THE CONSISTENCY STRENGTH OF THE PERFECT SET PROPERTY FOR UNIVERSALLY BAIRE SETS OF REALS. (10th June 2022)

• Record Type: Journal Article
• Authors: SCHINDLER, RALF; WILSON, TREVOR M.
• Is Part Of: Journal of symbolic logic. Volume 87:Number 2(2022)
• Page Start: 508
• Page End: 526
• Publication Date: 2022-06-10
• Subjects: 03E15 -- 03E55; large cardinals -- universally Baire; Logic, Symbolic and mathematical -- Periodicals; 511.3
• DOI: 10.1017/jsl.2019.63
• Languages: English
• ISSNs: 0022-4812

Abstract

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 virtually Shelah for supercompactness (VSS). These cardinals resemble Shelah cardinals and Shelah-for-supercompactness cardinals but are much weaker: if $0^\sharp $ exists then every Silver indiscernible is VSS in L . We also show that the statement $\operatorname {\mathrm {uB}} = {\boldsymbol {\Delta }}^1_2$, where $\operatorname {\mathrm {uB}}$ is the pointclass of all universally Baire sets of reals, is equiconsistent modulo ZFC with the existence of a $\Sigma _2$ -reflecting VSS cardinal.