First-order Gödel logics are a family of infinite-valued logics where the sets of truth values V are closed subsets of [0, 1] containing both 0 and 1. Different such sets V in general determine different Gödel logics GV (sets of those formulas which evaluate to 1 in every interpretation into V). In a new paper with Matthias Baaz and Norbert Preining, we shown that GV is axiomatizable iff V is finite, V is uncountable with 0 isolated in V, or every neighborhood of 0 in V is uncountable. We give complete axiomatizations for each of these cases. The r.e. prenex, negation-free, and existential fragments of all first-order Gödel logics are also characterized.
The paper’s in the arXiv.