The Special Aronszajn Tree Property

TL;DR

This paper demonstrates that under the assumption of a proper class of supercompact cardinals, it is consistent to have, for every regular cardinal, the existence of special Aronszajn trees at the successor cardinal.

Contribution

It proves the consistency of the existence of special Aronszajn trees at all regular cardinals' successors assuming supercompact cardinals.

Findings

Existence of special Aronszajn trees at all regular cardinals' successors.

All such Aronszajn trees can be made special.

Consistency results under large cardinal assumptions.

Abstract

Assuming the existence of a proper class of supercompact cardinals, we force that for every regular cardinal $\kappa$, there are $\kappa^+$-Aronszajn trees and all such trees are special.