Grothendieck Universes and Indefinite Extensibility

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Abstract
This essay endeavors to define the concept of indefinite extensibility in the setting of category theory. I argue that the generative property of indefinite extensibility for set-theoretic truths in the category of sets is identifiable with the elementary embeddings of large cardinal axioms. A modal coalgebraic automata's mappings are further argued to account for both reinterpretations of quantifier domains as well as the ontological expansion effected by the elementary embeddings in the category of sets. The interaction between the interpretational and objective modalities of indefinite extensibility is defined via the epistemic interpretation of two-dimensional semantics. The semantics can be defined intensionally or hyperintensionally. By characterizing the modal profile of $\Omega$-logical validity, and thus the generic invariance of mathematical truth, modal coalgebraic automata are further capable of capturing the notion of definiteness for set-theoretic truths, in order to yield a non-circular definition of indefinite extensibility.
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First archival date: 2017-06-14
Latest version: 57 (2022-09-20)
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