Perfect Subtree Property for Weakly Compact Cardinals

TL;DR

This paper explores the consistency strength of the Perfect Subtree Property for weakly compact cardinals, linking it to inner models with Woodin cardinals and the Axiom of Determinacy for the reals.

Contribution

It establishes the failure of the Perfect Subtree Property under certain inner model assumptions and connects its nonexistence to the consistency of AD_R + DC.

Findings

The statement fails strongly if no inner model with a Woodin cardinal exists.

The nonexistence of a certain tree implies the consistency of AD_R + DC.

The Perfect Subtree Property for weakly compact cardinals has deep implications in set theory.

Abstract

We investigate the consistency strength of the statement: $\kappa$ is weakly compact and there is no tree on $\kappa$ with exactly $\kappa^{+}$ many branches. We show that this statement fails strongly (in the sense that there is a sealed tree with exactly $\kappa^{+}$ many branches) if there is no inner model with a Woodin cardinal. Moreover, we show that for a weakly compact cardinal $\kappa$ the nonexistence of a tree on $\kappa$ with exactly $\kappa^{+}$ many branches and, in particular, the Perfect Subtree Property for $\kappa$, implies the consistency of $AD_{\mathbb{R}} + DC$.