A Simple Proof of Completeness and Cut-admissibility for Propositional Gödel Logic

We provide a constructive, direct and simple proof of the completeness of the cut-free part of the hypersequential calculus HG for Gödel logic (thereby proving both completeness of the calculus for its standard semantics, and the admissibility of the cut rule in the full calculus). We then extend the results and proofs to derivations from assumptions, showing that such derivations can be confined to those in which cuts are made only on formulas which occur in the assumptions. The article is self-contained, and no previous knowledge concerning HG (or even Gödel logic) is needed for understanding it.