More structural consequences of 𝐴𝐷

Ketchersid, Richard

doi:10.1090/conm/533/10504

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Cited by 6 publications

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“…For each λ, let F λ be the club filter on P ω1 (λ ω ). Woodin's model is defined as 4 Woodin shows that if there is a proper class of Woodin cardinals which are limits of Woodin cardinals, then C + satisfies the AD and ω 1 is supercompact. We note that in Takeuti's model, AD fails.…”

Section: Introductionmentioning

confidence: 99%

On supercompactness of $ω_1$

Ikegami,

Trang

2019

Preprint

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This paper studies structural consequences of supercompactness of ω1 under ZF. We show that the Axiom of Dependent Choice (DC) follows from "ω1 is supercompact". "ω1 is supercompact" also implies that AD + , a strengthening of the Axiom of Determinacy (AD), is equivalent to AD R . It is shown that "ω1 is supercompact" does not imply AD. The most one can hope for is Suslin co-Suslin determinacy. We show that this follows from "ω1 is supercompact" and Hod Pair Capturing (HPC), an inner-model theoretic hypothesis that imposes certain smallness conditions on the universe of sets. "ω1 is supercompact" on its own implies that every Suslin co-Suslin set is the projection of a determined (in fact, homogenously Suslin) set. "ω1 is supercompact" also implies all sets in the Chang model have all the usual regularity properties, like Lebesgue measurability and the Baire property.

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Section: Introductionmentioning

confidence: 99%