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Mar 10, 2020 at 23:43 vote accept Dmytro Taranovsky
Mar 10, 2020 at 20:33 comment added Gabe Goldberg When $\varphi$ is a $\Sigma_1$ formula, I think you'll get an elementary substructure if and only if $\alpha$ is not admissible by the analysis from Steel's "Scales in $L(\mathbb R)$," Corollary 2.8 and Theorem 2.9. I don't know about the case when $\varphi$ is more complex, but there is probably a generalization. My guess would be it fails for all ordinals inside a $\Sigma_1$-gap, and holds for ordinals $\beta$ at the end of a gap if and only if $\beta$ fails to be strongly $\Pi_n$-reflecting for some $n<\omega$. (See Definition 3.1 of Steel.) But I never looked closely at the ends of gaps.
Mar 10, 2020 at 3:26 comment added Dmytro Taranovsky Thank you. I will accept the answer. Do you happen to know though of conditions that would give a positive answer? One possibility is for $α$ (or $α$ and $β$ for the extension) to have definable countable cofinality.
Mar 10, 2020 at 3:18 history edited Gabe Goldberg CC BY-SA 4.0
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Mar 10, 2020 at 1:56 history edited Gabe Goldberg CC BY-SA 4.0
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Mar 10, 2020 at 1:49 history answered Gabe Goldberg CC BY-SA 4.0