which is a conclusion that Cantor accepted because it resonated with his theology. — sime

Are you suggesting that is a reason for rejecting his conclusions? Either way, I would suggest that we leave Cantor's theology as a matter between Cantor and his God.

Why do you think the proposition that the natural numbers is countable does not contradict the proposition that the natural numbers are infinite, in the way I explained? — Metaphysician Undercover

It depends, as I explained earlier, how you define "countable". I don't say that it's just all just a matter of definitions, but it's probably a good idea to get those agreed so that we can be sure we are talking about the real issues. As it is, we don't agree and so we never get to identify and discuss the real issues.

That's exactly right. To say that 2 is between 1 and 3 is to say that it serves as a medium. However, in the true conception and use of numbers, 1, 2, 3, is conceived as a unified, continuous idea. This unity is what allows for the simple succession representation which you like to bring up. No number is between any other number, they are conceived as a continuous succession. To say that 2 comes between 1 and 3 is a statement of division, rather than the true representation of 1, 2, 3, as a unity, in the way that the unified numbering system is conceived and applied. — Metaphysician Undercover

I'm not sure what you mean by "serves as a medium". I accept you are right to observe that the numbers are defined as a succession. (I don't know why you call the successor function a representation of something, but let that pass...) But the point of a succession is that every step (apart, perhaps, from 0) has a predecessor and a successor. That is what it means to say that n is between n-1 and n+1. It is not wrong to say that 2 unites 1 and 3 and it is not wrong to say that 2 divides 1 and 3. But it is wrong not to say both.

"Infinite" means limitless, boundless. The natural numbers are defined as infinite, endless. limitless. All measurement is base on boundaries. To say a specific parameter is infinite, means that it cannot be measured. Counting is a form of measurement. Therefore the natural numbers cannot be counted. To propose that they are countable, is contradictory, because to count them requires a boundary which is lacking, by definition. — Metaphysician Undercover

This just turns on your definition of what it is to count something.
Using a ruler to measure a (limited) distance means counting the units. Obviously, we need enough numbers to count any distance we measure. So having an infinite number of numbers is not a bug, but a feature. It guarantees that we can measure (or count) anything we want to measure or count.
I maintain that if you can start to count some things, they are countable. You maintain that things are countable only if you can finish counting them., It's a rather trivial disagreement about definitions. But I do wonder how it is possible to start counting if I can only start if I can finish.

- I've been meaning to return to this for a while now, but just haven't had time. You're already juggling multiple interlocutors; hopefully this won't be interpreted as "piling on".

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. — Metaphysician Undercover

I see what you are saying here. I was coming at this from a slightly different angle.

I take it that you are aware that there are several different axiomatizations of set theory. Some examples are: ZFC, ZF, Z, CZF, IZF and various Finitistic and even Ultrafinitistic systems.

The argument about measurement that you provided in your reply is interesting, and I can see how it is relevant to question of whether (or in what sense) a countably infinite set can be said to "exist". But the word "exists" can have different meanings depending on the context. Within the context of ZFC set theory, to say that a countably infinite set "exists" doesn't imply that it exists in some Platonic heaven. That's not to say that you couldn't interpret it in a Platonic way, just that nothing in ZFC itself forces this interpretation.

Now, I see that a few others on the thread have raised a similar point and that you have not been convinced. That's fair. I doubt that I will be able to convince you either, but I will try to explain how I see it and then you can let me know what you think.

The way I (and many others) interpret the word "exists" with respect to ZFC set theory is something like "there is a derivation from the axioms of ZFC using the inference rules of classical first-order logic, of the formula ∃x P(x)". Or, more compactly, ZFC ⊢ ∃x P(x).

So to say that "a countably infinite set exists" is just to say "ZFC ⊢ ∃x CountablyInfinite(x)". The actual derivation follows very simply from the axiom of infinity in combination with the definition of "countably infinite".

In my view, accepting this does not mean that you have to believe that countably infinite sets "exist" in any other sense, whether that be in a Platonic heaven, in the mind of God, or as an actual collection of objects somewhere within the physical universe.

What are your thoughts on this?

(The finitude of an object's exact position in position space, becomes infinite when described in momentum space, and vice versa. Zeno's paradox is dissolved by giving up the assumption that either position space or momentum space is primal) — sime

i don't consider this to be a solution, because the result is the uncertainty principle. What you indicate is two distinct concepts of space which are incompatible, "position space", and "momentum space".

It depends, as I explained earlier, how you define "countable". I don't say that it's just all just a matter of definitions, but it's probably a good idea to get those agreed so that we can be sure we are talking about the real issues. As it is, we don't agree and so we never get to identify and discuss the real issues. — Ludwig V

We went through the common definition of "countable" provided by jgill, and the contradiction remained. So I really don't know what type of definition of "countable" you might be thinking of.

I'm not sure what you mean by "serves as a medium". — Ludwig V

"Medium is commonly defined as "something in a middle position". If something is between two things, it is distinct from each of the two as in the middle.

But the point of a succession is that every step (apart, perhaps, from 0) has a predecessor and a successor. That is what it means to say that n is between n-1 and n+1. It is not wrong to say that 2 unites 1 and 3 and it is not wrong to say that 2 divides 1 and 3. But it is wrong not to say both. — Ludwig V

Yes, every step has a successor, but the succession is described as a continuous process. No individual step can serve as a division between the prior step and the posterior, as each is continuity, not a division. To say that one step is a division would produce two distinct successions, one prior one posterior. then the one which served as the divisor would have no place in either of the two successions.

So I dont't understand what you are saying here, especially what you mean by "2 divides 1 and 3". One divided by two produces a half, and three divided two produces one and a half. But it doesn't make sense to say that two acts as a division between one and three in the way that you propose.

This just turns on your definition of what it is to count something.
Using a ruler to measure a (limited) distance means counting the units. Obviously, we need enough numbers to count any distance we measure. So having an infinite number of numbers is not a bug, but a feature. It guarantees that we can measure (or count) anything we want to measure or count.
I maintain that if you can start to count some things, they are countable. You maintain that things are countable only if you can finish counting them., It's a rather trivial disagreement about definitions. But I do wonder how it is possible to start counting if I can only start if I can finish. — Ludwig V

So it looks like you disagree with my premise that counting is a form of measurement. Since you claim that starting to count something is sufficient to claim that it is countable, then if we maintain consistency for other forms of measurement, puling out the tape measure would be sufficient to claim that the item is measurable. Since this is obviously not true, it seems you are claiming that counting is not a form of measurement at all. How would you define "countable"?

Are you suggesting that is a reason for rejecting his conclusions? Either way, I would suggest that we leave Cantor's theology as a matter between Cantor and his God. — Ludwig V

I'm saying that in the presence of an inconsistency between ZFC and computable notions of mathematics, coupled with the obvious uselessness of of non-constructive cardinal analysis, the theological origin of ZFC becomes conspicuous.

i don't consider this to be a solution, because the result is the uncertainty principle. What you indicate is two distinct concepts of space which are incompatible, "position space", and "momentum space". — Metaphysician Undercover

yes, there are two incompatible bases for describing dynamics, and in line with your proposal, what looks like a hypertask iwhen measured in one basis, is a mere task in the other (where measurement is understood as destructive interference).

I presently suspect that the structure of the uncertainty principle, that concerns non-commutative measurements, is a logical principle derivable from Zeno's arguments, without needing to appeal to Physics.

So I dont't understand what you are saying here, especially what you mean by "2 divides 1 and 3". One divided by two produces a half, and three divided two produces one and a half. But it doesn't make sense to say that two acts as a division between one and three in the way that you propose. — Metaphysician Undercover

I'm sorry. I should have said "separates", not "divides".

So it looks like you disagree with my premise that counting is a form of measurement. — Metaphysician Undercover

Can you think of a form of measurement that is not counting - apart from guessing or "judging"?

Since you claim that starting to count something is sufficient to claim that it is countable, then if we maintain consistency for other forms of measurement, puling out the tape measure would be sufficient to claim that the item is measurable. — Metaphysician Undercover

I disagree. Since this is not an argument, it seems inappropriate to reply.

I'm saying that in the presence of an inconsistency between ZFC and computable notions of mathematics, coupled with the obvious uselessness of of non-constructive cardinal analysis, the theological origin of ZFC becomes conspicuous. — sime

OK. Obviously I'm not in a position to comment.

Again, there seems to me to be a bunch of errors in what you have said here. The core one seems to be equating P(N) with the decidable sets.

The statement “We can construct an injection P(N)→N via Turing machine encoding of decidable sets”
would mean every subset of N can be uniquely encoded by a natural number. But that is equivalent to saying ∣P(N)∣≤∣N∣, which directly contradicts Cantor’s theorem. So if the statement were true, Cantor’s theorem would already be false.

There are undecidable subsets of N. We cannot construct an injection P(N)→N via Turing machine encoding of decidable sets

I'll stop there. I can't see that your account works.

, Given Meta's rejection of quantification, and now of numbers being ordered, it's about as clear as it could be that for Meta there is very little left of mathematics.

There is a point at which one's interlocutor's commitments collapse the subject matter under discussion.

That's where we are at with Meta.

As Frank points out,

It really comes down to which view best accommodates what we do with math. — frank

And Meta's view undermines most of mathematics, despite what we do with it.

Meta treats the ∃ of quantification, a logical move within the game of maths that understands there is a symbol n in the domain of discourse that satisfies P according to the rules of the theory, as if it implies n exists as an abstract object independent of language, symbols, or human conventions. That's just a muddle. At the core he perhaps does not understand the difference between syntax, semantics and ontology.

Given that Meta asserts that 2 is not between 1 and 3, I think I'm done here. I don't see any gain in showing further absurdities in his position.

And Meta's view undermines most of mathematics, despite what we do with it. — Banno

A nominalist would provide an argument for why we can use math without committing to abstract objects. I guess Meta is a math skeptic.

've been meaning to return to this for a while now, but just haven't had time. — Esse Quam Videri

I'm glad you're back.

The argument about measurement that you provided in your reply is interesting, and I can see how it is relevant to question of whether (or in what sense) a countably infinite set can be said to "exist". — Esse Quam Videri

The issue of platonism is more about the existence of any bijection in general, and the question of whether a measurement exists without requiring someone to measure it. It's a form of naive realism, which in our conventions and educational habits, we tend to take for granted. We look at an object, a tree, a mountain, etc., and we assume that it has a corresponding measurement, without requiring that someone measures it. Then, when someone goes to measure it, the correct measurement is assumed to be the one which presumably corresponds with the supposed independent measurement. This type of realism requires platonism because there must be independent numbers and measuring standards which exist independently from any mind, in order that the object has a measurement before being measured.

But if we understand that numbering conceptions, and measuring conceptions are products of the human mind, then it's impossible that an independent object could have a measurement prior to the measurement being made by a human being. This rules out the possibility that the natural numbers could have a measurement, or be countable because we know that human beings could not count them all.

The argument about measurement that you provided in your reply is interesting, and I can see how it is relevant to question of whether (or in what sense) a countably infinite set can be said to "exist". But the word "exists" can have different meanings depending on the context. Within the context of ZFC set theory, to say that a countably infinite set "exists" doesn't imply that it exists in some Platonic heaven. That's not to say that you couldn't interpret it in a Platonic way, just that nothing in ZFC itself forces this interpretation. — Esse Quam Videri

So we could say, that numbers "exist" in a way other than platonic realism, but we must consider what would be meant by this. We need to ask, what is the criteria for existence. Consider the difference between the following two statements. 1. "the set of natural numbers between 0 and 5". 2. "{1,2,3,4.}". We might say at first glance that they both say the very same thing, and they both necessitate the existence of those four integers, but this would not be correct. That is because the first is a formula, and the existence of the named integers requires that the formula be carried out correctly. So we need to respect this difference, the existence of the integers in the first example is conditional, or contingent, on "correctness", and in the existence is necessary. And when we say 1, 2, 3, ..., or use the successor function, the existence of those numbers is conditional, contingent on correctness. Now we have the problem mentioned above, the natural numbers cannot have a measurement, because the procedure cannot be carried out to the end, "correctly".

The inclination might be to deny that distinction between necessary existence and conditional existence, which I provided. But we cannot do this because we need to account for the reality of human error. A formula does not necessitate the existence of numbers because error may arise in a number of different ways. The formula might be carried out incorrectly, or it might be a mistaken formula in the first place. So we might add, "the designated numbers exist if the the formula is properly formulated, and if it is carried out correctly. But that makes it conditional.

So to say that "a countably infinite set exists" is just to say "ZFC ⊢ ∃x CountablyInfinite(x)". The actual derivation follows very simply from the axiom of infinity in combination with the definition of "countably infinite". — Esse Quam Videri

So you are talking about a conditional existence. The supposed existence of the natural numbers, is dependent on the correct procedure. The issue with the definition of "countably infinite" is that the procedure cannot be carried out. The formula states something which is impossible to correctly finish, therefore the numbers cannot exist.

Furthermore, platonism doesn't solve the problem because the infinite is defined as being impossible, so the numbers cannot even exist in a platonic realm. That would be like saying that the full extension of pi exists in a platonic realm, when this has been demonstrated to be impossible.

I presently suspect that the structure of the uncertainty principle, that concerns non-commutative measurements, is a logical principle derivable from Zeno's arguments, without needing to appeal to Physics. — sime

I agree. Many people conclude that calculus solved Zeno's paradoxes. I've argued elsewhere, that all calculus provided was a workaround, which was sufficient for a while, until the problem reappeared with the Fourier transform.

I'm sorry. I should have said "separates", not "divides". — Ludwig V

I don't think this makes any difference.

Can you think of a form of measurement that is not counting - apart from guessing or "judging"? — Ludwig V

Sure, I believe measuring is fundamentally a form of ordering. So most comparisons which are intended to produce an order are instances of measuring. Get a bunch of people, compare their heights, and order them accordingly. That's a form of measuring.

As Frank points out,
It really comes down to which view best accommodates what we do with math.
— frank
And Meta's view undermines most of mathematics, despite what we do with it. — Banno

You mention "what we do with math", but are neglecting something very important, "what we can't do with math". This is the limitations, like the uncertainty principle mention above. We do a lot with math, sure, but there is a lot more we would be able to do if we work out some of the bugs. Then there's the even worse problem of the many things that people believe we do with mathematics, which we really don't. Some people think that calculus has solved Zeno's paradoxes. It has not. Some people think that mathematics allows us to determine the velocity of an object at a single instant in time. It doesn't. Some people think that mathematics has provided a way to make infinite numbers countable. It has not. That's what I'm talking about. To have the attitude that math is perfect, ideal, therefore it is wrong to subject it to philosophical skepticism is the real problem.

I guess Meta is a math skeptic. — frank

I like to apply a healthy dose of skepticism to any so-called knowledge. Nothing escapes the skeptic's doubt.

I guess Meta is a math skeptic.
— frank

I like to apply a healthy dose of skepticism to any so-called knowledge. Nothing escapes the skeptic's doubt. — Metaphysician Undercover

I suppose so, but the GPS in your phone was designed using math invented by Descartes. It's so weird that your GPS works even though math does not exist. :confused: