# Naive understanding of choice axiom [duplicate]

It's written in many resources that if we consider only finite sets, that choice axiom can be skipped and be proven from other ZF axioms.

I remember I've read the following explanation. If $A$ is a finite set, there is exists some natural $n, |A|=n$, therefore exists some bijection $f:\{1,\cdots,n\}\to A$. And we can choose an element from $A$ by choosing $f(1)$

But there is a problem with this explanation.

Actually there are $n!$ of such bijections. So how can we choose one $f$ of them without this axiom of choice?!

UPD: actually I can simplify my question heavily.

Let's consider one-point set $A, |A|=1$. How can I get an element from this set without using axiom of choice? We know that $A=\{x\}$ for some $x$. But we don't know how to obtain it. What allows us to just say Let's pick $x$ from $A$?

## marked as duplicate by Cameron Buie, Community♦Jul 1 '16 at 12:01

• The existential claim is entirely the Axiom of Choice! Sure there is an inductive algorithm, since we can assume that we may pick an element from a given set: pick $x_0$ from $A_0$, pick $x_1$ from $A_1$ ... This is entirely unnecessary though – basket Jul 1 '16 at 9:14