Khudairi (Bowen), Hasen (Tim) (2019) Modal Ω-Logic. [Preprint]

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Abstract

This essay examines the philosophical significance of $\Omega$-logic in Zermelo-Fraenkel set theory with choice (ZFC). The duality between coalgebra and algebra permits Boolean-valued algebraic models of ZFC to be interpreted as coalgebras. The modal profile of $\Omega$-logical validity can then be countenanced within a coalgebraic logic, and $\Omega$-logical validity can be defined via deterministic automata. I argue that the philosophical significance of the foregoing is two-fold. First, because the epistemic and modal profiles of $\Omega$-logical validity correspond to those of second-order logical consequence, $\Omega$-logical validity is genuinely logical, and thus vindicates a neo-logicist conception of mathematical truth in the set-theoretic multiverse. Second, the foregoing provides a modal-computational account of the interpretation of mathematical vocabulary, adducing in favor of a realist conception of the cumulative hierarchy of sets.

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Item Type:	Preprint
Creators:	Creators Email ORCID Khudairi (Bowen), Hasen (Tim) hasen.khudairi@gmail.com 0000-0003-1726-6123
Additional Information:	In Matteo Vincenzo D'Alfonso & Don Berkich (eds.), On the Cognitive, Ethical, and Scientific Dimensions of Artificial Intelligence, Springer. pp. 65-82. 2019
Keywords:	Ω-Logic; Modal Logic; Logical Consequence; Large Cardinals; Coalgebra; Automata; Neo-Logicism; Set-theoretic Realism
Subjects:	Specific Sciences > Mathematics > Epistemology Specific Sciences > Mathematics > Foundations Specific Sciences > Mathematics > Logic Specific Sciences > Computation/Information
Depositing User:	Harrison Payne
Date Deposited:	20 Dec 2022 18:42
Last Modified:	19 Jan 2024 21:24
Item ID:	21573
Subjects:	Specific Sciences > Mathematics > Epistemology Specific Sciences > Mathematics > Foundations Specific Sciences > Mathematics > Logic Specific Sciences > Computation/Information
Date:	2019
URI:	https://philsci-archive.pitt.edu/id/eprint/21573

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• Modal Ω-Logic: Automata, Neo-Logicism, and Set-Theoretic Realism. (deposited 09 Aug 2018 22:54)

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