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.