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Almost Kurepa Suslin trees and destructibility of the Guessing Model Property

Chris Lambie-Hanson, Šárka Stejskalová

Abstract

Building on recent work of Krueger and the second author, we prove the consistency of the Guessing Model Principle at $ω_2$ together with the existence of an almost Kurepa Suslin tree. In particular, it is consistent that the Guessing Model Principle holds but is destructible by a ccc forcing of size $ω_1$. We also prove the consistency of the existence of a weak Kurepa tree together with the failure of the Kurepa Hypothesis and a certain guessing model principle that, for example, implies the tree property at $ω_2$.

Almost Kurepa Suslin trees and destructibility of the Guessing Model Property

Abstract

Building on recent work of Krueger and the second author, we prove the consistency of the Guessing Model Principle at together with the existence of an almost Kurepa Suslin tree. In particular, it is consistent that the Guessing Model Principle holds but is destructible by a ccc forcing of size . We also prove the consistency of the existence of a weak Kurepa tree together with the failure of the Kurepa Hypothesis and a certain guessing model principle that, for example, implies the tree property at .
Paper Structure (17 sections, 33 theorems, 33 equations)

This paper contains 17 sections, 33 theorems, 33 equations.

Table of Contents

  1. Introduction
  2. Notation and conventions
  3. Preliminaries
  4. Trees and the guessing model property
  5. Preservation lemmas
  6. Mitchell forcing
  7. Automorphisms of trees
  8. Guessing models with an almost Kurepa Suslin tree
  9. A Mitchell variation
  10. Technical lemmas
  11. Dense subsets of derived trees
  12. Approximation and the proof of Theorem A
  13. Weak Kurepa trees
  14. Forcing for adding a weak Kurepa tree
  15. The consistency of $\sf wKH$ with $\neg\sf KH$ and $\mathsf{GMP}^{\omega_2}(\omega_2)$
  16. ...and 2 more sections

Key Result

Corollary 1.1

Suppose that there exists an inaccessible cardinal. Then there exists a forcing extension in which In particular, it is consistent that $\neg\sf wKH$ holds and yet is destructible by a ccc forcing of cardinality $\omega_1$.

Theorems & Definitions (100)

  • Corollary 1.1
  • Definition 2.1
  • Definition 2.2
  • Definition 2.3
  • Definition 2.4
  • Definition 2.5
  • Remark 2.13
  • Proposition 2.14
  • Proposition 2.15
  • Definition 2.16: Consistency
  • ...and 90 more