The Axiom of choice

Question

I'm a little lost with this proof:

If every set is equipotent to an ordinal, then we have the axiom of choice

And I want to know if someone can help or maybe give me a hint of how to proceed.

Answer 1

HINT:

• If you have a choice function for $\wp(X)\setminus\{\varnothing\}$ and a bijection $f:Y\to X$, it’s straightforward to construct a choice function for $\wp(Y)\setminus\{\varnothing\}$.
• If $\alpha$ is an ordinal, there is a very simple choice function for $\wp(\alpha)\setminus\{\varnothing\}$.

Answer 2

HINT:

1. To show that the axiom of choice holds it suffices to show that for every non-empty $X$, there exists a choice function on $\mathcal P(X)\setminus\{\varnothing\}$.

2. If $(A,\leq)$ is a well-ordered set then there is a choice function for $\mathcal P(A)\setminus\{\varnothing\}$ definable from $\leq$.

3. If $A$ is equipotent with an ordinal then $A$ can be well-ordered.