Logo
ScienceStack
Table of Contents
Fetching ...
Sign In

Chang models over derived models with supercompact measures

Takehiko Gappo, Sandra Müller, Grigor Sargsyan

TL;DR

The paper constructs Chang-type determinacy models over derived hod-derived models to obtain rich structure above $\Theta$ and, crucially, to realize $\omega_1$ with strong large-cardinal properties. By developing the Chang model over the derived model, $\mathsf{CDM}^{+}$, inside a symmetric collapse of a hod mouse, it proves $\mathsf{AD}^{+}+\mathsf{AD}_{\mathbb{R}}$ and establishes $\omega_1$ as $<{\delta}_{\infty}$-supercompact for a suitably large regular ${\delta}_{\infty}>\Theta$, with $\Theta$ regular and even measurable under stronger hypotheses. The construction relies on a directed system of genericity iterations yielding a robust direct-limit model $\mathcal{M}_{\infty}(\mathcal{Q},\eta)$ and club-filter-based ultrafilters, enabling $\Theta$-measurability and $\omega_1$-supercompactness within $\mathsf{CDM}^{+}$. These results extend Woodin's generalized Chang model by providing a determinacy framework with supercompactness properties tied to a derived-model backbone, offering weaker consistency-strength assumptions than traditional Woodin-limit frameworks and broadening the landscape of determinacy with large-cardinal features.

Abstract

Based on earlier work of the third author, we construct a Chang-type model with supercompact measures extending a derived model of a given hod mouse with a regular cardinal $δ$ that is both a limit of Woodin cardinals and a limit of ${<}δ$-strong cardinals. The existence of such a hod mouse is consistent relative to a Woodin cardinal that is a limit of Woodin cardinals. We argue that our Chang-type model satisfies $\mathsf{AD}_{\mathbb{R}} + Θ$ is regular + $ω_1$ is ${<}δ_{\infty}$-supercompact for some regular cardinal $δ_{\infty}>Θ$. This complements Woodin's generalized Chang model, which satisfies $\mathsf{AD}_{\mathbb{R}}+ω_1$ is supercompact, assuming a proper class of Woodin cardinals that are limits of Woodin cardinals.

Chang models over derived models with supercompact measures

TL;DR

The paper constructs Chang-type determinacy models over derived hod-derived models to obtain rich structure above and, crucially, to realize with strong large-cardinal properties. By developing the Chang model over the derived model, , inside a symmetric collapse of a hod mouse, it proves and establishes as -supercompact for a suitably large regular , with regular and even measurable under stronger hypotheses. The construction relies on a directed system of genericity iterations yielding a robust direct-limit model and club-filter-based ultrafilters, enabling -measurability and -supercompactness within . These results extend Woodin's generalized Chang model by providing a determinacy framework with supercompactness properties tied to a derived-model backbone, offering weaker consistency-strength assumptions than traditional Woodin-limit frameworks and broadening the landscape of determinacy with large-cardinal features.

Abstract

Based on earlier work of the third author, we construct a Chang-type model with supercompact measures extending a derived model of a given hod mouse with a regular cardinal that is both a limit of Woodin cardinals and a limit of -strong cardinals. The existence of such a hod mouse is consistent relative to a Woodin cardinal that is a limit of Woodin cardinals. We argue that our Chang-type model satisfies is regular + is -supercompact for some regular cardinal . This complements Woodin's generalized Chang model, which satisfies is supercompact, assuming a proper class of Woodin cardinals that are limits of Woodin cardinals.
Paper Structure (13 sections, 23 theorems, 52 equations)

This paper contains 13 sections, 23 theorems, 52 equations.

Table of Contents

  1. Introduction
  2. Motivation behind higher models of determinacy
  3. Beyond Woodin's derived model theorem
  4. Determinacy and supercompactness of $\omega_1$
  5. Summary of our main result
  6. Construction
  7. Setup
  8. Genericity iterations
  9. Supercompact measures on $\wp_{\omega_1}(\alpha)$
  10. Proof of Determinacy
  11. The regularity of $\Theta$
  12. The measurability of $\Theta$
  13. Value of $\delta_{\infty}$

Key Result

Theorem 1.1

Assume that $V=L(\wp(\mathbb{R}))$ and $\Theta\mathsf{reg}$ holds. If $G\subseteq\mathbb{P}_{\mathrm{max}}*\mathrm{Add}(\omega_3, 1)$ is $V$-generic, then $V[G]\models\mathsf{ZFC}+\mathsf{MM}^{++}(\mathfrak{c})$.

Theorems & Definitions (56)

  • Theorem 1.1: Woodin, Wo10
  • Theorem 1.2: Blue--Larson--Sargsyan, Nairian
  • Theorem 1.3: Woodin, Wo21
  • Conjecture 1.4
  • Theorem 1.5
  • Conjecture 1.6
  • Definition 2.1
  • Theorem 2.2: CPMPdm_of_self_it
  • proof
  • Definition 2.3
  • ...and 46 more