Suslin tree preservation and club isomorphisms

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.