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The special Aronszajn tree property at $\aleph_2$ and $GCH$

David Asperó, Mohammad Golshani

Abstract

Starting from the existence of a weakly compact cardinal, we build a generic extension of the universe in which $GCH$ holds and all $\aleph_2$-Aronszajn trees are special and hence there are no $\aleph_2$-Souslin trees. This result answers a well-known open question from the 1970's.

The special Aronszajn tree property at $\aleph_2$ and $GCH$

Abstract

Starting from the existence of a weakly compact cardinal, we build a generic extension of the universe in which holds and all -Aronszajn trees are special and hence there are no -Souslin trees. This result answers a well-known open question from the 1970's.

Paper Structure

This paper contains 6 sections, 26 theorems, 99 equations.

Table of Contents

  1. introduction
  2. Models with markers and edges
  3. Definition of the forcing and its basic properties
  4. Basic properties of $\langle{\mathbb{Q}}_\beta\,\mid\,\beta\leq\kappa^{+}\rangle$
  5. The chain condition
  6. Completing the proof of Theorem \ref{['main theorem']}

Key Result

Theorem \oldthetheorem

Suppose $\kappa$ is a weakly compact cardinal. Then there exists a set-generic extension of the universe in which $\hbox{GCH}$ holds, $\kappa=\aleph_2$, and the special Aronszajn tree property at $\aleph_2$ (and hence Souslin's Hypothesis at $\aleph_2$) holds.

Theorems & Definitions (69)

  • Definition \oldthetheorem
  • Theorem \oldthetheorem
  • Remark \oldthetheorem
  • Definition \oldthetheorem: Models with markers
  • Lemma \oldthetheorem
  • proof
  • Definition \oldthetheorem: Edge
  • Definition \oldthetheorem: Generalized edge
  • Definition \oldthetheorem: Closedness under copying
  • Definition \oldthetheorem: Pure side conditions forcing
  • ...and 59 more