Timeline for Pointwise definable models of determinacy
Current License: CC BY-SA 4.0
6 events
when toggle format | what | by | license | comment | |
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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 |