> But in the third, it doesn't, it can't, and the reals are countable
Isn't that a bit of an overreach? It seems that in Constructivism, Cantor's diagonalization doesn't work as a proof. But that doesn't mean that the reals are countable, just that this proof isn't valid.
Well, you have to be careful about what countable means.
If countable means "put into a one to one correspondence with" then the reals are not countable because there are pairs of reals that can be proven to be real whose equality or inequality can't be proven. So in such cases you can overcount, or undercount, but you can never exactly pick each real only once.
But the set of possible constructions can be enumerated. We may not know which actually are real numbers, and which pairs are equal. But we can list all of the possibilities. So there is a countable list of things that contains them all..multiple times..along with other stuff.
Isn't that a bit of an overreach? It seems that in Constructivism, Cantor's diagonalization doesn't work as a proof. But that doesn't mean that the reals are countable, just that this proof isn't valid.