The consistency strength of the perfect set property for universally Baire sets of reals

TL;DR

This paper explores the logical strength of the perfect set property for universally Baire sets, linking it to the existence of a new type of large cardinal called virtually Shelah cardinals, which are weaker than Shelah cardinals.

Contribution

It introduces virtually Shelah cardinals and establishes their equiconsistency with the perfect set property for universally Baire sets, connecting set-theoretic principles with large cardinal hypotheses.

Findings

The perfect set property for universally Baire sets is equiconsistent with virtually Shelah cardinals.

Existence of a virtually Shelah cardinal is consistent with the perfect set property.

The statement $ ext{uB} = {f riangle}^1_2$ is equiconsistent with a $ ext{Σ}_2$-reflecting virtually Shelah cardinal.

Abstract

We show that the statement "every universally Baire set of reals has the perfect set property" is equiconsistent modulo ZFC with the existence of a cardinal that we call a virtually Shelah cardinal. These cardinals resemble Shelah cardinals but are much weaker: if $0^\sharp$ exists then every Silver indiscernible is virtually Shelah in $L$. We also show that the statement $\text{uB} = {\bf\Delta}^1_2$, where $\text{uB}$ is the pointclass of all universally Baire sets of reals, is equiconsistent modulo ZFC with the existence of a $\Sigma_2$-reflecting virtually Shelah cardinal.