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The uniqueness of the core model

Benjamin Siskind

TL;DR

This work addresses the potential non-uniqueness of core models arising from different premice by introducing an abstract notion of a model that resembles the core model and proving uniqueness results in two regimes: (i) when there is a proper class of measurable cardinals and no inner model with a $\text{Woodin cardinal}$, and (ii) below $0^\textparagraph$ with no Woodin cardinal, where Jensen- and Schindler-indexed core models are shown to coincide. The method hinges on defining uniform, definable embeddings that witness how candidate models resemble the core model, and on rigidity arguments that prevent nontrivial automorphisms. The results establish that the canonical core model is indexing-invariant under the stated hypotheses, aligning Jensen-Steel’s $K$ with Schindler’s $J$ below $0^\textparagraph$ and reinforcing the role of covering, universality, and definability in inner model theory. Overall, the paper provides a robust, abstraction-driven route to core-model uniqueness, reducing dependence on the particular premouse indexing used.

Abstract

The Jensen-Steel core model is a canonical inner model which plays a fundamental role in the meta-mathematics of set theory. Its definition depends on exactly which hierarchy of fine-structural models of set theory, premice, one uses. Each such hierarchy involves somewhat arbitrary decisions and working with different hierarchies ostensibly leads to different versions of the core model. We show that in some contexts, abstract properties of the core model uniquely determine it; that is, there is at most one inner model with these properties.

The uniqueness of the core model

TL;DR

This work addresses the potential non-uniqueness of core models arising from different premice by introducing an abstract notion of a model that resembles the core model and proving uniqueness results in two regimes: (i) when there is a proper class of measurable cardinals and no inner model with a , and (ii) below with no Woodin cardinal, where Jensen- and Schindler-indexed core models are shown to coincide. The method hinges on defining uniform, definable embeddings that witness how candidate models resemble the core model, and on rigidity arguments that prevent nontrivial automorphisms. The results establish that the canonical core model is indexing-invariant under the stated hypotheses, aligning Jensen-Steel’s with Schindler’s below and reinforcing the role of covering, universality, and definability in inner model theory. Overall, the paper provides a robust, abstraction-driven route to core-model uniqueness, reducing dependence on the particular premouse indexing used.

Abstract

The Jensen-Steel core model is a canonical inner model which plays a fundamental role in the meta-mathematics of set theory. Its definition depends on exactly which hierarchy of fine-structural models of set theory, premice, one uses. Each such hierarchy involves somewhat arbitrary decisions and working with different hierarchies ostensibly leads to different versions of the core model. We show that in some contexts, abstract properties of the core model uniquely determine it; that is, there is at most one inner model with these properties.
Paper Structure (4 sections, 34 theorems, 6 equations)

This paper contains 4 sections, 34 theorems, 6 equations.

Table of Contents

  1. Introduction
  2. Preliminaries
  3. With measurable cardinals
  4. Below $0^\text{\textparagraph}$

Key Result

Theorem 2.3

Assume there is no inner model with a Woodin cardinal. Let $W$ be an inner model of ZFC. Then $K^W$ is iterable.

Theorems & Definitions (56)

  • Definition 2.1
  • Definition 2.2
  • Theorem 2.3
  • Theorem 2.4
  • Definition 2.5
  • Theorem 2.6
  • Theorem 2.7
  • Theorem 2.8
  • Definition 2.9
  • Theorem 2.10
  • ...and 46 more