On supercompactness of $ω_1$

Abstract

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.

On some applications of strongly compact Prikry forcing

“…Remark 2. Ikegami and Trang [ §2, [IT19]] defined that an ultrafilter U on P κ (X) is normal if for any set A ∈ U and f : A → P κ (X) with ∅ = f (σ) ⊆ σ for all σ ∈ A, there is an x 0 ∈ X such that for U-measure one many σ in A, x 0 ∈ f (σ). They note that their definition of normality is equivalent to the closure under diagonal intersections in ZF, while it may not be equivalent to the definition of normality in our sense without AC.…”