The $\mathsf{HOD}$ Hypothesis and a supercompact cardinal

TL;DR

The paper investigates the $\mathsf{HOD}$ Hypothesis in the presence of a supercompact cardinal, proving that if $\kappa$ is supercompact and the hypothesis holds, then there is a proper class of regular cardinals below $\kappa$ that are measurable in $\mathsf{HOD}$. The core technique combines a new equivalent formulation of supercompactness with an $\mathsf{HOD}$-partition argument to produce $\mathsf{HOD}$-measurable cardinals locally, from which Woodin's Local Universality Theorem follows as a corollary. This work highlights a strong local reflection of large cardinal strength from $\mathsf{V}$ to $\mathsf{HOD}$ under the hypothesis and contrasts $\mathsf{HOD}$-supercompactness with ordinary supercompactness in this setting. It also discusses forcing and inner model theory implications and remains open regarding equivalences under weaker assumptions and connections to weak extender models.

Abstract

In this paper, we prove that: if $κ$ is supercompact and the $\mathsf{HOD}$ Hypothesis holds, then there is a proper class of regular cardinals in $V_κ$ which are measurable in $\mathsf{HOD}$. Woodin also proved this result. As a corollary, we prove Woodin's Local Universality Theorem. This work shows that under the assumption of the $\mathsf{HOD}$ Hypothesis and supercompact cardinals, large cardinals in $\mathsf{V}$ are reflected to be large cardinals in $\mathsf{HOD}$ in a local way, and reveals the huge difference between $\mathsf{HOD}$-supercompact cardinals and supercompact cardinals under the $\mathsf{HOD}$ Hypothesis.

Theorems & Definitions (41)

• Proposition 3.1
• Theorem 3.2
• proof
• Definition 3.3
• Definition 3.4
• Definition 3.5
• Theorem 3.6
• Definition 3.7
• Theorem 3.8
• Definition 3.9