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Absoluteness via resurrection

Giorgio Audrito & Matteo Viale

Journal of Mathematical Logic 17 (2):1750005 (2017)

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Incompatible bounded category forcing axioms.David Asperó & Matteo Viale - 2022 - Journal of Mathematical Logic 22 (2).details Journal of Mathematical Logic, Volume 22, Issue 02, August 2022. We introduce bounded category forcing axioms for well-behaved classes [math]. These are strong forms of bounded forcing axioms which completely decide the theory of some initial segment of the universe [math] modulo forcing in [math], for some cardinal [math] naturally associated to [math]. These axioms naturally extend projective absoluteness for arbitrary set-forcing — in this situation [math] — to classes [math] with [math]. Unlike projective absoluteness, these higher bounded category forcing (...) Download Export citation Bookmark 1 citation
First‐order undefinability of the notion of transfinitely uplifting cardinals.Kentaro Fujimoto - 2021 - Mathematical Logic Quarterly 67 (1):105-115.details Audrito and Viale introduced the new large cardinal notion of an (α)‐uplifting cardinal (for an ordinal α). We shall show that this notion cannot be defined (or expressed) in the standard first‐order language of set theory for every tranfinite α. Download Export citation Bookmark
Second order arithmetic as the model companion of set theory.Giorgio Venturi & Matteo Viale - 2023 - Archive for Mathematical Logic 62 (1):29-53.details This is an introductory paper to a series of results linking generic absoluteness results for second and third order number theory to the model theoretic notion of model companionship. Specifically we develop here a general framework linking Woodin’s generic absoluteness results for second order number theory and the theory of universally Baire sets to model companionship and show that (with the required care in details) a $$\Pi _2$$ -property formalized in an appropriate language for second order number theory is forcible (...) Download Export citation Bookmark 1 citation
What Model Companionship Can Say About the Continuum Problem.Giorgio Venturi & Matteo Viale - 2024 - Review of Symbolic Logic 17 (2):546-585.details We present recent results on the model companions of set theory, placing them in the context of a current debate in the philosophy of mathematics. We start by describing the dependence of the notion of model companionship on the signature, and then we analyze this dependence in the specific case of set theory. We argue that the most natural model companions of set theory describe (as the signature in which we axiomatize set theory varies) theories of $H_{\kappa ^+}$, as $\kappa (...) Download Export citation Bookmark 2 citations

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