Can a set(in ZFC) be well-ordered with any order type equipotent to it?

Question

Let $D$ be a set with cardinality $\aleph_\alpha$, and give an ordinal number $\beta$ between $\omega_\alpha$ and $\omega_{\alpha+1}$, can $D$ be well-ordered with order type $\beta$?

Answer 1

Note that $\omega_\alpha\leq\beta<\omega_{\alpha+1}$ means that there is a bijection between $\beta$ and $\omega_\alpha$.

If $D$ has a bijection with $\omega_\alpha$ then it has a bijection with $\beta$ as well. Let $f\colon D\to\beta$ be such bijection, we define $\prec$ on $D$: $$x\prec y\iff f(x)<f(y)$$

It is not hard to see that $f$ is now an order isomorphism, so $(D,\prec)$ has order type $\beta$, as wanted.

Also note that we didn't use the axiom of choice at any point. We assumed that $D$ was well-ordered to begin with, so the axiom of choice was not needed to have that.

Answer 2

What does it mean to be equipotent? It means that there exists a bijection $f:D\to\beta$. Then simply define $R:=\{(a,b)\mid f(a)\le f(b) \}$. This is going to give a well-ordering of type $\beta$.