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arXiv · 1807.02213

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

Ralf Schindler·Trevor M. Wilson
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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 $\text{uB} = {\bfΔ}^1_2$, where $\text{uB}$ is the pointclass of all universally Baire sets of reals, is equiconsistent modulo ZFC with the existence of a $Σ_2$-reflecting virtually Shelah cardinal.

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Ralf Schindler, Trevor M. Wilson. 2018-07-06. The consistency strength of the perfect set property for universally Baire sets of reals. https://arxiv.org/abs/1807.02213

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