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The following is an exercise in Just/Weese:

Prove that for every set $X$ there exists exactly one initial ordinal $\kappa$ such that $X \approx \kappa$.

An ordinal is called initial ordinal if it is not equipotent with any smaller ordinal.

Can you tell me if my proof is correct? Thank you.

Assume there are two initial ordinals, $\kappa \neq \kappa'$ with $\kappa \approx X \approx \kappa'$. Wlog let $\kappa < \kappa' $. Then this is a contradiction to $\kappa'$ being the smallest ordinal in bijection with $X$.

To show existence assume AC so that $X$ can be well-ordered. Let $o$ be the order type of $X$. Define $S = \{o' \mid o' \in \mathbf{ON}, o' \approx o \} \subseteq \mathbf{ON}$. Since $\mathbf{ON}$ is a well-order, $S$ has a minimal element $s_0$. This is the initial ordinal of $X$.

The exercise is rated difficult so I expect this proof to be flawed but I can't seem to spot the mistake.

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I see nothing wrong with your overall reasoning. Though the contradiction in your uniqueness proof is to $\kappa^\prime$ being an initial ordinal, since it is equipotent with a strictly smaller ordinal. –  Arthur Fischer Dec 9 '12 at 10:56
    
@ArthurFischer Thank you. Then perhaps the rating of the exercise is a typo. –  Rudy the Reindeer Dec 9 '12 at 10:57
    
The proof seems fine to me. –  Michael Greinecker Dec 9 '12 at 10:58
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Most of the PG-rated exercises in Just-Weese just require having the correct idea about the concepts they are about. If you get the right insight quickly, they may seem very easy and solutions write themselves. If you don't, they may seem very hard. (This is all to say that if a PG exercise seems easy to you, it is likely because you are understanding the underlying concepts.) –  Arthur Fischer Dec 9 '12 at 11:21
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I have read a large chunk of the second volume, though I cannot say I've been through all of it. It continues with most of the conversational style of the first, however it is much more densely written, with even more exercises. Overall, I think it is a good second text in set theory, and should prepare you well for going through the classic text of Kunen. –  Arthur Fischer Dec 9 '12 at 12:56

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Your proof is correct.

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