# On supercompactness of ω1

Daisuke Ikegami, Nam Trang

Research output: Contribution to journalArticlepeer-review

## 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 ADR. 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.

03E55, 03E60, 03E45

Original language English Unknown Journal Published - 2019 Apr 3

## Keywords

• Axiom of Determinacy
• Large cardinal properties
• Supercompactness
• ω

• General