From rational Gödel logic to ultrametric logic

Abstract

This article is devoted to systematic studies of some extensions of first-order Gödel logic. The first extension is first-order rational Gödel logic which is an extension of first-order Gödel logic, enriched by countably many nullary logical connectives. By introducing some suitable semantics and proof theory, it is shown that first-order rational Gödel logic has a weak version of the completeness property, i.e. any (strongly) consistent theory is satisfiable. Furthermore, two notions of entailment and strong entailment are defined and their relations with the corresponding notion of proof is studied. In particular, an approximate entailment-compactness is shown. Next, by adding a binary predicate symbol d to first-order rational Gödel logic, ultrametric logic is introduced. This serves as a suitable framework for analyzing structures which carry an ultrametric d together with some functions and predicates which are uniformly continuous with respect to the ultrametric d . Some model theory is developed and to justify the relevance of this model theory, the Robinson joint consistency theorem is proven.