Khudairi, Hasen (2017) Modal Ω-Logic: Automata, Neo-Logicism, and Set-Theoretic Realism. [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, Hasen hasen.khudairi@gmail.com 0000-0003-1726-6123
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:	Hasen Khudairi
Date Deposited:	21 May 2021 01:36
Last Modified:	21 May 2021 01:36
Item ID:	19070
Subjects:	Specific Sciences > Mathematics > Epistemology Specific Sciences > Mathematics > Foundations Specific Sciences > Mathematics > Logic Specific Sciences > Computation/Information
Date:	2017
URI:	http://philsci-archive.pitt.edu/id/eprint/19070

Available Versions of this Item

• Modal Ω-Logic: Automata, Neo-Logicism, and Set-Theoretic Realism. (deposited 09 Aug 2018 22:54)
  • Modal Ω-Logic: Automata, Neo-Logicism, and Set-Theoretic Realism. (deposited 17 May 2021 16:43)
    • Modal Ω-Logic: Automata, Neo-Logicism, and Set-Theoretic Realism. (deposited 21 May 2021 01:36) [Currently Displayed]
      • Modal Ω-Logic: Automata, Neo-Logicism, and Set-Theoretic Realism. (deposited 03 Jun 2021 18:41)
        • Modal Ω-Logic: Automata, Neo-Logicism, and Set-Theoretic Realism. (deposited 23 Jun 2021 19:04)
          • Modal Ω-Logic: Automata, Neo-Logicism, and Set-Theoretic Realism. (deposited 29 Jun 2021 19:34)
            • Modal Ω-Logic: Automata, Neo-Logicism, and Set-Theoretic Realism. (deposited 09 May 2022 22:47)
            • Modal Ω-Logic: Automata, Neo-Logicism, and Set-Theoretic Realism. (deposited 22 Jul 2021 16:35)
              • Modal Ω-Logic. (deposited 25 Nov 2022 17:02)
              • Modal Ω-Logic: Automata, Neo-Logicism, and Set-Theoretic Realism. (deposited 22 Nov 2022 16:03)
                • Modal Ω-Logic: Automata, Neo-Logicism, and Set-Theoretic Realism. (deposited 20 Dec 2022 18:42)

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