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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!

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  • $\begingroup$ "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? $\endgroup$ – Andrés E. Caicedo Oct 29 '13 at 1:57
  • $\begingroup$ @AndresCaicedo I might have. Sorry about that. I will fix my wording. $\endgroup$ – Maria Oct 29 '13 at 2:25
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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.

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  • $\begingroup$ 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? $\endgroup$ – Maria Oct 29 '13 at 11:37
  • $\begingroup$ @Maria: Yes. Exactly. $\endgroup$ – Asaf Karagila Oct 29 '13 at 11:42
  • $\begingroup$ Thanks again Asaf! This was, yet again, one of those problems where I thought about it too much and made it more difficult than it really is. $\endgroup$ – Maria Oct 29 '13 at 14:00
  • $\begingroup$ That's a big part of understanding things. $\endgroup$ – Asaf Karagila Oct 29 '13 at 21:37

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