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Modal Ω-Logic: Automata, Neo-Logicism, and Set-theoretic Realism

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
Creators:
CreatorsEmailORCID
Khudairi, Hasenhasen.khudairi@gmail.com
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 > Computer Science > Artificial Intelligence
Depositing User: Hasen Khudairi
Date Deposited: 19 May 2017 14:34
Last Modified: 19 May 2017 14:34
Item ID: 13047
DOI or Unique Handle: https://philpapers.org/archive/HASMLA-3.pdf
Subjects: Specific Sciences > Mathematics > Epistemology
Specific Sciences > Mathematics > Foundations
Specific Sciences > Mathematics > Logic
Specific Sciences > Computer Science > Artificial Intelligence
Date: 9 January 2017
URI: http://philsci-archive.pitt.edu/id/eprint/13047

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