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Generic Generators

Grigor Sargsyan

TL;DR

The paper introduces a novel framework to tackle Hod Pair Capturing (HPC) by defining generic generators for hod pair (and pure extender) constructions and by formulating two principles, ${\sf{Direct\ Limit\ Independence\ Hypothesis}}$ ($\text{DLIH}$) and ${\sf{Bounded\ Direct\ Limits}}$ (${\sf{BDL}}$), to control how powersets behave across generic iterations. It develops the machinery of fully backgrounded $K^c$-constructions, amenable linear iterations, and derived-model techniques to produce and analyze generic generators, aiming to transfer set-theoretic complexity between frameworks while preserving determinacy structures. Key contributions include formalizing $pe$- and $hp$-generic generators, establishing comparison principles for amenable iterations, and outlining a program to obtain Woodin cardinals that are limits of Woodin cardinals, and eventually superstrong cardinals, via $L[\vec{E}]$- and backgrounded constructions. The work sketches conjectural routes (e.g., LEC, CWLD) to deduce HPC from inner-model inductions and supports its program with detailed structural analyses of powerset computation in the generic-generator regime. Overall, it integrates core model induction with hod-mouse techniques to propose a concrete path toward resolving HPC under robust large-cardinal hypotheses.

Abstract

The goal of this paper is to present an approach to Hod Pair Capturing (HPC). $HPC$ is the most outstanding open problem of descriptive inner model theory. More specifically, we introduce two principles, the Direct Limit Independence and the Bounded Direct Limits, and show that they together imply HPC.

Generic Generators

TL;DR

The paper introduces a novel framework to tackle Hod Pair Capturing (HPC) by defining generic generators for hod pair (and pure extender) constructions and by formulating two principles, () and (), to control how powersets behave across generic iterations. It develops the machinery of fully backgrounded -constructions, amenable linear iterations, and derived-model techniques to produce and analyze generic generators, aiming to transfer set-theoretic complexity between frameworks while preserving determinacy structures. Key contributions include formalizing - and -generic generators, establishing comparison principles for amenable iterations, and outlining a program to obtain Woodin cardinals that are limits of Woodin cardinals, and eventually superstrong cardinals, via - and backgrounded constructions. The work sketches conjectural routes (e.g., LEC, CWLD) to deduce HPC from inner-model inductions and supports its program with detailed structural analyses of powerset computation in the generic-generator regime. Overall, it integrates core model induction with hod-mouse techniques to propose a concrete path toward resolving HPC under robust large-cardinal hypotheses.

Abstract

The goal of this paper is to present an approach to Hod Pair Capturing (HPC). is the most outstanding open problem of descriptive inner model theory. More specifically, we introduce two principles, the Direct Limit Independence and the Bounded Direct Limits, and show that they together imply HPC.
Paper Structure (8 sections, 15 theorems, 3 equations)

This paper contains 8 sections, 15 theorems, 3 equations.

Table of Contents

  1. The largest generated segment
  2. Generic generators
  3. Production of generic generators
  4. Amenable linear iterations
  5. Computing the powerset operation: Towards a Woodin cardinal that is a limit of Woodin cardinals
  6. A Woodin cardinal that is a limit of Woodin cardinals
  7. Computing the powerset operation: towards superstrong cardinals
  8. Concluding remarks

Key Result

Theorem 7

Let $\kappa\geq \aleph_3$ be a regular cardinal. Assume that $\kappa$ is countably closed in the sense that for all $\eta<\kappa$, $\eta^{\aleph_0}< \kappa$. Suppose that both $\square(\kappa)$ and $\square_\kappa$ fail. If the certified $K^c$ exists in $V^{Coll(\kappa, \kappa)}$ then there is a sub

Theorems & Definitions (66)

  • Definition 1: No Long Extender
  • Remark 2
  • Conjecture 3: Covering with Chang Models
  • Definition 4: $\sf{Hod\ Pair\ Capturing}$
  • Conjecture 5
  • Conjecture 6
  • Theorem 7: JSSS
  • Conjecture 8: OIMT
  • Definition 9
  • Remark 10
  • ...and 56 more