# Math Help - Must a countable set be indexed with the aid of Axiom of Choice?

1. ## Must a countable set be indexed with the aid of Axiom of Choice?

that is, is it true that we can not prove the existence of function $f: \mathbb N\to A$ such that $ran f=A$ without invocation of Axiom of Choice, where $\mathbb N$ is the set of all natural numbers and $A$ is countably infinite?

2. Well, I understand: to simply prove the existence of such an index does not need Axiom of Choice, but singling out one such index out of a set of indices needs Axiom of Choice. So the first sentence "Let $A_n=\{a_{1,n},a_{2,n},a_{3,n},...\}$" of the proof of Th2.25 of Apostol's "Mathematical Analysis" is actually using Axiom of Choice implicitly.

3. I haven't seen the proof. But I think you're correct if the writer assumes that A can be well-ordered.

Let