On Infinitary G\"odel logics

Nicholas Pischke

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arXiv · 1708.07897

On Infinitary G\"odel logics

Nicholas Pischke

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Abstract

We study propositional and first-order G\"odel logics over infinitary languages which are motivated semantically by corresponding interpretations into the unit interval [0,1]. We provide infinitary Hilbert-style calculi for the particular (propositional and first-order) cases with con-/disjunctions of countable length and prove corresponding completeness theorems by extending the usual Lindenbaum-Tarski construction to the infinitary case for a respective algebraic semantics via complete linear Heyting algebras. We provide infinitary hypersequent calculi and prove corresponding cut-elimination theorems in the Sch\"utte-Tait-style. Initial observations are made regarding truth-value sets other than [0,1].

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Nicholas Pischke. 2017-08-25. On Infinitary G\"odel logics. https://arxiv.org/abs/1708.07897

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