It all depends on how one defines "countable" — jgill

Exactly. "Countable" means something very specific within the formalism. The critique provided amounts to a rejection of that notion, not a derivation of contradiction from within the system.

Exactly. "Countable" means something very specific within the formalism. The critique provided amounts to a rejection of that notion, not a derivation of contradiction from within the system. — Esse Quam Videri

That's right, "countable" means something very specific. But as I've demonstrated, the meaning of it, as defined, contradicts the meaning of 'the natural numbers extend endlessly'. That's where the problem lies. The natural numbers have been in use for a long time, with a very specific formulation allowing for infinite, or endless, extension. Then, "countable" was introduced as a term with a definition which contradicts the infinite extension of the natural numbers.

Please see my reply to jgill below.

It all depends on how one defines "countable" — jgill

As usual, I agree with you jgill. Here's the definition you provided: "capable of being put into one-to-one correspondence with the positive integers".

Nothing is capable of being put into one-to-one correspondence with all of the positive integers. We might say that the system was designed this way, to be unlimited in its capacity to measure quantitative value, 'to count'. That's why the system was formulated to extend infinitely. The positive integers derive their extraordinary usefulness from being extendable indefinitely, to be capable of counting any possible quantity. Notice, infinite possibility covers anything possible. To allow that the integers themselves may be counted. or to designate that something may be put into one-to-one correspondence with them all, is to say that there is a capacity which extends beyond them, i.e. that capacity to count them. This is to limit their usefulness as unable to measure that specific capacity. To limit the usefulness of the integers is counterproductive to the various disciplines which use mathematics.

Nothing is capable of being put into one-to-one correspondence with all of the positive integers. — Metaphysician Undercover

I will attempt to clarify once more for the sake of the thread.

This statement of yours is neither a theorem, nor a definition nor a logical consequence of anything from within the formal system. This is a philosophical assertion grounded in a procedural interpretation of "capable" that is foreign to the mathematics. All you are saying here is that the impossibility follows from your definition of "capable", and that you think your definition is the right definition. This is an external critique. At no point have you derived a contradiction from within the system. Therefore, nothing you have said so far justifies the claim that the system is inconsistent.

I apologize if this comes off as rude, but this has been spelled out multiple times now from multiple different users. I think that if we still can't agree, then we have probably reached a principled stopping point that no further clarification is likely to resolve.

How do you know that you will be able to produce all of the outputs? — Magnus Anderson

In other words, the problem is that you'll never finish.

Under this view, there are no functions on any infinite set. Not even f(n)=1. No functions on segments of the real line.

You could also demand that to be a set "in the stronger sense" you have to be able to finish listing its elements, and under that definition N cannot be a set.

Which, whatever. It's your sandbox, do as you like.

As usual, I agree with you jgill. Here's the definition you provided: "capable of being put into one-to-one correspondence with the positive integers".

Nothing is capable of being put into one-to-one correspondence with all of the positive integers. We might say [ ... ] — Metaphysician Undercover

I'm just wondering if you think somewhere in the rest of the paragraph (following the bolded sentence) you have provided an argument in its support. Is this the post you will have in mind when someone asks and you claim to have demonstrated that "Nothing is capable of being put into one-to-one correspondence with all of the positive integers"? Because it's just an assertion of incredulity followed by a lot of chitchat. (I think you have in your mind somewhere an issue of conceptual priority, but it's not an actual argument.)

Notice, infinite possibility covers anything possible. — Metaphysician Undercover

Sigh. You can't even pretend to be listing the reals and putting them into a one-to-one correspondence with the naturals. Rather the whole point of this kind of talk about transfinite cardinalities is that they are not all the same.



"Countable" is just a word, of course, and it doesn't bother us that it has been given a technical definition. Maybe "list-orderable" would be clearer.

Not only does none of this bother me, it has all the charm of good mathematics. Cantor's diagonal proof is simple, clear, and convincing. Even better is the zig-zag demonstration that the rationals are countable. ( (I think a more common presentation is just ordering pairs by diagonal after diagonal, but I saw it done first zig-zagging and it's stuck with me.) I think that was even more thrilling for me. In the natural ordering, in between any two, there are an infinite number -- how can they not be bigger than the naturals?! And then you see how they can be rearranged so that there is always a unique next rational. It's brilliant and convincing. People who don't ever see this, or who reject it for semantic reasons, are missing out on some lovely examples of the sort of thinking we should all aspire to.

Well said.

an infinite number — an-salad

That phrase is incoherent.

“An” refers to a discreet, limited, individuated, measured, unit.

So what does “an infinite number” point to or refer to? Certainly not some thing; certainly not some number.

We would be better off using the concept of infinity as an adverb, to describe a process. Instead of saying “there are an infinite number of natural numbers” we should say “we can count off natural numbers infinitely” or something similar.

Grammar police.

No infinite number of fractions (or infinite number of anything) exists like countable, quantifiable, distinguishable things exist. Existence sets a limit. Minds can take a whole, existing, limited thing, and then subject it to a mental process of division into fractions, infinitely. Or minds can multiply wholes infinitely, constructing bigger new wholes, infinitely. But at each step along the way, infinity is nowhere in site, and has never been reached, as it never will, infinitely.

So it is confusing to therefore assert at the get go “there is an infinite number of X.”

Saying any individual thing (like the universe, or God, or the pieces of an apple) is infinite, makes no sense, because it misunderstands where infinity exists, which is in the mind, as it constructs its descriptions and definitions of things and processes.

The notion some infinities are bigger than other infinities sounds romantic and poetic and is a curiosity - but no one can identify a discreet, whole, individuated, existing infinity to then compare it to a distinct, separately individuated whole other infinity. Infinity doesn’t quantify a single thing. It’s an adverb, tied to an existing process that theoretically is never finished processing.

It's brilliant and convincing. — Srap Tasmaner

Yes!

The diagonal argument and its friends are amongst the most beautiful and impressive intellectual presentations. I pity those who do not see this. The exercise here is to show folk something extraordinary; but it seems that there are a small but vocal minority who for whatever reason cannot see.

It's as if someone were to say "A circle is a plain figure with every point equidistant from a given point", and you were to insist that such a thing cannot be spoke of until it is shown not to involve an inherent contradiction...

Why not work with the definition unless some contradiction is shown?

And in the cases of infinite sets, you have not shown a contradiction.

And that's not true.

The only thing that you have shown is that you can take any element from N and uniquely pair it with an element from N0. — Magnus Anderson

This is perverse. That is exactly what has been shown. That each element of ℕ can be paired with an element of ℕ₀, and that each element of ℕ₀ can be paired with an element of ℕ. The bijection is fully established.

And a circle contains an uncountably infinite number of points. Oh well, no more analytic geometry.

The onus of proof is always on the one making the claim. If you're making the claim that bijection between N and N0 exists, you have to show it, and that means, you have to show that such a bijection is not a contradiction in terms. That's what it means to show that something exists in mathematics. — Magnus Anderson

The very first line of the proof does exactly what you ask for here. A function maps a each individual in one domain with an individual in the other. Hence:

The function is Well-defined: For every $n \in \mathbb{N}$, we have $n \ge 1$, so $n-1 \ge 0$. Hence $f(n) \in \mathbb{N}_0$, and the function is well-defined.

If there is some other contradiction, then that is your claim, and up to you to demonstrate.

Oh well, no more analytic geometry. — Srap Tasmaner

Indeed. And not just that. Much of modern maths would be unavailable or need reworking, with no apparent gain.

Magnus's position appears incoherent, in that he makes use of ℕ and other infinities while disavowing the relations between them. Meta is perhaps more consistent in apparently simply rejecting any infinities - or something like that.

This statement of yours is neither a theorem, nor a definition nor a logical consequence of anything from within the formal system. This is a philosophical assertion grounded in a procedural interpretation of "capable" that is foreign to the mathematics. All you are saying here is that the impossibility follows from your definition of "capable", and that you think your definition is the right definition. This is an external critique. At no point have you derived a contradiction from within the system. Therefore, nothing you have said so far justifies the claim that the system is inconsistent. — Esse Quam Videri

Esse, please read what is written. I took the definition from a mathematics site, provided by a mathematician, jgill. The definition was "capable of being put into one-to-one correspondence with the positive integers". Please, for the sake of an honest discussion, recognize the word "capable" in that definition. And please recognize that your diatribe about my use of the concept "capable" is completely wrong, and out of place.

"Capable" is not a concept foreign to mathematics. Mathematicians employ the concept of "capable" with the concept of "countable", and surprise, there it is in that definition. You have no argument unless you define "countable" in a way other than capable of being counted. Are you prepared to argue that "countable" means something other than capable of being counted for a mathematician.

Or, are you proposing that mathematicians have their own special definition of "capable", designed so as to avoid this contradiction. Are you proposing that they have a meaning of "capable" which applies to things which are impossible, allowing that mathematicians are "capable" of doing something which they understand to be impossible? If so, then let's see this definition of "capable" which allows them to be capable of doing what they know is impossible to do.

I'm just wondering if you think somewhere in the rest of the paragraph (following the bolded sentence) you have provided an argument in its support. Is this the post you will have in mind when someone asks and you claim to have demonstrated that "Nothing is capable of being put into one-to-one correspondence with all of the positive integers"? Because it's just an assertion of incredulity followed by a lot of chitchat. (I think you have in your mind somewhere an issue of conceptual priority, but it's not an actual argument.) — Srap Tasmaner

Sorry Srap, it seems you haven't been following the discussion. I suggest you start at the beginning.

As I have mentioned before, the interpretation I have used for years is that infinity means boundlessness, not a cardinal number. As for transfinite entities of greater cardinalities than the reals, I have encountered only one theorem in functional analysis that requires their use - and even there by altering the hypotheses a tad one escapes that situation.

There is a "point at infinity" in complex analysis that arises when the complex plane is mapped onto the Riemann sphere. But it is simply the north pole of the sphere.

I wonder if and when physics will find uses for transfinite objects. Perhaps it already has.

Here's an example to consider Esse. Would you say that someone is "capable" of producing the entire decimal extension of pi? If not, then why would you say that something is "capable" of being put into one-to-one correspondence with all of the positive integers? Or do you equivocate on your meaning of "capable"?

Sorry Srap, it seems you haven't been following the discussion. I suggest you start at the beginning. — Metaphysician Undercover

God forbid you repeat yourself ...

You can list them in a sequence, 1/1,1/2, 1/3, 2/3, 1/4, and so on, and so you can count them - line them up one-to-one with the integers.
— Banno

That's funny. Why do you think that you can line them all up? That seems like an extraordinarily irrational idea to me. You don't honestly believe it, do you?

Do you think anyone can write out all the decimal places to pi? If not, why would you think anyone can line up infinite numbers? — Metaphysician Undercover

The key word in all this seems to be "all". You might as well bold it each time you use it.

Now, it's a known fact that you can line up all the rationals, in the sense of "fact", "can", "all", and even "you" that matters to mathematics. You disagree, and so far as I can tell only because anyone who tried to do this would never finish. Which --

Okay but when you said

Nothing is capable of being put into one-to-one correspondence with all of the positive integers. — Metaphysician Undercover

what are you referring to with this phrase, "all the positive integers"? I know what I would mean by that phrase; I genuinely do not know what you mean.

The key word in all this seems to be "all". You might as well bold it each time you use it. — Srap Tasmaner

I don't see anything special about that word. Why do you think I should embolden it?

You disagree, and so far as I can tell only because anyone who tried to do this would never finish. — Srap Tasmaner

That's right, we know, by the defining features of the system, that no one could ever finish this task. It is impossible, by definition.

So, tell me how it is that you claim "it's a known fact that you can line up all the rationals"? Has someone produced this line of all the rationals, to prove this fact? Of course not, because it is also a known fact that this is impossible to do, because no one could ever finish. What's with the contradiction?

what are you referring to with this phrase, "all the positive integers"? I know what I would mean by that phrase; I genuinely do not know what you mean. — Srap Tasmaner

I probably mean the very same thing as you're thinking. Jgill raised the the issue of the meaning of "countable", and provided a reference. The definition from that referred page was: "capable of being put into one-to-one correspondence with the positive integers". So, think of what a "positive integer" is, a whole number greater than zero, and imagine all of them. Now do you know what I mean?

imagine all of them. Now do you know what I mean? — Metaphysician Undercover

How on earth do you imagine all the natural numbers?

If you re-read my reply carefully you will see that I did not say that mathematicians do not use the word "capable", but that they use it in a different way.

"A is countable" means "∃f such that f is a bijection between A and ℕ". That's it. There is nothing procedural in this definition. That was the point I was trying to make.

Then, "countable" was introduced as a term with a definition which contradicts the infinite extension of the natural numbers. — Metaphysician Undercover

This is just one example of the way in which, when you change one feature of a language-game (conceptual structure), you often have to change the meaning of other terms within that structure.
So, "countable" in the context of infinity cannot possibly mean the same as "countable" in normal contexts. In the context of infinity, it means that you can start counting the terms and count as many as you like, and there is no term that cannot be included in a count; the requirement that it be possible to complete the count is vacuous, since there is no last term. It's not a problem.

So, tell me how it is that you claim "it's a known fact that you can line up all the rationals"? Has someone produced this line of all the rationals, to prove this fact? Of course not, because it is also a known fact that this is impossible to do, because no one could ever finish. What's with the contradiction? — Metaphysician Undercover

Well, perhaps it needs putting in a slightly different way. For example, how about "there is no rational that you cannot place on the number line"?
When you define the successor function - Successor(n)=(n+1) - you can see that there will never be a last number, and you don't have to try to write all the numbers down to do it. You can also see that each and every number is defined - or perhaps better, there is no number that is not defined. So you do not need to complete the task in order to see that the conclusion is true. In the relevant mathematical system, that is a proof that they all exist and can be located on the number line.
Is there something going on in the background here about actual and potential infinities?

Thanks. I hope nothing I've said is at odds with this? Good feedback.

I suppose that while transfinite numbers are not much used in physics, continuum cardinality and so on are present as background commitments. So if the space-time manifold in General Relativity is continuous, then I suppose transfinite cardinals are included by default in that formalisation; or so I believe. Of course, quantum theories would involve granularity, but this is entering into speculative physics, a can of worms.

There are oddities. In particular, I've had discussions previously with "finitist" folk who denied limits and such, and so were unable to make sense of differential calculus, and so in turn were led to denying corresponding physical entities such as instantaneous velocity. @Metaphysician Undercover has been known to do something along these lines.

As I have mentioned before, the interpretation I have used for years is that infinity means boundlessness, not a cardinal number. — jgill

I've thought about that, but always assumed that someone would then demand how I explain "unbounded but finite", which, I'm led to believe is also possible. I've sometimes used "there is no last term". Is there any problem with that? (Mathematically, I'm sympathetic layman.)

I wonder if and when physics will find uses for transfinite objects. Perhaps it already has. — jgill

I certainly woudn't bet against that. I'm only deterred from betting in favour by the fact that it could take a long, long time before it happened.

I suppose that while transfinite numbers are not much used in physics, continuum cardinality and so on are present as background commitments. So if the space-time manifold in General Relativity is continuous, then I suppose transfinite cardinals are included by default in that formalisation; or so I believe. — Banno

the existence and meaningfulness of transfinite cardinals rests upon the Axiom of Choice, but that principle also implies unfettered resource duplication via the Banach Tarski Paradox, which goes way what is needed to define mathematical continuity, and well beyond the physical requirements and assumptions of general relativity, not to mentioning violating energy conservation.

By definition, the computational content of physics (i.e. physical inferential semantics) cannot rest upon choice axioms, because they represent what cannot be computed. In practice, physical continuity doesn't refer to an idealised continuum in the sky, but only to the ability to construct or measure vanishingly small changes in output in response to vanishingly small changes in input, for which multiple alternative languages are available, that don't carry the metaphysical baggage.

Moreover, the standard dogma of the transfinite cardinals, harms theoretical physics, by denying the ability speak of potentially infinite sets that naturally describe the content of a physical process better than Dedekind-infinite completed set balony.

Once AOC is relinquished, the unreal "beauty" of the ideal cardinals is replaced with the ugly and uncertain truth of equivalence classes of set bijections that are generally undecidable, and only potentially infinite, such they cannot hide their complexity behind a veil of cardinal representatives obeying a simple cardinal arithmetic.

In addition, it cannot even be proven, without begging the question, that transfinite induction up to ε0 is sound, since ε0 might not be well-founded. Hence a theoretical physics claim cannot rest upon transfinite induction. In practice, a "proof" that PA is consistent essentially amounts to beating the skeptic into submission with pseudo-religious dogma about the metaphysical "truth" of transfinite induction as decided by classical mathematicians who don't ultimately care about the physical truth and use-value of such theories.

Only the boring transfinite ordinals up to ε0 are empirically and computationally meaningful.

How on earth do you imagine all the natural numbers? — Srap Tasmaner

I can't, neither can you. Get the point?

If you re-read my reply carefully you will see that I did not say that mathematicians do not use the word "capable", but that they use it in a different way. — Esse Quam Videri

I know you said this, but I do not believe you . The concept of "capable" is very straight forward with very little ambiguity. It means having the ability for. So, if you read through to the end of my post, I requested that you provide this special definition of "capable", which you claim mathematicians are using.

"A is countable" means "∃f such that f is a bijection between A and ℕ". That's it. There is nothing procedural in this definition. That was my point. — Esse Quam Videri

You are wrong again Esse. "Countable" is defined as a form of "capable" which is defined as "ability for". Therefore it is very clear that something procedural is referred to by "countable". Producing a bijection is a procedure. That is the point Magnus took up with Banno. You might obscure this fact with reference to 'function", and insist on a separation between "function" and "procedure" or employ a variety of other terms to veil this reality, but all this amounts to is a dishonest attempt to obscure the facts, deception.

Why do you keep insisting on things which you really ought to know are wrong? That is the problem. Instead of acknowledging, 'oh yeah, there are some problems with mathematical principles, and this is one of them', you go off and try to hide the problem. You see, in philosophy we meet these sorts of problems all the time, everywhere, in metaphysics, theology, free will, mathematics, physics, biology, etc.. Philosophers are critical, and look for these issues, that is critical thinking. Those things always pop up, because knowledge evolves, and what was once cutting edge becomes old, a then the problems get exposed. The faster knowledge progresses the more these issue get overlooked, and they multiply.

Now, philosophical criticism seems to be expected in some fields, relative to ancient ideas like metaphysics, theology, etc.. When a philosopher demonstrates problems in an ancient concept of God for example, this does not surprise anyone. However, in my experience on this forum, there are certain fields, mathematics and physics, for example, where criticism is regarded as unacceptable. It's like the dogma takes hold of the people, and is adhered to in such a religious manner, that criticism (heresy) must not be allowed. Those who faithfully uphold these principles seem to be programmed to disallow criticism. When problems are pointed out, they deny that their chosen dogma and ideology could even have such issues, and use whatever means possible to hide those features.

The critical point here is that these issues, which we as philosophers point out (inconsistencies and contradictions), are not unusual in human knowledge. They are common, widespread, extending throughout all the fields of knowledge. They are nothing to be ashamed of. We all make mistakes, and the human species in general is a growing and learning culture. The real problem arises from failure to recognize mistakes as mistakes, when they are exposed and the ensuing denial. That ought to elicit shame.

This is just one example of the way in which, when you change one feature of a language-game (conceptual structure), you often have to change the meaning of other terms within that structure.
So, "countable" in the context of infinity cannot possibly mean the same as "countable" in normal contexts. In the context of infinity, it means that you can start counting the terms and count as many as you like, and there is no term that cannot be included in a count; the requirement that it be possible to complete the count is vacuous, since there is no last term. It's not a problem. — Ludwig V

Let's say that any language game is always evolving. Someone will dream up a new idea, or a new rule, in one's own private mind, and propose it to the others. They start using it, and if the others accept it, it becomes integrated into the game. If the new rule is not consistent with what's already existing then the others ought to notice this, point it out, and rectify the situation. Adopting it for use, would appear to justify it, and if it is inconsistent with some existing rule, that would be a faulty justification. It's analogous to someone offering you a proposal, and instead of thinking about it, to determine if you really agree, you just accept it, and carry on.

Obviously there is a problem in the concept of "countable". I submit that your proposal would not solve the problem. You are suggesting that when it becomes evident that the recently accepted rule is really contradictory to a previously existing rule, and ought not have been accepted in the first place, that we ought to just alter the definition of the offending word in one of the rules. But this is still not acceptable within a logical system because it amounts to equivocation. What this would do is simply obscure the obvious problem, contradiction, with a less obvious problem equivocation. Then all the problems created by what is really a contradiction would be obscured, hidden and more difficult to determine. This would amount to intentional deception, to recognize a problem of contradiction, then try to hide it behind equivocation. That's like taking a shotgun to your problem, blowing it to smithereens, so that you're left with a multitude of little problems instead of one big one.

For example, how about "there is no rational that you cannot place on the number line"? — Ludwig V

How does this make sense to you? To "place on the number line" is a procedural expression, to use Esse's word. We know that it is impossible to make the procedure of placing all the rationals on the number line. Therefore the proper conclusion and procedural statement is exactly opposite to what you propose: "there will always be rationals which you cannot place on the number line".

It is not my intention to obscure the facts. I am engaging honestly with you - and in good faith - even if it may not seem like it to you.

Here are the facts as I understand them:

The formal definition I provided to you (or similar variation) is the one you will find in many of the standard textbooks on Real Analysis, Set Theory and Discrete Mathematics that discuss countably infinite sets. This is why it confuses me when you say that you don't believe that this is the standard formal definition of "countably infinite".

Likewise, and for the same reason, I am also confused by your insistence that the definitional existence of a bijection requires that the bijection be temporally or procedurally executable. Within the global mathematics community it is commonly understood and accepted that procedural execution is not a requirement for definitional existence. This is why you will not find such a requirement listed in the aforementioned textbooks. This is also why I previously stated that adding this requirement would amount to something like an external constructivist critique of the dominant paradigm.

I hope that this helps clarify my perspective on this. I understand that you may not agree with the criticisms that I have offered, but they are based on sincere and honest confusion regarding your claims, given my current understanding of academic mathematics. I am certainly open to being mistaken on these points, but it's currently hard to see how given that these are fairly basic observations about how mathematics is currently done. Thanks.

The formal definition I provided to you (or similar variation) is the one you will find in many of the standard textbooks on Real Analysis, Set Theory and Discrete Mathematics that discuss countably infinite sets. This is why it confuses me when you say that you don't believe that this is the standard formal definition of "countably infinite". — Esse Quam Videri

You don't seem to understand the problem. "Countably" implies a procedure which you continue to deny. When we looked at the definition of "countable" it is defined by "capable", which implies "able to" perform a specified procedure. Then you claimed that mathematicians use a different definition of "capable" which doesn't imply the ability to perform a procedure. That's when I accused you of intentionally trying to obscure the issue, instead of facing the reality of it.

Likewise, and for the same reason, I am also confused by your insistence that the definitional existence of a bijection requires that the bijection be temporally or procedurally executable. Within the global mathematics community it is commonly understood and accepted that procedural execution is not a requirement for definitional existence. This is why you will not find such a requirement listed in the aforementioned textbooks. This is also why I previously stated that adding this requirement would amount to something like an external constructivist critique of the dominant paradigm. — Esse Quam Videri

Well, it appears like "the global mathematics community" is mistaken then. When something is defined in terms of the capability to perform a procedure, and then it's understood that actually being able to perform that procedure is "not a requirement" for fulfilling the criteria of that definition, then this is obviously a mistaken understanding. Don't you agree? And please, live up to your claim of "open to being mistaken on these points".

I am very much open to be mistaken. I have had numerous discussions with mathematicians on this forum, and have learned a lot, altering my perspective on many things. This issue though, as I see it, is so simple, clear, and obvious, that it would require a substantial argument to prove that I am mistaken here. But that substantial argument has not been forthcoming. People simply assert that I am mistaken, and ridicule me for arguing against "the global mathematics community", as a form of appealing to authority, rather than actually addressing the matter with clear principles.

Allow me to apologize if my previous replies came off as an attempt to ridicule you. That was not my intention.

I see that what I've said so far has not convinced you. That's understandable. That said, I'm not sure I have the ability to express my critique any more clearly than I already have. I say that not in an attempt to blame you for misunderstanding me, but more as an acknowledgement of my own limitations in that regard. I still stand by my arguments, but I'm not sure how to productively move the discussion forward from here. Thanks.

Allow me to apologize if my previous replies came off as an attempt to ridicule you. That was not my intention. — Esse Quam Videri

No, my apology too, I didn't intend to imply that you have done this, in particular. But I might mention@Banno, and a few other members in the past.

I see that what I've said so far has not convinced you. That's understandable. That said, I'm not sure I have the ability to express my critique any more clearly than I already have. I say that not in an attempt to blame you for misunderstanding me, but more as an acknowledgement of my own limitations in that regard. I still stand by my arguments, but I'm not sure how to productively move the discussion forward from here. Thanks. — Esse Quam Videri

As I said, I've learned a lot in my past discussions, so I'll offer you a perspective which you may be able to make sense of. Let's suppose that bijections simply exist without needing to be carried out as a procedural thing. This might be what's intended with the term "function".

For example, imagine that there is forty chairs in a room somewhere. There is simply an existing bijection between the chairs and the integers, so that the count is already made without having to be counted. It's just a brute fact that there is forty chairs there, without anyone counting them. This is a form of realism known as Platonic realism. The numbers simply exist, and have those relations, which we would put them into through our methods, but it is not required that we put them into those relations for the relations to exist.

I discussed this before, with @Banno I believe, in a discussion about the nature of measurement. The example was a jar of marbles. Our common intuition is that when there is a jar of marbles, or something like this, there is a measurement, a count, associated with it, the number of marbles which are in the jar. "Truth", or the correct count, would be to produce a count which corresponds with this already existing relation. You can see that this is completely different from a procedural "correct count". The procedural correctness is produced by performing the procedure correctly according to the rules, and the answer then is the correct answer without any necessary assumption of an independent measuring system (Platonic Ideals) already related, "truth".

The reason this issue came up, is because of the so-called measurement problem in quantum physics. In quantum physics it has been demonstrated that there cannot be an already existing independent measurement. So measurement is not a case of producing the result which corresponds with the already existing relation, it must be a matter of correctly carrying out the procedure.

Therefore, I argue that this is actually the true nature of "measurement" in all cases, that the correct answer is always a matter of carrying out the prescribed procedure correctly. Consequently I also argue that the Platonic realism which supports the other, intuitive notion of measurement, that there are independent numbers, which are already associated with things, as the true measurement, is misleading. This issue becomes very evident in the notion of infinity.

There's a category error that involves thinking that because we can't start at one and write down every subsequent natural number, they don't exist.

1 is a number, and every number has a successor. That's enough to show that the natural numbers exist.