[pdonis]

These concerns don't apply to the claim that the set of real numbers is uncountable. Cantor's diagonal proof is constructive: given any countable set of real numbers, it tells you how to construct a real number that is not in the set. That is sufficient to show that the set of real numbers cannot be countable. Also, even though many real numbers cannot be written down with a finite set of symbols, Cantor's diagonal proof can be.

[ummonk]

You're assuming you've been able to construct all those real numbers in the first place, using arbitrary imaginary cauchy sequences (i.e. cauchy sequences that cannot be constructed but rather rely on some magic axiom of infinite choice).

[papandada]

Such an interesting criticism. All of this thread has been enlightening. I sat through traditional undergrad analysis and set theory classes and never imagined there was another option, other than things bothering me for decades since in my subconscious.

[btilly]

Yes, you can write down Cantor's diagonal proof. But if you're careful about it, you find that the diagonalized thing is not a real number. For example a program to list programs that can be proven (by some set of axioms) to create Cauchy sequences can be used to create a Cauchy sequence, but you can't prove that that sequence is a Cauchy sequence without running into Gödel's incompleteness theorem.