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Non-Galvin Filters

Tom Benhamou, Shimon Garti, Moti Gitik, Alejandro Poveda

Abstract

We address the question of the consistency strength of certain filters and ultrafilters which fail to satisfy the Galvin property. We answer questions \cite[Questions 7.8,7.9]{TomMotiII}, \cite[Question 5]{NegGalSing} and improve theorem \cite[Theorem 2.3]{NegGalSing}.

Non-Galvin Filters

Abstract

We address the question of the consistency strength of certain filters and ultrafilters which fail to satisfy the Galvin property. We answer questions \cite[Questions 7.8,7.9]{TomMotiII}, \cite[Question 5]{NegGalSing} and improve theorem \cite[Theorem 2.3]{NegGalSing}.
Paper Structure (14 sections, 30 theorems, 60 equations)

This paper contains 14 sections, 30 theorems, 60 equations.

Table of Contents

  1. introduction
  2. Notation
  3. The failure of Galvin's property for ultrafilters
  4. Non-Galvin ultrafilters
  5. A non Galvin ultrafilter concentrating on singulars
  6. The failure of Galvin's property for filters
  7. Many non-Galvin filters
  8. Many non-Galvin filters at successors of singulars containing the club filter
  9. Failure of Galvin's property for the club filter
  10. A measure concentrating on failures of Galvin's property
  11. A Prikry-type poset forcing the failure of Galvin's property
  12. On consistency strength
  13. Improving the consistency strength of the global failure
  14. The exact consistency strength for the local failure

Key Result

Lemma 2.1

Assume that: Then ${\rm Gal}(\mathscr{U},\kappa,\kappa^+)$.

Theorems & Definitions (80)

  • Lemma 2.1
  • proof
  • Theorem 2.2
  • proof
  • Theorem 2.3
  • proof
  • Theorem 2.4
  • Lemma 2.5
  • Proposition 2.6
  • Definition 2.7
  • ...and 70 more