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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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    Unrelated: is your name supposed to be a pig-Latinization of "Barwise"?2012-07-13

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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.)2013-06-07