Let $\mathbf{L}_{\varsigma}$ be the level of ordinal $\varsigma$ of Gödel's constructible universe $\mathbf{L}$. Let $\Sigma_{3}$-KP be Kripke-Platek set theory with infinity and $\Sigma_{3}$-$collection$ and $\Sigma_{3}$-$comprehension$. Let $\sigma$ be the least ordinal such that $\mathbf{L}_{\sigma}$ models $\Sigma_{3}$-KP. If $\sigma$ is not a $\Sigma_{3}$-admissible ordinal, then what is it?
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asked Nov 15, 2014 at 0:23
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3$\begingroup$ Isn't this by definition $\Sigma_3$-admissible ordinal? $\endgroup$– WojowuCommented Nov 15, 2014 at 12:35
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$\begingroup$ That may well be, and I thought so as well. Perhaps I misunderstood a comment in a communication. May someone else confirm, so that I perhaps should delete the question? Does someone have references to literature on this? $\endgroup$– Frode Alfson BjørdalCommented Nov 15, 2014 at 16:58
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Some references to literature: In "Reflection and Partition Properties of Admissible Ordinals" (Annals of Math. Logic vol. 22, iss. 3, 1982), Kranakis defines a $\Sigma_n$-admissible ordinal to be an $\alpha$ is closed under rudimentary functions and satisfies the $\Sigma_n$-collection schema. If you are defining a $\Sigma_3$-admissible ordinal in this way, then $\sigma$ is $\Sigma_3$-admissible, as $L_\sigma$ is closed under rud. functions (by $L_\sigma\vDash\mathrm{KP}$) and $L_\sigma$ satisfies $\Sigma_3$-separation by definition. In "An Introduction to the Fine Structure of the Constructible Hierarchy" (1974), Devlin uses the same definition of "$\Sigma_n$-admissible ordinal", except calling the $\Sigma_n$-collection schema by the name of "$\Sigma_n$-replacement" (although the schema listed is what is usually now called the $\Sigma_n$-collection schema).
There are characterizations of the ordinals in your question in terms of elementary substructures which may shed light on their relative size. Let $\delta$ be least such that $L_\delta$ is $\Sigma_3$-admissible. Once it is shown that $L_\delta$ and $L_\sigma$ satisfy the axiom $V=HC$ ("every set is hereditarily countable"), theorem 2.2 of Kranakis's paper is applicable to show that $\sigma$ is least such that the ordinals $\xi$ where $\{\xi<\sigma\mid L_\xi\prec_{\Sigma_3}L_\sigma\}$ is cofinal in $\sigma$, and theorem 2.3 is applicable to show that $\delta$ is least such that $\delta$ is $\Pi_2$-reflecting on $\{\xi<\delta\mid L_\xi\prec_{\Sigma_2}L_\sigma\}$, where $\prec_\Gamma$ denotes the elementary substructure relation with elementarity restricted to a class of formulas $\Gamma$.
For the reader interested in large countable ordinals $\alpha$ such that $L_\alpha$ satisfies extensions of KP like the one you mention, some relevant papers include W. Marek's "Some comments on the paper by Artigue, Isambert, Perrin and Zalc" (Fundamenta Mathematicae vol. 101, iss. 3, 1978) and R. Gostanian's "Constructible Models of Subsystems of ZF" (Journal of Symbolic Logic vol. 45, no. 2, 1980) and some of the work in S.-D. Friedman's "Parameter-Free Uniformization." (Proceedings of the American Mathematical Society vol. 136, no. 9, 2008), in addition to Kranakis's.