provide you with the k-algorithm which can output a k-interval of arbitrarily narrow width. This k-algorithm can be described by the k-string RRRLR. Therefore the k-area between the k-lines is the k-string RRRLR.

Of course, k-algorithms are just one type of algorithm. If people were to adopt my view I wouldn't expect them to transition to using the S-B algorithm. I could have used decimals to describe algorithms and the decimal which would describe the equivalent algorithm is 2.9.

The reason why I say that my proposition is philosophical and not mathematical is because if you take away all of the k's you basically have standard calculus.

Quoting TonesInDeepFreeze

Are my proposed algorithms that different from Cauchy sequences?
— keystone
Indeed they are! I EXPLAINED this.

I think it's a matter of perspective by what one means by 'that different'. What I believe though is that if this approach ever gets formalized it's going to use a lot of similar language as Cauchy sequences.

Quoting TonesInDeepFreeze

Your use of 'line' is only a figure of speech. It's not a line. It has nothing on it

k-lines are associated with k-functions that describe their infinite potential.

Quoting TonesInDeepFreeze

So what? It used infinitisitic methods. Set theory provided axioms to make those methods rigorously derived from axioms.

Maybe one day you will see set theory as the mathematics of the bottom of the S-B tree…the bottom which you (rightfully) claim doesn't exist. Perhaps it is you who wants to eat your cake and have it too.

Quoting TonesInDeepFreeze

There is an algorithm, call it the 'k-S-B algorithm', that generates rows, starting with the base row, then to the next row that is row 0, ad infinitum. The k-S-B algorithm recursively exhausts all "turn decisions" of R and L.

No, I realized after contemplating about our debate on outputting something after an infinite loop that it wasn't necessary. k-algorithms only take natural numbers (corresponding to rows) as input and always output a k-interval in finite time.

Quoting keystone

Meanwhile, I've asked you three times now whether you understand this post:
https://thephilosophyforum.com/discussion/comment/806060
But you still say not a word about it.
— TonesInDeepFreeze

Sorry, I thought I was answering this question indirectly but let me be more clear. The successive outputs of a k-algorithm do not converge to any object. Ever. The S-B algorithm does not terminate (or to someone who believes in actual infinity - there is no bottom of the S-B tree).

It's good that you've conceded that there is no convergence to an object, that the algorithm does not terminate, and there is no last row. And it seems that the post helped you to that. But I don't know actually know what role the post had in that. I was asking whether you understand the post, which includes the various aspects of its explanations. Knowing your answer would let me know how much communication is taking place here.

AMAZING obtuseness right here:

Quoting keystone

Maybe one day you will see set theory as the mathematics of the bottom of the S-B tree…the bottom which you (rightfully) claim doesn't exist. Perhaps it is you who wants to eat your cake and have it too.

And this answers the question: No you did NOT understand the post mentioned in my post above.

I went out of my way to distinguish between standard mathematics and keystone musings.

In standard mathematics, there ARE objects to serve as limits. I gave an exact example of that. In standard mathematics, with infinite sets, there IS a limit to the sequence of successive finite approximations of Phi. But in keystone musings, without infinite sets, there is NOT. I get to say, "Phi is the limit", because set theory proves there IS such a limit. You do not, because your framework PRECLUDES that infinitistic limit.

Your admonishment about this is a product of you getting completely backwards.

But at least it's good to have an answer: No, you did not understand that post.

Quoting keystone

To continue to clean up some of the language:

I'm not inclined to indulge you with a formulation that is more complicated than it needs to be. I offered you a more simple outline. You can follow up on it if you like.

Quoting keystone

Your use of 'line' is only a figure of speech. It's not a line. It has nothing on it
— TonesInDeepFreeze

k-lines are associated with k-functions that describe their infinite potential.

Wow, you just turn on a dime away from what you say previously. I've explicated enough. Go back and read my posts and think about them rather than driving right over them.

Quoting keystone

The k-line becomes important when exploring higher dimensions.

You haven't even figured out the first "dimension". But carry on, though I will very likely not be subscribing.

Quoting keystone

I think it's a matter of perspective by what one means by 'that different'.

What? I listed the CRUCIAL, ESSENTIAL ways in which they are different. Rather than recognize that, you cop out with "it's a matter of perspective what one means by 'that different'."

Quoting keystone

I believe though is that if this approach ever gets formalized it's going to use a lot of similar language as Cauchy sequences.

So what? Lots of things use similar language, but say RADICALLY different things.

Reply to keystone

You are still confused. You still SKIP the MAIN points I post. You SKIP over the explanations about how you're mathematically wrong (not wrong to eschew standard mathematics, but wrong about the implications of your OWN framework). Often enough, your notions are incoherent. You make ridiculous jejune arguments (see posts above). And you lie about me. You're a sinkhole.

Quoting TonesInDeepFreeze

I was asking whether you understand the post, which includes the various aspects of its explanations. Knowing your answer would let me know how much communication is taking place here.

I suppose you would like me to paraphrase so you can judge my comprehension. Fair enough. Earlier on in the discussion, I incorrectly claimed that the S-B paths converged to a limit. But all that there is in the tree are nodes and paths. Each node corresponds to a rational number. You used sqrt(2) but let me use phi for simplicity since it is RRL. The nodes along the phi path are 1, 2, 3/2, 5/3, … But none of the nodes along the path are the limit, the limit in fact is an irrational number. In this sense, the rational numbers are incomplete because the limit of some sequences of rational numbers are not rational numbers themselves. So then you asked, what object on the tree does this limit correspond to? Such an object must be a part of the system for it to be complete. Reals do not have this problem. The limit of every sequence of real numbers is also a real number.

One way to rephrase this is to say that in the rationals there are some bounded sets for which there is no least upper bound. Essentially what this means is that if I were to have a set of rational numbers as described by the red coloring, there are many upper bounds (in blue) but there is no least upper bound (member of blue) because the number that we need is sqrt(2), which is not a rational number. Again, reals do not have this problem. "Every non-empty subset of X with an upper bound has a least upper bound in X."



I think this may be an important point to you because you are stressing the importance of completeness to calculus. If in my view the number system is incomplete, how is it possible to have the continuum needed to do much of calculus?

Quoting TonesInDeepFreeze

I get to say, "Phi is the limit", because set theory proves there IS such a limit. You do not, because your framework PRECLUDES that infinitistic limit.

I understand this. I wasn't challenging this. What I'm suggesting is that by starting with uncountably infinite objects (corresponding to real numbers) you are effectively starting with the 'bottom of the tree'. And that agreeing to the former and not the latter is wanting your cake and having it too. I do not challenge the fact that one can prove a lot of things using a system built on infinite sets.

Of course, I'm sure you do not see set theory as starting from the bottom of the tree. And that's why I mused that maybe one day…

Quoting TonesInDeepFreeze

So what? Lots of things use similar language, but say RADICALLY different things.

Okay, fine. It's significantly different. I suppose I'm coming from the applied side of mathematics and I don't see how my view changes anything in my day to day computations. But yes, if a mathematician were to go from a system based on actual infinities to one based on potential infinites there will be significant differences in the formalization. I should not understate this.

Okay, let me retrace my steps responding more thoroughly to some of your comments from your earlier post.

Quoting TonesInDeepFreeze

A continuum requires that the ordering is complete, meaning that for every bounded set, there is a least upper bound.

A continuum defined by numbers requires that the ordering is complete. I'm not proposing this. I'm attacking this from the other direction - numbers defined by a continuum. The ordering of numbers in this system does not need to be complete.

Quoting TonesInDeepFreeze

You don't get to incorrectly say what I'm willing to imagine. I don't even have notion of "points are fundamental", let alone that I won't imagine that it's not the case….
Please don't make pronouncements about what I am willing to imagine.

Okay, I take back my comment. But let me ask this. Would you agree to either of the following?
1) A continuum is defined completely by numbers.
2) A line is made up entirely of points.
I'm sure you will not agree to the exact phrasing, but perhaps you could say it in your words.

Quoting TonesInDeepFreeze

Cauchy sequences have a limit. But if we somehow defined the limit of an algorithm, then that would be infinitistic

I don't think there's a need to define the limit of an algorithm.

Quoting TonesInDeepFreeze

our use of 'line' is only a figure of speech. It's not a line. It has nothing on it; it's a placeholder only - as YOU said. It's not a line in the sense of geometry or analytic geometry.

In my response yesterday I wanted to show you that the k-line is not just a placeholder. By moving up a dimension, it becomes clear that even the uncut line holds a lot of information. For example, if you draw a plot including the k-lines y=1 and y=2, the y=2 line will be above the k-line y=1 even if they do not have any points on them.

Quoting TonesInDeepFreeze

And at no output does the cutting remotely resemble the continuum. First, at every output, there are only finitely many cuts and thus only finitely many rationals described. Second, there are no irrationals described. That is VERY different from the continuum that has both rational reals and irrational reals and altogether not just finitely many, but uncountably many, and proving a continuum.

You keep coming back to this point, which comes from a very number-centric view. I agree that a number system based on rational numbers cannot be continuous. I am not proposing that. I'm proposing an algorithm (e.g. program-based, equation-based) system. The restrictions on number systems do not apply to algorithm systems.

Quoting TonesInDeepFreeze

Your "line", the k-line, has NOTHING on it, as YOU said. So 'continuous' is not even applicable. And there is no infinite set of cuts on the k-line that comes after all the rows. You just now admitted that.

Over and over you repeat the same point, as if I'm not understanding you. I understand what you're saying. I just thought my last response most concisely addressed this point, but I'm responding here because I actually value your input and you think I skip over points. In my view the equation (line) is fundamental. In the standard view the number (point) is fundamental. I say start with an equation (line) and then create numbers (points). The standard view is to start with infinite numbers (points) and then create the equation (line). You are wrong to say that the k-line has nothing on it. It has an equation (algorithm). And I wanted to go up a dimension to demonstrate this.

Quoting TonesInDeepFreeze

Length is the absolute value of a difference. Even without irrationals, we have length with just rationals. Uncountability is not required to define length. Sheesh!

True. But in higher dimensions what about arc length.

Quoting TonesInDeepFreeze

But even if it worked out, calculus needs more than just lengths. It seems you don't know what calculus is. Do you?

In my post yesterday, I tried to demonstrate to you that my thinking extends beyond length to include, for example, area.

Quoting TonesInDeepFreeze

You keep wanting to have both only finite objects but also objects that exist only as provided as an end of an infinite process, while refusing in different forms to recognize that there is no such end hence no such objects.

At one point I was arguing this, but now everything that I'm working with is finite. The k-line fits entirely on the page (and so does the higher dimensional plot), there is no infinite tree, I'm not waiting infinite time for the k-algorithm to spit something out. What objects are you referring to?

Quoting TonesInDeepFreeze

We don't need the k-line. It is extraneous to capturing the information we want. We can just say a row is the set of cuts.

We don't need cuts. They are extraneous to capturing the information we want. We can just mention the fractions and their ordering.

I think the reason you want all that is to give the illusion that it amounts to a kind of pseudo-"continuum". But it doesn't. Essentially it's a big red herring. Toss out the red herring and simplify as I showed you, which is basically what you proposed yesterday.

It is impossible to formalize a number system that completely describes a continuum without actual infinity. Such a system would be dead on arrival. I truly appreciate you drafting out some ideas, but I don't think that's the right way to go. I don't think that my algorithmic/interval/line approach is a red herring. Rather, I see it as the only viable path a finitist can go to make sense of calculus.

So when you say that you’re unwilling to go to higher dimensions because I haven’t worked out the first dimension what I’m hearing is that you’re unwilling to deal with functions because I haven’t worked out the number system. And what I’m trying to show you is that you have it backwards. The algorithms (e.g. functions) come first and the numbers follow. I’m building an algorithm system, not a number system. If you’d consider my last message you might understand this.

So in sum, I am responding to your main points, I’m just not spelling it out.

Reply to keystone

PERSONAL MOTIVATION

What attracted me to the S-B tree in this thread is that we can take reals to be sequences of nodes. Unlike with equivalence classes of Cauchy sequences, we can see particular Cauchy sequences that we can use to define each particular real. (You wanted to use paths instead, but either nodes or paths should work.) Then I was interested in how that might be developed to derive the needed notions of ordering, addition and multiplication.

Then you changed your proposal to taking generating algorithms themselves as the reals. That interested me too, since, if I'm not mistaken, it is a notion in the subject of computable analysis, which I don't know enough about but piques my curiosity. And, again, that raises the question of how to define the ordering and addition and multiplication.

Then you added more apparatus that doesn't seem to me to improve the more basic and original goal that was not being addressed. Then you went further about "higher dimensions". I'm not sufficiently interested in whatever that's about to invest time and energy on it, while instead my curiosity is with the original questions of defining ordering and the operations.

But, of course, you should continue to post whatever interests you, notwithstanding my own disinterest in it.

INTUITION / FORMULATION

I don't think there's just a single roadmap to creating mathematical theories. But my guess is that a mathematician first has an intuition. Then she develops that intuition - in both depth and extent. Then she figures out how to formalize the ideas and to prove the important theorems.

So while the mathematician is still in the pre-formalized stage, deepening and extending the intuitions, she is putting herself into a kind of "intellectual debt". That is, the mathematician eventually is going to have to "pay" for the intuitive commitments with the hard cash of formalizing them.

When we formalize, it's usually along these lines:

We state the syntax of the primitive symbols, then the terms (nouns), the predicates (adjectives), formulas and sentences (statements), inference rules (logic), axioms (basic premises), and deduction (proof). Then definitional axioms (definitions) are added and theorems (the mathematical content) are deduced (proven).

Also, we state a formal semantics that provides for the meaning of the syntactical objects and also provides a means for proving that certain sentences are not theorems.

In the late 19th and early 20th centuries, different mathematicians developed ideas about how intuitions about 'number', 'is infinite', etc. could be formalized. Most of those intuitions among mathematicians, even when differing, offered an essential consensus. This eventually led to ZFC set theory as the standard theory. (But we only need (Z+DC)\regularity.)

But there was dissent. From finitists (stricter than Hilbertian finitism that still allows use of infinite sets as formally handled), constructivists and intuitionists, and predicativists. For the most part, those mathematicians were not very much concerned with formalizing their alternative mathematics. However, eventually much of alternative mathematics has been formalized. And the range of alternatives has wonderfully burgeoned. Now there's a truly amazing, densely populated gamut of alternative mathematics, and it's been formalized. And there's reverse mathematics, which figures out how to have the desired theorems but from weaker axioms.

(Z+DC)\REGULARITY (a set theory)

With (Z+DC)\Regularity we can formulate mathematics including number theory, analysis, topology, geometry, abstract algebra, graph theory, computability, probability, statistics, game theory ... on and on ... and mathematical logic itself.

(Z+DC)\Regularity addresses formalizing analysis this way:

The logic is first order predicate logic with identiity.

The only primitive is 'is a member of'.

The axioms are:

Extensionality: For any sets x and y, they are the same set if they have the same members.

Schema of Separation. For any "formalizable property" P, for any set x, there is the set of all members of x having property P.

Pairs: For any sets x and y, there is the set whose only members are x and y.

Union: For any set x, there is the set of all members of members of x.

Power Set. For any set x, there is the set of all subsets of x.

We prove the existence of a unique set that has no members, called '0'.

We prove that for any set x, there is the set whose only member is x, called '{x}'.

We prove that for any sets x and y, there is the set whose members are all the members of x and all the members of y, called 'xuy'.

Infinity: There is a set w such that 0 is a member of w, and for any set x, if x is a member of w then xu{x} is a member of w.

We develop the reals this way:

We define 'is a natural number'

We prove that there is a set whose members are all and only the natural numbers.

We define 'equivalence class' (per an equivalence relation).

We define 'is an integer' as 'is an equivalence class of natural numbers'.

We define 'is a rational' as 'is an equivalence class of integers'.

We define 'converges'.

We define 'is a Cauchy sequence (of rationals)'.

We define 'is a real' as 'is an equivalence class of Cauchy sequences'.

We define '<', '+', '*' for reals.

We define 'is a complete ordered field'.

We prove that the reals with <, +, * is a complete ordered field.

We define 'is isomorphic with'.

We prove that all complete ordered fields are isomorphic with the reals.

We define 'the continuum' as 'the reals along with <'.

Then we develop differentiation and integration to provide mathematics for things like speed, acceleration, etc.

SET THEORY and the S-B TREE

It is crucial to recognize that the S-B Tree is also itself developed in set theory. Thus, in set theory, we can construct and deduce from the S-B tree while also having all of the developments I described above.

So, in set theory, there is both the tree that doesn't have a final row or "row infinity" and the continuum. This is not having our cake and eating it too. Whatever we have comes from proofs from the axioms. The axioms are productive enough to proof the existence of many things including: the continuum, the S-B tree, finite algorithms, etc.

k-MUSINGS

You are in an intuition stage. If you ever followed through to write some mathematics, then you would confront the debt you're accumulating and pay it off with rigorous formulations. But, in the meantime, one still needs discipline to not just mouth a bunch of incoherent mental picture stories. Even with intuitions, one would like not to commit to informal contradictions (unless one wants to base the proposal in a paraconsistent logic). Which is to say, crankery is a dead end.

Quoting keystone

I suppose you would like me to paraphrase so you can judge my comprehension.

I just wanted to know whether you understand.

Quoting keystone

I incorrectly claimed that the S-B paths converged to a limit.

I like nodes better than paths for this.

In set theory, every denumerable sequence of nodes converges to a limit.

In k-musings, there is no limit for the sequences to converge to, and there are no denumerable sequences anyway.

But the rest of your paraphrase is good.

Quoting keystone

I think this may be an important point to you because you are stressing the importance of completeness to calculus.

It is core to standard analysis. I don't claim that there can't be viable alternatives to standard mathematics.

But I stressed the lack of such limits in k-musings because you kept posting as if those limits exist in k-musings.

Quoting keystone

What I'm suggesting is that by starting with uncountably infinite objects (corresponding to real numbers) you are effectively starting with the 'bottom of the tree'. And that agreeing to the former and not the latter is wanting your cake and having it too.

What is the 'bottom'? What are the 'former' and 'latter'?

Quoting keystone

A continuum defined by numbers

^^^ A structure isomorphic with the continuum may be made with non-numbers. Anyway, in set theory, every object is a set.

Quoting keystone

[in k-musings] numbers defined by a continuum. The ordering of numbers in this [k-]system does not need to be complete.

You don't have a system. You have some ideas.

And you still misunderstand what I posted. You have it backwards. In set theory, defining the ordering does not require proving completeness. Rather we define the ordering and then prove completeness.

Quoting keystone

Would you agree to either of the following?
1) A continuum is defined completely by numbers.
2) A line is made up entirely of points.

(1) I don't know what sense of 'defined' you mean. I said what the continuum is:

c (the continuum) is the set of reals with the standard ordering

The set of reals is the carrier set for c.

A continuum is any structure (a carrier set and an ordering) isomorphic with c.

And a continuum may have a carrier set whose members are not any kind of number.

*** You seem to have a notion that we have to distinguish numbers. No, every object is a set. There's not even a definition of 'is a number'. Though there are definitions of 'is a natural number', 'is a rational number', 'is a real number', etc. But we don't need the word 'number' there. For that matter, we could instead say 'is a zatural', 'is a zational', 'is a zeal'. There's no special force in saying 'number'.

(2) In set theory, 'point' and 'line' can be defined (we don't have to take them as primitives such as in axiomatic geometry). A line is a certain kind of set, and its members are called its 'points'.

I don't know the purpose of this exercise.

Quoting keystone

I don't think there's a need to define the limit of an algorithm.

What? Here's the context:

Quoting TonesInDeepFreeze

Are my proposed algorithms that different from Cauchy sequences?
— keystone

Indeed they are! I EXPLAINED this. I don't understand what you don't understand in my explanation.

(1) An algorithm is finite. A Cauchy sequence is denumerable. And an equivalence class of Cauchy sequences has the uncountable cardinality of the set of equivalence classes of Cauchy sequences.*

* I think that sentence is right.

(2) There are only denumerably many algorithms, but uncountably many equivalence classes of Cauchy sequences.

(3) Cauchy sequences have a limit. But if we somehow defined the limit of an algorithm, then that would be infinitistic (unless some actual rigorous workaround could be formulated).

You asked whether algorithms are so very different from Cauchy sequences. One of the differences I mentioned is that Cauchy sequences have limits, but even IF we defined limits of algorithms (of the kind of algorithms that approximate irrationals), then they would be infinitistic (so they would not comport with your finitism).

Obviously, I'm not suggesting that you countenance limits for algorithms. I'm only answering your query about how algorithms are different from Cauchy sequences.

Quoting keystone

very number-centric view.

Oh, get out of here already with that nonsense. See ^^^ and *** above.

Quoting keystone

Your "line", the k-line, has NOTHING on it, as YOU said. So 'continuous' is not even applicable. And there is no infinite set of cuts on the k-line that comes after all the rows. You just now admitted that.
— TonesInDeepFreeze

Over and over you repeat the same point, as if I'm not understanding you. I understand what you're saying.

I repeated it because you repeated contradictions of your own stipulations. I just went with what you literally wrote when you laid out the proposal. Then you say there's some other explanation. At this point, I'm not interested. I carefully read your earlier proposal - I noted each of your stipulations and definitions. Then later what you WROTE (notwithstanding what you might have MEANT) contradicted that. It's inconsiderate to ask a reader to not be able to take each stipulation and definition as having some constancy - to have to continually start all over again to keep up to your later explanations as to what you meant when you didn't write what you meant originally. I'm done with that.

Quoting TonesInDeepFreeze

Then you added more apparatus that doesn't seem to me to improve the more basic and original goal that was not being addressed. Then you went further about "higher dimensions". I'm not sufficiently interested in whatever that's about to invest time and energy on it, while instead my curiosity is with the original questions of defining ordering and the operations.

Going to higher dimensions helps explain my position and would more clearly demonstrate why 'more apparatus' improves my position. But yes, it would take time to try and understand what I'm saying and time is limited. You're right, I've taken this discussion beyond the original question and it is reasonable for you to not want to come along with me. You've already been generous with your time. Thanks.

Quoting TonesInDeepFreeze

So while the mathematician is still in the pre-formalized stage, deepening and extending the intuitions, she is putting herself into a kind of "intellectual debt". That is, the mathematician eventually is going to have to "pay" for the intuitive commitments with the hard cash of formalizing them.

I agree. I suppose as this conversation evolved I wanted to bounce my pre-formalized idea off of someone to see whether it was worth me investing in formalizing it. Although I'm disappointed, I acknowledge that it is reasonable for you to not want to discuss it until it is formalized.

Quoting TonesInDeepFreeze

So, in set theory, there is both the tree that doesn't have a final row or "row infinity" and the continuum. This is not having our cake and eating it too. Whatever we have comes from proofs from the axioms. The axioms are productive enough to proof the existence of many things including: the continuum, the S-B tree, finite algorithms, etc.

Yes, the proofs come from the axiom and unless I can prove the axioms to be inconsistent there's no point discussing my musing.

Quoting TonesInDeepFreeze

You are in an intuition stage. If you ever followed through to write some mathematics, then you would confront the debt you're accumulating and pay it off with rigorous formulations. But, in the meantime, one still needs discipline to not just mouth a bunch of incoherent mental picture stories. Even with intuitions, one would like not to commit to informal contradictions (unless one wants to base the proposal in a paraconsistent logic). Which is to say, crankery is a dead end.

I agree with the first sentence. I also agree that many times I have not been clear. While I would have deeply appreciated you trying to truly understand what I'm trying to say, I fully acknowledge that it is reasonable for you to not want to invest the time into it.

Quoting TonesInDeepFreeze

You seem to have a notion that we have to distinguish numbers. No, every object is a set.

I think I understand what you're saying. Start with a small set of axioms and everything else follows. Numbers follow. Points follow. You do not want to discuss this on an intuitive level and want to stick with formalities. This is not how one would explain what should be a simple concept to a grade schooler but that's fine if that's how you want to approach it. And I think that's how you have to approach it because when discussing the standard position on an intuitive level many paradoxes arise. The standard position doesn't gel with our intuitions and so we must stick with the formalization. But you and I have been down this road before.

Quoting TonesInDeepFreeze

I don't know the purpose of this exercise.

If you are not willing to discuss basics on an intuitive level then there is no purpose of this exercise.

----------------------------

All in all, it's clear that we want to have different discussions. You want to talk in terms of formalities and I want to talk in terms of intuitions. Neither of us are able to talk on the other person's level. While I'm extremely disappointed that this conversation has come to an end, I once again want to thank you for your time and insights. As I mentioned before, I've gotten great value from this discussion. Thank you.

Reply to keystone

I didn't say that I'll only consider formalizations. I have been interested in the earlier proposals though not formalized. Rather, I said that I'm not inclined now to study your latest revisions.

Quoting keystone

I wanted to bounce my pre-formalized idea off of someone to see whether it was worth me investing in formalizing it.

Somehow, I don't believe you. To formalize you'd have to know what formalization IS. Be honest: Learning what goes into an axiomatic formulation is not a goal for you.

Quoting keystone

the proofs come from the axiom[s]

So, hopefully, you understand now that there's no "cake and eating it too" about the S-B tree and Cauchy sequences in set theory, or generally in set theory having both finite algorithms and infinite sets.

Quoting keystone

unless I can prove the axioms to be inconsistent there's no point discussing my musing.

Perhaps you meant 'consistent' there. First you have to have primitives, formation rules, inference rules, and axioms. Then you can address whether the axioms are consistent. But it's not required to prove their consistency.

Quoting keystone

While I would have deeply appreciated you trying to truly understand what I'm trying to say, I fully acknowledge that it is reasonable for you to not want to invest the time into it.

I understood what you said in the earlier proposals. And I showed you the respects in which it was incoherent until eventually a couple of coherent proposals did emerge (though still clouded with certain stubborn misconceptions you've had).

Quoting keystone

when discussing the standard position on an intuitive level many paradoxes arise.

No, set theory shows how the paradoxes with the naive notion of sets are avoided.

Quoting keystone

The standard position doesn't gel with our intuitions

Axiomatic set theory is quite intuitive to me. I listed the axioms for you. I find each of them to be eminently intuitive.

Quoting TonesInDeepFreeze

I didn't say that I'll only consider formalizations. I have been interested in the earlier proposals though not formalized. Rather, I said that I'm not inclined now to study your latest revisions.

Fair enough.

Quoting TonesInDeepFreeze

Somehow, I don't believe you. To formalize you'd have to know what formalization IS. Be honest: Learning what goes into an axiomatic formulation is not a goal for you.

Formalization is the way for an idea to be treated seriously. Learning how to get there would require a huge investment of time and money. If the idea seems very promising and my finances were just right I would pursue it. Let's not debate my motivations.

Quoting TonesInDeepFreeze

So, hopefully, you understand now that there's no "cake and eating it too" about the S-B tree and Cauchy sequences in set theory, or generally in set theory having both finite algorithms and infinite sets.

If ZFC is consistent then there's no cake and eating it too.

Quoting TonesInDeepFreeze

No, set theory shows how the paradoxes with the naive notion of sets are avoided.

We've been down this road already.

Quoting TonesInDeepFreeze

Set theory is quite intuitive to me. I listed the axioms for you. I find each of them to be eminently intuitive.

Given that everything fits nicely together for you and in your view the paradoxes are addressed, I can see how you're not motivated to pursue a potential infinity solution.

-----------------

Let's leave it at that. Thanks again!

Quoting keystone

huge investment of time and money

Not a lot of money. A few good books.

Quoting keystone

If ZFC is consistent then there's no cake and eating it too.

Right. No Kate and Edith too.

Quoting keystone

No, set theory shows how the paradoxes with the naive notion of sets are avoided.
— TonesInDeepFreeze

We've been down this road already.

Yes, and down that road we arrive at a concept free of the paradoxes of naive set theory.

Quoting keystone

Given that everything fits nicely together for you and in your view the paradoxes are addressed, I can see how you're not motivated to pursue a potential infinity solution.

What? I've said over and over and over that I am open to learning about alternative mathematics - including strong finitism and, constructivism and intuitionism. I don't know enough about them, but I know vastly more about them than you do. And I even went through a lot of posts in this thread alone to consider your own proposals. Your snipe is ridiculous.

Every time you blatantly lie about me, your integrity shrinks and shrinks. But you just can't resist ...

Quoting keystone

Let's not debate my motivations.

While you lie about mine.

Quoting keystone

Let's leave it at that.

Let US? No, I don't need to take direction from you as to whether I comment or not.

It's odd that a thread on the S-B expansion arises in this forum. S-B is an outlier in number theory - the Wiki page is classified as low interest. And since a primary interpretation of its generation are continued fractions of a special type, one immediately loses easy addition and subtraction - which decimal expansions of the reals have.

The value of continued fraction representations is that chopping them off at various levels give rational approximations to what one is expanding. This is easily seen when expanding a real number, like the Golden Ratio. But where it is of greater value is expanding a complex or real function as a continued fraction, providing rational functions (one polynomial over another) as approximations to the expanded functions.

[math]Tan(z)=\frac{z}{1-\frac{{{z}^{2}}}{3-\frac{{{z}^{2}}}{5-\ddots }}}[/math]

Although I have little knowledge of this kind of number theory (S-B) I feel the line of inquiry expressed here would be of little interest to the mathematical community. But I could be wrong.

Edit: After doing an internet search for "area" in S-B defined in the Wiki article on Farey sequences, I probably am wrong about the interest shown in S-B by mathematicians. Embarrassingly so as I find that two former colleagues of mine have included it in their book. :yikes:

Quoting keystone

Formalization is the way for an idea to be treated seriously.

Most importantly, it is the way to confirm (to yourself or to anybody) that you have actual mathematics free of any hand-waving.