What does Axiom of Choice mean

Question

So this a fundamental assumption in mathematics. Can someone explain informally what it actually is please.

My guess is that its when we say in proofs that "Let $x \in X$". But I am not sure.

Answer 1

For the deepest results about partially ordered sets we need a new set-theoretic tool; ... We begin by observing that a set is either empty or it is not, and if it is not, then, by the definition of the empty set, there is an element in it. This remark can be generalized. If $X$ and $Y$ are sets, and if one of them is empty, then the Cartesian product $X\times Y$ is empty. If neither $X$ nor $Y$ is empty, then there is an element $x$ in $X$, and there is an element $y$ in $Y$; it follows that the ordered pair $(x,y)$ belongs to the Cartesian product $X\times Y$, so that $X\times Y$ is not empty. The preceding remarks constitute the cases $n=1$ and $n=2$ of the following assertion: if $\{ X_i\}$ is a finite sequence of sets, for $i$ in $n$, say, then a necessary and sufficient condition that their Cartesian product be empty is that at least one of them be empty. The assertion is easy to prove by induction on $n$. .... The generalization to infinite families of the non-trivial part of the assertion in the preceding paragraph (necessity) is the following important principle of set theory.

Axiom of choice: The Cartesian product of a non-empty family of non-empty sets is non-empty.

Paul Halmos Naive Set Theory page 59.

Answer 2

Suppose you have a set $X$ (which may be infinite which is when things get exciting) and another set $Y$. Suppose that for each $x\in X$ you associate a non-empty subset $Y_x$ of $Y$, then there is a function $f:X\to Y$ such that $f(x)\in Y_x$. This seemingly intuitive "fact" (and in fact, trivial for finite $X$) is the Axiom of Choice. To be specific, the existence of $f$ is what the Axiom tells you. $f$ is a function that for each $x$ "chooses" in $f(x)\in Y_x$.

Answer 3

Informally the axiom of choice says if you have a bunch of sets (possibly infinitely many) then you can choose one element from each set and build a new set out of them.

That you can do such a thing sounds at first incredibly intuitive and obvious. But over time mathematicians have realized that it is exactly that assumption that leads to a bunch of really disturbing results. Therefore some people have concluded that we shouldn't allow that assumption. However if you don't there are also a bunch of basic things that become much harder if not impossible to prove.

It's an ongoing debate with no end in sight.