Mathematics Stack Exchange is a question and answer site for people studying math at any level and professionals in related fields. Join them; it only takes a minute:

Sign up
Here's how it works:
  1. Anybody can ask a question
  2. Anybody can answer
  3. The best answers are voted up and rise to the top

Possible Duplicate:
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?

share|cite|improve this question

marked as duplicate by Cameron Buie, Carl Mummert, Chris Eagle, Ross Millikan, Thomas Andrews Dec 11 '12 at 20:39

This question has been asked before and already has an answer. If those answers do not fully address your question, please ask a new question.

It's not even clear why you call this a "counter-example," since it is an example of choice working (although an example that doesn't need the Axiom of Choice to work.) – Thomas Andrews Dec 11 '12 at 20:40
up vote 8 down vote accepted

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.

share|cite|improve this answer

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).

share|cite|improve this answer

Not the answer you're looking for? Browse other questions tagged or ask your own question.