Club isomorphisms on higher Aronszajn trees

TL;DR

This paper proves the consistency, assuming an ineffable cardinal, that all normal countably closed 2-Aronszajn trees are club isomorphic, extending known results from 1 and relating to the 2-Suslin hypothesis.

Contribution

It generalizes the club isomorphism property from 1 to 2-Aronszajn trees under certain large cardinal assumptions.

Findings

All normal countably closed 2-Aronszajn trees are club isomorphic assuming an ineffable cardinal.

The result implies the nonexistence of 2-Suslin trees.

Extends methods for the 2-Suslin hypothesis to higher cardinals.

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 $\textsf{PFA}$. 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.