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

  1. #1
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    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?
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  2. #2
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    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.
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  3. #3
    Senior Member Dinkydoe's Avatar
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    I haven't seen the proof. But I think you're correct if the writer assumes that A can be well-ordered.

    Let
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