# Relative Consistency proof for the Axiom of Choice

I've been reading about relative consistency proofs in Kunen's latest Set Theory text and going over which Axioms hold in certain classes. To get my hands dirty, I've been working on an exercise that involves the class $HWO$ of "hereditarily well-orderable sets".

$HWO$ is the class of all $x \in WF$ such that trcl$(x)$ can be well-ordered. $WF$ is the proper class of all well-founded sets, and trcl$(x)$ is the transitive closure of $x$.

I was able to show that the axioms of Extensionality, Foundation, Comprehension, Pairing, Union, and Infinity hold in $HWO$. In addition to these axioms, it is also asked to show that the Axiom of Choice holds in $HWO$ (not assuming the Axiom of Choice). I'm having trouble showing that the Axiom of Choice holds in $HWO$. Any help would be greatly appreciated. Thanks in advance!

• "it is also asked to show that the Axiom of Choice holds in HWO. I'm not sure how to go about showing this or why this axiom holds in HWO." Did you say twice the same thing? Oct 29, 2013 at 1:57
• @AndresCaicedo I might have. Sorry about that. I will fix my wording. Oct 29, 2013 at 2:25

HINT: Suppose that $F\in HWO$ is a family of non-empty sets. Use the fact that $\bigcup F\subseteq\operatorname{trcl}(F)$ to define a choice function.
• From your hint, since $\bigcup F \subseteq \mbox{trcl}(F)$, $\bigcup F$ is well-ordered since any subset of a well-ordered set is well-ordered. Wouldn't it follow that we can find a choice function since $\bigcup F$ is well-ordered? Oct 29, 2013 at 11:37