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 Ω-logic in Zermelo-Fraenkel set theory with choice (ZFC). The dual isomorphism between algebra and coalgebra permits Boolean-valued algebraic models of ZFC to be interpreted as coalgebras. The modal profile of Ω-logical validity can then be countenanced within a coalgebraic logic, and Ω-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 Ω-logical validity correspond to those of second-order logical consequence, Ω-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 | ||||||
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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 |
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Depositing User: | Hasen Khudairi | ||||||
Date Deposited: | 17 May 2021 16:43 | ||||||
Last Modified: | 17 May 2021 16:43 | ||||||
Item ID: | 19047 | ||||||
Subjects: | Specific Sciences > Mathematics > Epistemology Specific Sciences > Mathematics > Foundations Specific Sciences > Mathematics > Logic Specific Sciences > Computation/Information |
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Date: | 2017 | ||||||
URI: | http://philsci-archive.pitt.edu/id/eprint/19047 |
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Modal Ω-Logic: Automata, Neo-Logicism, and Set-Theoretic Realism. (deposited 09 Aug 2018 22:54)
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Modal Ω-Logic: Automata, Neo-Logicism, and Set-Theoretic Realism. (deposited 17 May 2021 16:43)
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Modal Ω-Logic: Automata, Neo-Logicism, and Set-Theoretic Realism. (deposited 21 May 2021 01:36)
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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)
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Modal Ω-Logic: Automata, Neo-Logicism, and Set-Theoretic Realism. (deposited 03 Jun 2021 18:41)
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Modal Ω-Logic: Automata, Neo-Logicism, and Set-Theoretic Realism. (deposited 21 May 2021 01:36)
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Modal Ω-Logic: Automata, Neo-Logicism, and Set-Theoretic Realism. (deposited 17 May 2021 16:43)
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