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

  • Hasen Khudairi
Chapter
Part of the Philosophical Studies Series book series (PSSP, volume 134)

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.

Keywords

Modal Ω-logic Ω-logical Validity Modal Coalgebraic Automata Neo-Logicism Set-theoretic Realism 

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Copyright information

© Springer Nature Switzerland AG 2019

Authors and Affiliations

  • Hasen Khudairi
    • 1
  1. 1.Arché Philosophical Research CentreUniversity of St AndrewsScotlandUK

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