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Outward compactness

Peter Holy, Philipp Lücke, Sandra Müller

TL;DR

The paper introduces outward compactness for strong logics and builds a framework of Ψ-measures of closeness to classify large cardinals between measurability and extendibility. It establishes precise L^2-characterizations linking outward compactness to measurability and extends these ideas to abstract logics, yielding a Makowsky-style equivalence for Vopěnka’s Principle. It further analyzes the naturalness of such characterizations, distinguishing robust Ψ_ext-based accounts from more fragile ones and relating fragments of VP to Ψ-large cardinals for classes, including the Ord is Woodin scenario. The work culminates in a broad, unified program connecting second-order logic, abstract logics, and prominent large-cardinal principles, and it closes with several open questions about optimal formulations and potential canonical characterizations.

Abstract

We introduce and study a new type of compactness principle for strong logics that, roughly speaking, infers the consistency of a theory from the consistency of its small fragments in certain outer models of the set-theoretic universe. We refer to this type of compactness property as outward compactness, and we show that instances of this type of principle for second-order logic can be used to characterize various large cardinal notions between measurability and extendibility, directly generalizing a classical result of Magidor that characterizes extendible cardinals as the strong compactness cardinals of second-order logic. In addition, we generalize a result of Makowsky that shows that Vopěnka's Principle is equivalent to the existence of compactness cardinals for all abstract logics by characterizing the principle "Ord is Woodin" through outward compactness properties of abstract logics.

Outward compactness

TL;DR

The paper introduces outward compactness for strong logics and builds a framework of Ψ-measures of closeness to classify large cardinals between measurability and extendibility. It establishes precise L^2-characterizations linking outward compactness to measurability and extends these ideas to abstract logics, yielding a Makowsky-style equivalence for Vopěnka’s Principle. It further analyzes the naturalness of such characterizations, distinguishing robust Ψ_ext-based accounts from more fragile ones and relating fragments of VP to Ψ-large cardinals for classes, including the Ord is Woodin scenario. The work culminates in a broad, unified program connecting second-order logic, abstract logics, and prominent large-cardinal principles, and it closes with several open questions about optimal formulations and potential canonical characterizations.

Abstract

We introduce and study a new type of compactness principle for strong logics that, roughly speaking, infers the consistency of a theory from the consistency of its small fragments in certain outer models of the set-theoretic universe. We refer to this type of compactness property as outward compactness, and we show that instances of this type of principle for second-order logic can be used to characterize various large cardinal notions between measurability and extendibility, directly generalizing a classical result of Magidor that characterizes extendible cardinals as the strong compactness cardinals of second-order logic. In addition, we generalize a result of Makowsky that shows that Vopěnka's Principle is equivalent to the existence of compactness cardinals for all abstract logics by characterizing the principle "Ord is Woodin" through outward compactness properties of abstract logics.
Paper Structure (13 sections, 43 theorems, 42 equations)

This paper contains 13 sections, 43 theorems, 42 equations.

Table of Contents

  1. Introduction
  2. $\Psi$-large cardinals
  3. Strong cardinals
  4. Supercompact cardinals
  5. Extendible cardinals
  6. $\omega_1$-strongly compact cardinals
  7. $\mathcal{L}^2$-characterizations
  8. On the naturalness of $\mathcal{L}^2$-characterizations
  9. Fragments of Vopěnka's Principle
  10. Vopěnka's Principle
  11. Ord is Woodin
  12. Abstract logics
  13. Open questions

Key Result

Theorem 1.1

A cardinal $\kappa$ is a strong compactness cardinal for $\mathcal{L}^2$ if and only if there is an extendible cardinal less than or equal to $\kappa$.

Theorems & Definitions (110)

  • Theorem 1.1: Magidor MR0295904
  • Definition 1.2
  • Definition 1.3
  • Proposition 1.4
  • proof
  • Theorem 1.5
  • proof
  • Claim
  • proof : Proof of the Claim
  • Remark 1.6
  • ...and 100 more