Suslin tree preservation and club isomorphisms

TL;DR

This paper constructs a set-theoretic model where Suslin trees coexist with a property that all non-Suslin Aronszajn trees are club isomorphic, and explores forcing extensions affecting Suslin trees' properties.

Contribution

It introduces a model with a Suslin tree and a novel club isomorphism property among Aronszajn trees, and analyzes forcing extensions altering Suslin trees' freeness.

Findings

Existence of a model with a Suslin tree and club isomorphism among certain Aronszajn trees

Forcing extensions can make Suslin trees n-free while rendering derived trees special

Demonstrates interplay between Suslin trees and forcing in set-theoretic models

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.