We study propositional and first-order G?del 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 Schütte–Tait style. Initial observations are made regarding truth-value sets other than 0, 1.
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