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Building Models of Determinacy from Below

Obrad Kasum, Grigor Sargsyan

TL;DR

The paper develops an $L$-like construction that produces the minimal inner model of $ extsf{AD}_ ext{R}^+$ with $oldsymbol{ riangle}$-regularity, by iterating a determinacy-pointclass operator $ extscr{O}$ to generate a sequence $(oldsymbol{ riangle}_eta)$. It proves that any inner determinacy model containing all reals and satisfying $ extsf{AD}^+ + extsf{AD}_ ext{R} + extsf{NMLW}$ has its powerset of reals equal to some $oldsymbol{ riangle}_eta$, and when a model with $oldsymbol{ riangle}$-regularity exists, $L(oldsymbol{ riangle}_eta)$ is the minimal such model. The construction hinges on a robust HOD-mice/HOD-analysis framework, a choiceless club-filter apparatus, and a realizability strategy that renders fragments of the generator definable inside $ extsf{L}(oldsymbol{ riangle},K_ ext{∞}^ ext{ω})$. The results illuminate how to reach determinacy models below Hod mice with Woodin-limit complexities, and establish an explicit, absolute route to identifying the powerset of $ eals$ in these minimal models, with potential extensions bound by the Chang-plus barrier. Overall, the work advances a constructive pathway to determinacy-models below large-cardinal thresholds and clarifies the role of oracle-like devices in extending determinacy hierarchies.

Abstract

We present an $L$-like construction that produces the minimal model of $\mathsf{AD}_\mathbb{R}+$"$Θ$ is regular". In fact, our construction can produce any model of $\mathsf{AD}^++\mathsf{AD}_\mathbb{R}+V=L(P(\mathbb{R}))$ in which there is no hod mouse with a measurable limit of Woodins.

Building Models of Determinacy from Below

TL;DR

The paper develops an -like construction that produces the minimal inner model of with -regularity, by iterating a determinacy-pointclass operator to generate a sequence . It proves that any inner determinacy model containing all reals and satisfying has its powerset of reals equal to some , and when a model with -regularity exists, is the minimal such model. The construction hinges on a robust HOD-mice/HOD-analysis framework, a choiceless club-filter apparatus, and a realizability strategy that renders fragments of the generator definable inside . The results illuminate how to reach determinacy models below Hod mice with Woodin-limit complexities, and establish an explicit, absolute route to identifying the powerset of in these minimal models, with potential extensions bound by the Chang-plus barrier. Overall, the work advances a constructive pathway to determinacy-models below large-cardinal thresholds and clarifies the role of oracle-like devices in extending determinacy hierarchies.

Abstract

We present an -like construction that produces the minimal model of " is regular". In fact, our construction can produce any model of in which there is no hod mouse with a measurable limit of Woodins.
Paper Structure (12 sections, 43 theorems, 50 equations)

This paper contains 12 sections, 43 theorems, 50 equations.

Table of Contents

  1. Introduction
  2. Statement of the Result
  3. Outline of the Proof
  4. History
  5. Future Prospects
  6. Some Technical Remarks
  7. HOD Mice and HOD Analysis
  8. Club Filter
  9. Description of a Generator
  10. Realizability Strategy
  11. Adding Sequences and Measures
  12. Synthesis

Key Result

Corollary 1

If there exists an inner model containing all reals and satisfying $\mathsf{AD}_\mathbb{R}+$"$\Theta$ is regular", then there exists $\alpha<\sh$ such that $L(\De_\alpha)$ is the minimal model of $\mathsf{AD}_\mathbb{R}+$"$\Theta$ is regular".

Theorems & Definitions (101)

  • Corollary
  • proof
  • Corollary 1.1
  • proof
  • Definition 1.5
  • Definition 1.6
  • Lemma 2.4
  • Theorem 2.6
  • proof
  • Definition 2.7
  • ...and 91 more