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a little question about admissible sets: Is every $\mathfrak{M}$-admissible ordinals an admissible ordinal ? where $\mathfrak{M}$ is a $L$-structure over $L=\{R_1,\dots,R_k \}$.

Thanks.

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Yes, admissibility relativizes downward. For a transitive structure to be admissible, it must be amenable, and satisfy $\Sigma_0$-collection. Both are conditions that keep holding if you "remove parameters". This is obvious for amenability. For collection, a little argument is needed. If you have access to Devlin's "Constructibility" book, this is at the beginning of II.7. (Sorry, I do not currently have time to flesh out details.) –  Andres Caicedo Jul 13 '12 at 16:03
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Unrelated: is your name supposed to be a pig-Latinization of "Barwise"? –  Quinn Culver Jul 13 '12 at 16:36

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Removing this question from the unanswered list:

Yes, admissibility relativizes downward. For a transitive structure to be admissible, it must be amenable, and satisfy $\Sigma_0$-collection. Both are conditions that keep holding if you "remove parameters". This is obvious for amenability. For collection, a little argument is needed. If you have access to Devlin's "Constructibility" book, this is at the beginning of II.7. (Andres Caicedo)

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(Yes, I assume the question was in the context of Barwise's book, where urelements are allowed, which makes the arguments somewhat more involved than in the pure sets case.) –  Andres Caicedo Jun 7 '13 at 2:45

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