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Suslin tree preservation and club isomorphisms

John Krueger

TL;DR

This work addresses whether a Suslin tree can coexist with a strong club-isomorphism property for Aronszajn trees. It develops a Suslin-tree preservation framework for forcing that builds club isomorphisms via $\mathbb{Q}(T,U)$ while ensuring $S$ remains Suslin, and connects the result to $PFA(S)$. The main contributions include a model where a Suslin tree exists and any two normal Aronszajn trees, neither containing a Suslin subtree, are club isomorphic, and a forcing construction showing that a free Suslin tree can be made $n$-free while higher-dimensional derived trees become special. These results illustrate the malleability of freeness under forcing and extend Abraham–Shelah’s work, reflecting the landscape shaped by Todorcevic’s forcing axioms without requiring large cardinals.

Abstract

We construct a model of set theory in which there exists a Suslin tree and satisfies that any two normal Aronszajn trees, neither of which contains a Suslin subtree, are club isomorphic. We also show that if $S$ is a free normal Suslin tree, then for any positive integer $n$ there is a c.c.c. forcing extension in which $S$ is $n$-free but all of its derived trees of dimension greater than $n$ are special.

Suslin tree preservation and club isomorphisms

TL;DR

This work addresses whether a Suslin tree can coexist with a strong club-isomorphism property for Aronszajn trees. It develops a Suslin-tree preservation framework for forcing that builds club isomorphisms via while ensuring remains Suslin, and connects the result to . The main contributions include a model where a Suslin tree exists and any two normal Aronszajn trees, neither containing a Suslin subtree, are club isomorphic, and a forcing construction showing that a free Suslin tree can be made -free while higher-dimensional derived trees become special. These results illustrate the malleability of freeness under forcing and extend Abraham–Shelah’s work, reflecting the landscape shaped by Todorcevic’s forcing axioms without requiring large cardinals.

Abstract

We construct a model of set theory in which there exists a Suslin tree and satisfies that any two normal Aronszajn trees, neither of which contains a Suslin subtree, are club isomorphic. We also show that if is a free normal Suslin tree, then for any positive integer there is a c.c.c. forcing extension in which is -free but all of its derived trees of dimension greater than are special.
Paper Structure (4 sections, 18 theorems, 12 equations)

This paper contains 4 sections, 18 theorems, 12 equations.

Table of Contents

  1. Introduction
  2. Preliminaries
  3. Preserving a Suslin Tree
  4. Consistency Results

Key Result

Theorem 2.1

Suppose that $T$ is an Aronszajn tree and $\{ x_\alpha : \alpha < \omega_1 \}$ is a collection of pairwise disjoint finite subsets of $T$. Then there exist $\alpha < \beta$ such that every element of $x_\alpha$ is incomparable in $T$ with every element of $x_\beta$.

Theorems & Definitions (34)

  • Theorem 2.1: Baumgartner BDISS
  • Theorem 2.2: Abraham-Shelah AS2, Miyamoto MIYAMOTO
  • Theorem 2.3
  • proof : Proof (Sketch)
  • Lemma 2.4
  • proof
  • Lemma 2.5
  • proof
  • Lemma 2.6
  • proof
  • ...and 24 more