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Negating the Galvin Property

Tom Benhamou, Shimon Garti, Alejandro Poveda

Abstract

We prove that Galvin's property consistently fails at successors of strong limit singular cardinals. We also prove the consistency of this property failing at every successor of a singular cardinal. In addition, the paper analyzes the effect of Prikry-type forcings on the strong failure of the Galvin property and explores stronger forms of this property in the context of large cardinals

Negating the Galvin Property

Abstract

We prove that Galvin's property consistently fails at successors of strong limit singular cardinals. We also prove the consistency of this property failing at every successor of a singular cardinal. In addition, the paper analyzes the effect of Prikry-type forcings on the strong failure of the Galvin property and explores stronger forms of this property in the context of large cardinals
Paper Structure (20 sections, 33 theorems, 73 equations)

This paper contains 20 sections, 33 theorems, 73 equations.

Table of Contents

  1. Introduction
  2. Preliminaries
  3. Notations
  4. Clubs and Filters
  5. Forcing preliminaries
  6. Radin forcing with interleaved collapses
  7. Measure sequences and weak constructing pairs
  8. The main poset
  9. Properties of the main forcing
  10. The consistency of the local and global failure
  11. Local failure of Galvin's property
  12. Global failure of Galvin's property
  13. The impossibility of the ultimate global failure
  14. Galvin's number at successor of singular cardinals
  15. The strong failure and Prikry-type generic extensions
  16. ...and 5 more sections

Key Result

Theorem 1

Assume the $\mathsf{GCH}$ holds. Suppose that $\kappa<\lambda$ are infinite cardinals with $\kappa$ regular. Then, there is a forcing extension of the set-theoretic universe containing a family $\mathcal{C}$ of clubs at $\kappa^+$ with $|\mathcal{C}|=\lambda$, that witnesses the following property: Moreover, $2^\kappa=2^{\kappa^+}=\lambda$ holds in this model provided $\lambda\geq{\rm cf}(\lambda

Theorems & Definitions (122)

  • Theorem : Abraham-Shelah
  • Definition 1.1: Galvin's property
  • Definition 1.2: Combinatorial properties associated to filters
  • Definition 1.3: Cohen forcing and Lévy Collapse
  • Definition 1.4: Basic properties
  • Definition 1.5: Abraham-Shelah MR830084
  • Definition 1.6: Prikry forcing
  • Definition 1.7: Weak constructing pair
  • Remark 1.8
  • Definition 1.9
  • ...and 112 more