[aaachilless]

I don't think I buy your argument, but I'm absolutely not a mathematician, so I could have missed something. Why I don't buy it:

It seems you didn't need to redefine the reals as programs to make this argument--the reals are classically defined as Cauchy sequences of rationals and (if the argument were valid) you could make the same claim using these sequences instead of programs.

I don't think the actual philosophical question is whether the reals are countable, it's whether the reals "exist". You can't define the reals such that they're countable because if you did, they wouldn't be the same structure.

[btilly]

Here is what you have missed.

The difference between the classical version and the constructive version that I wrote down is in what kind of rules you can use to construct a Cauchy sequence. My version is something that can be written down in a Turing machine. The classical version is that rules are anything that can follow a set of axiomatic rules - and includes rules that we cannot, even in principle, think of trying to evaluate.

Therefore in the classical version, you can create a rule that depends on being able to evaluate every other rule - including rules similar to itself. That is exactly what Cantor's argument does. In the constructive version you can only create rules using operations that we can guarantee will finish. Which doesn't include verifying the incalculable truth or falsity of arbitrary other rules.

Moving on, though, existence is critical. The key philosophical difference that drove the debate was pure existence proofs. Is proof by contradiction a valid method of reasoning with infinite sets? In particular, can you establish the existence of something by proving a contradiction if it does not exist? In what sense does the thing proven to exist actually exist? What if we prove that it exists but have no way to find it? What it we prove that it exists, have no way to find it, and no way to verify that we have found it?

These are not hypothetical questions. See the https://en.wikipedia.org/wiki/Robertson%E2%80%93Seymour_theo... for a class of graph theory problems, all of which have polynomial time algorithms to solve. But we have no way to find those solutions. And in some cases it is impossible for us to verify whether a purported solution is a solution even if we were handed it.

So..do you believe in the existence of things in an infinite set that are impossible to find and impossible to verify?

[MichaelBurge]

And then there's the Axiom of Choice and all that it brings. There's a Hamel Basis for the reals which is provably impossible to construct(since it's equivalent to AC).

Do you believe in the existence of things that are provably impossible to construct?

[btilly]

As it happens, I don't.

That said I'm quite able to quote and do mathematics that depends on axioms that I do not accept. I can be quite the formalist when it is convenient. But I privately think of it as formal bullshit. And don't like seeing people who haven't accepted it beaten over the head until they do.

[Smaug123]

For that matter, do you believe in really, really big integers?

[MichaelBurge]

There are some pretty big integers out there:

http://www.scottaaronson.com/blog/?p=2725

I think the current record is 1919 states, so if you run a certain Turing Machine for BB(1919) steps and it doesn't halt then you know that ZFC is consistent. Godelian considerations might make it reasonable to say that integers BB(1919) or larger don't exist.

I get skeptical of integers so large that you need weird Turing Machines to enumerate their digits whose halting proof is independent of set theory, but I don't yet disbelieve in their existence.