In Matteo Vincenzo D'Alfonso & Don Berkich (eds.), On the Cognitive, Ethical, and Scientific Dimensions of Artificial Intelligence. Springer. pp. 65-82 (2019)

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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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DOI 10.1007/978-3-030-01800-9_4
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Attitudes de Dicto and de Se.David Lewis - 1979 - Philosophical Review 88 (4):513-543.
Warrant for Nothing (and Foundations for Free)?Crispin Wright - 2004 - Aristotelian Society Supplementary Volume 78 (1):167–212.
Models and Reality.Hilary Putnam - 1980 - Journal of Symbolic Logic 45 (3):464-482.

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