A semantic proof of strong cut-admissibility for first-order Gödel logic

We provide a constructive direct semantic proof of the completeness of the cut-free part of the hypersequent calculus HIF for the standard first-order Gödel logic (thereby proving both completeness of the calculus for its standard semantics, and the admissibility of the cut rule in the full calculus). The results also apply to derivations from assumptions (or “non-logical axioms”), showing in particular that when the set of assumptions is closed under substitutions, then cuts can be confined to formulas occurring in the assumptions. The methods and results are then extended to handle the (Baaz) Delta connective as well.