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  1. 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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  • Double helix in large large cardinals and iteration of elementary embeddings.Kentaro Sato - 2007 - Annals of Pure and Applied Logic 146 (2):199-236.
    We consider iterations of general elementary embeddings and, using this notion, point out helices of consistency-wise implications between large large cardinals.Up to now, large cardinal properties have been considered as properties which cannot be accessed by any weaker properties and it has been known that, with respect to this relation, they form a proper hierarchy. The helices we point out significantly change this situation: the same sequence of large cardinal properties occurs repeatedly, changing only the parameters.As results of our investigation (...)
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  • Structural reflection, shrewd cardinals and the size of the continuum.Philipp Lücke - 2022 - Journal of Mathematical Logic 22 (2).
    Journal of Mathematical Logic, Volume 22, Issue 02, August 2022. Motivated by results of Bagaria, Magidor and Väänänen, we study characterizations of large cardinal properties through reflection principles for classes of structures. More specifically, we aim to characterize notions from the lower end of the large cardinal hierarchy through the principle [math] introduced by Bagaria and Väänänen. Our results isolate a narrow interval in the large cardinal hierarchy that is bounded from below by total indescribability and from above by subtleness, (...)
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  • A model of the generic Vopěnka principle in which the ordinals are not Mahlo.Victoria Gitman & Joel David Hamkins - 2019 - Archive for Mathematical Logic 58 (1-2):245-265.
    The generic Vopěnka principle, we prove, is relatively consistent with the ordinals being non-Mahlo. Similarly, the generic Vopěnka scheme is relatively consistent with the ordinals being definably non-Mahlo. Indeed, the generic Vopěnka scheme is relatively consistent with the existence of a \-definable class containing no regular cardinals. In such a model, there can be no \-reflecting cardinals and hence also no remarkable cardinals. This latter fact answers negatively a question of Bagaria, Gitman and Schindler.
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  • Definable MAD families and forcing axioms.Vera Fischer, David Schrittesser & Thilo Weinert - 2021 - Annals of Pure and Applied Logic 172 (5):102909.
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  • The strong reflecting property and Harrington's Principle.Yong Cheng - 2015 - Mathematical Logic Quarterly 61 (4-5):329-340.
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  • Indestructibility properties of remarkable cardinals.Yong Cheng & Victoria Gitman - 2015 - Archive for Mathematical Logic 54 (7-8):961-984.
    Remarkable cardinals were introduced by Schindler, who showed that the existence of a remarkable cardinal is equiconsistent with the assertion that the theory of L\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$${L}$$\end{document} is absolute for proper forcing :176–184, 2000). Here, we study the indestructibility properties of remarkable cardinals. We show that if κ is remarkable, then there is a forcing extension in which the remarkability of κ becomes indestructible by all <κ-closed ≤κ-distributive forcing and all two-step iterations of (...)
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  • Forcing a set model of Z3 + Harrington's Principle.Yong Cheng - 2015 - Mathematical Logic Quarterly 61 (4-5):274-287.
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  • Equivalence relations which are borel somewhere.William Chan - 2017 - Journal of Symbolic Logic 82 (3):893-930.
    The following will be shown: Let I be a σ-ideal on a Polish space X so that the associated forcing of I+${\bf{\Delta }}_1^1$ sets ordered by ⊆ is a proper forcing. Let E be a ${\bf{\Sigma }}_1^1$ or a ${\bf{\Pi }}_1^1$ equivalence relation on X with all equivalence classes ${\bf{\Delta }}_1^1$. If for all $z \in {H_{{{\left}^ + }}}$, z♯ exists, then there exists an I+${\bf{\Delta }}_1^1$ set C ⊆ X such that E ↾ C is a ${\bf{\Delta }}_1^1$ equivalence (...)
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  • The strength of choiceless patterns of singular and weakly compact cardinals.Daniel Busche & Ralf Schindler - 2009 - Annals of Pure and Applied Logic 159 (1-2):198-248.
    We extend the core model induction technique to a choiceless context, and we exploit it to show that each one of the following two hypotheses individually implies that , the Axiom of Determinacy, holds in the of a generic extension of : every uncountable cardinal is singular, and every infinite successor cardinal is weakly compact and every uncountable limit cardinal is singular.
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  • Forcing and the Universe of Sets: Must We Lose Insight?Neil Barton - 2020 - Journal of Philosophical Logic 49 (4):575-612.
    A central area of current philosophical debate in the foundations of mathematics concerns whether or not there is a single, maximal, universe of set theory. Universists maintain that there is such a universe, while Multiversists argue that there are many universes, no one of which is ontologically privileged. Often forcing constructions that add subsets to models are cited as evidence in favour of the latter. This paper informs this debate by analysing ways the Universist might interpret this discourse that seems (...)
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  • On coding uncountable sets by reals.Joan Bagaria & Vladimir Kanovei - 2010 - Mathematical Logic Quarterly 56 (4):409-424.
    If A ⊆ ω1, then there exists a cardinal preserving generic extension [MATHEMATICAL DOUBLE-STRUCK CAPITAL L][A ][x ] of [MATHEMATICAL DOUBLE-STRUCK CAPITAL L][A ] by a real x such that1) A ∈ [MATHEMATICAL DOUBLE-STRUCK CAPITAL L][x ] and A is Δ1HC in [MATHEMATICAL DOUBLE-STRUCK CAPITAL L][x ];2) x is minimal over [MATHEMATICAL DOUBLE-STRUCK CAPITAL L][A ], that is, if a set Y belongs to [MATHEMATICAL DOUBLE-STRUCK CAPITAL L][x ], then either x ∈ [MATHEMATICAL DOUBLE-STRUCK CAPITAL L][A, Y ] or Y (...)
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  • Absoluteness via resurrection.Giorgio Audrito & Matteo Viale - 2017 - Journal of Mathematical Logic 17 (2):1750005.
    The resurrection axioms are forcing axioms introduced recently by Hamkins and Johnstone, developing on ideas of Chalons and Veličković. We introduce a stronger form of resurrection axioms for a class of forcings Γ and a given ordinal α), and show that RAω implies generic absoluteness for the first-order theory of Hγ+ with respect to forcings in Γ preserving the axiom, where γ = γΓ is a cardinal which depends on Γ. We also prove that the consistency strength of these axioms (...)
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