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Completeness of infinitary intuitionistic logics

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Mathematical, Foundational and Computational Aspects of the Higher Infinite

Completeness theorems for infinitary classical logics have been known for decades. When removing the law of excluded middle, however, the situation is more difficult to analyse even in the propositional case, as the main obstacle for studying infinitary intuitionistic logics is the huge variety of non-equivalent formulas that one can obtain. Completeness results for the propositional fragment and disjunctions and conjunctions of countable size have been obtained, but the general case has not been addressed. The purpose of this talk is to outline set-theoretical and category-theoretical techniques that allow the study of completeness theorems for infinitary intuitionistic logics in the general case, both for propositional and first-order logics, in terms of an infinitary Kripke semantics. We will also analyse to what extent the use of large cardinal axioms (more precisely, weakly compact cardinals) is necessary, and some applications of these completeness results will be presented.

This talk is part of the Isaac Newton Institute Seminar Series series.

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