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  1. Weakly remarkable cardinals, erdős cardinals, and the generic vopěnka principle.Trevor M. Wilson - 2019 - Journal of Symbolic Logic 84 (4):1711-1721.
    We consider a weak version of Schindler’s remarkable cardinals that may fail to be ${{\rm{\Sigma }}_2}$-reflecting. We show that the ${{\rm{\Sigma }}_2}$-reflecting weakly remarkable cardinals are exactly the remarkable cardinals, and that the existence of a non-${{\rm{\Sigma }}_2}$-reflecting weakly remarkable cardinal has higher consistency strength: it is equiconsistent with the existence of an ω-Erdős cardinal. We give an application involving gVP, the generic Vopěnka principle defined by Bagaria, Gitman, and Schindler. Namely, we show that gVP + “Ord is not ${{\rm{\Delta (...)
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  • Generic Vopěnka cardinals and models of ZF with few $$\aleph _1$$ ℵ 1 -Suslin sets.Trevor M. Wilson - 2019 - Archive for Mathematical Logic 58 (7-8):841-856.
    We define a generic Vopěnka cardinal to be an inaccessible cardinal \ such that for every first-order language \ of cardinality less than \ and every set \ of \-structures, if \ and every structure in \ has cardinality less than \, then an elementary embedding between two structures in \ exists in some generic extension of V. We investigate connections between generic Vopěnka cardinals in models of ZFC and the number and complexity of \-Suslin sets of reals in models (...)
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  • Generic Vopěnka cardinals and models of ZF with few $$\aleph _1$$ ℵ 1 -Suslin sets.Trevor M. Wilson - 2019 - Archive for Mathematical Logic 58 (7-8):841-856.
    We define a generic Vopěnka cardinal to be an inaccessible cardinal \ such that for every first-order language \ of cardinality less than \ and every set \ of \-structures, if \ and every structure in \ has cardinality less than \, then an elementary embedding between two structures in \ exists in some generic extension of V. We investigate connections between generic Vopěnka cardinals in models of ZFC and the number and complexity of \-Suslin sets of reals in models (...)
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  • Infinity and continuum in the alternative set theory.Kateřina Trlifajová - 2021 - European Journal for Philosophy of Science 12 (1):1-23.
    Alternative set theory was created by the Czech mathematician Petr Vopěnka in 1979 as an alternative to Cantor’s set theory. Vopěnka criticised Cantor’s approach for its loss of correspondence with the real world. Alternative set theory can be partially axiomatised and regarded as a nonstandard theory of natural numbers. However, its intention is much wider. It attempts to retain a correspondence between mathematical notions and phenomena of the natural world. Through infinity, Vopěnka grasps the phenomena of vagueness. Infinite sets are (...)
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  • The consistency strength of the perfect set property for universally baire sets of reals.Ralf Schindler & Trevor M. Wilson - 2022 - Journal of Symbolic Logic 87 (2):508-526.
    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. 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 (...)
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  • Virtual large cardinals.Victoria Gitman & Ralf Schindler - 2018 - Annals of Pure and Applied Logic 169 (12):1317-1334.
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