You are being very cryptic. Are you trying to say that the existence of uncountable sets requires the axiom of choice? If you are, that's false. If you aren't, I'm not sure what you are trying to say.
I'm definitely not trying to say that the existence of uncountable sets requires the axiom of choice. Cantor's diagonalization argument for the reals demonstrates otherwise.
I'm saying that to go from the uncountability of the reals to the idea that this implies that the infinity of the reals is larger, requires making some important philosophical assumptions. Constructivism demonstrates that uncountable need not mean more.
On the algorithm example, you could have asked what I was referring to.
The result that I was referencing follows from the https://en.wikipedia.org/wiki/Robertson%E2%80%93Seymour_theo.... The theorem says that any class of finite graphs which is closed under graph minors, must be completely characterized by a finite set of forbidden minors. Given that set of forbidden minors, we can construct a polynomial time test for membership in the class - just test each forbidden minor in turn.
The problem is that the theorem is nonconstructive. While it classically proves that the set exists, it provides no way to find it. Worse yet, it can be proven that in general there is no way to find or verify the minimal solution. Or even to provide an upper bound on the number of forbidden minors that will be required.
This need not hold in special cases. For example planar graphs are characterized by 2 forbidden minors.
For the toroidal graphs, as https://en.wikipedia.org/wiki/Toroidal_graph will verify, the list of known forbidden minors currently has 17,523 graphs. We have no idea how many more there will be. Nor do we have any reason to believe that it is possible to verify the complete list in ZFC. Therefore the polynomial time algorithm that Robinson-Seymour says must exist, does not seem to exist in any meaningful and useful way. Such as, for example, being findable or provably correct from ZFC.
