Axiom of Choice - Naive Counterexample

Question

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Axiom of choice question

I know there is a lot of discussion on the axiom of choice and, in fact, I attended once a lecture on it, but I still cannot understand the following: Let $A$ be a nonempty set. I want to pick an element from $A$. Since by hypothesis $A$ is nonempty, it must contain at least one element, $a$. So I pick this $a$. Where is the "gap" or the need for the axiom of choice?

Answer 1

Finite choice is simply true (in ZF say).

You can of course pick an element from a single set, you can do this yourself: go up to the set and select out one of the elements, just as you have described.

The problem comes when you need to make infinitely many different choices.

In this case the problem is not so clear, as you cannot select these yourself anymore.

Answer 2

Finite choice is in fact easy to prove without needing any axiom of choice; it's essentially a consequence of induction; what you say is in fact perfectly true.

For an infinite number of collections, one can no longer do that, and in many cases an explicit choice function cannot be constructed (and this results in philosophical conflicts, especially for some of those in the constructive camp).