system
Given the following post, respond as if you're commenting on Reddit:
human
ELI5:Axiom of Choice

Trying to wrap my head around this concept.  If someone could kindly duh it down for me I would be grateful.

Thanks!
gpt
A "choice function" is a function which takes as input any one of a collection of sets, and returns some element of that set. So if we have all of the nonempty sets of positive integers, we can define the function "f(S) = the smallest integer in S". For example, f({11,12,13}) = 11, while f({2, 4, 6, 8, ...}) = 2. This is a simple example of a choice function. On the other hand, "f(S) = the largest integer in S" is *not* a valid choice function, since the set of all even positive integers {2,4,6,8, ...} has no largest element, and so f({2,4,6,8...}) would be undefined.

But what if, instead of nonempty sets of positive integers, we're working with nonempty sets of real numbers? If we try to just copy over the same choice function, we'll find "the smallest real number in S" is undefined for a great many sets of real numbers, such as the set of all negative real numbers. Other basic attempts at defining a choice function will similarly run into problems with sufficiently weird sets of real numbers. It's far from clear whether a choice function should exist that is defined on all possible nonempty sets of real numbers.

The axiom of choice states that for *any* collection of nonempty sets (whether nonempty sets of positive integers, nonempty sets of real numbers, or anything else) there always exists at least one choice function. It doesn't say anything about what the function is or how it works - it just says that there *must* be some such function. Surprisingly, this axiom is necessary to resolve several basic-seeming conjectures in set theory (and occasionally other areas). Some of them even feel "obvious", like the statement: given two sets A and B, either they are the same size or one is smaller than the other. Others are deeply counterintuitive, like the Banach-Tarski paradox, which shows that (assuming the axiom of choice) it is possible to cut a sphere into five pieces, move and rotate the pieces, and end up with two spheres identical to the original. Regardless of this latter category, the axiom of choice is generally accepted by mathematicians - though there are still some who reject the axiom of choice or who work with alternate axioms which negate the axiom of choice (the most prominent being the axiom of determinacy).