Baaz, Matthias, Norbert Preining, and Richard Zach. 2007. “First-Order Gödel Logics.” Annals of Pure and Applied Logic 147 (1–2): 23–47. https://doi.org/10.1016/j.apal.2007.03.001.
First-order Gödel logics are a family of finite- or 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). It is 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. Complete axiomatizations for each of these cases are given. The r.e. prenex, negation-free, and existential fragments of all first-order Gödel logics are also characterized.
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