You should be able to solve this.

True or false: Are algebraic numbers countable?
You can't use axiom of choice.

>>10 That's not necessary a biyection:
1) It is a biyection iff the elements of X are disjoint.
2) Even with that, you are using AC in your proof. How do you know that the {gi : gi is a biyection between f(i) and N} is a set?