>>13951029To expand a bit on the last point: It's clear that all words can be enumerated, and all C++ codes for the form "int main{ ··· }" are among the words. If you don't adopt excluded middle (as e.g. in Russian constructivism), it's consistent that the real numbers are exactly only all the codable sequences. This makes use of the fact that not all subsets S of N are _effectively_ countable, but if you adopt LEM, your theory will formally claim all subsets S of N are countable.
Either way, even if the uncountable set of reals R is the surjective image of an (uncountable) subset S of N (S is subset of N, but there's no effective bijection between them, i.e. program that computes the association), then there's still always an injection from N to R or from Z to R.
