Club isomorphisms on higher Aronszajn trees

Abstract

We prove the consistency, assuming an ineffable cardinal, that any two normal countably closed ${\omega }_2$-Aronszajn trees are club isomorphic. This work generalizes to higher cardinals the property of Abraham-Shelah that any two normal ${\omega }_1$-Aronszajn trees are club isomorphic, which follows from $\text{PFA}$. The statement that any two normal countably closed ${\omega }_2$-Aronszajn trees are club isomorphic implies that there are no ${\omega }_2$-Suslin trees, so our proof also expands on the method of Laver-Shelah for obtaining the ${\omega }_2$-Suslin hypothesis.