From rational Godel logic to continuous ultrametric logic

This paper is devoted to systematic studies of some extensions of first-order G\"odel logic. The first extension is the first-order rational G\"odel logic which is an extension of first-order G\"odel logic, enriched by countably many nullary logical connectives. By introducing some suitable semantics and proof theory, it is shown that the first-order rational G\"odel logic has the completeness property, that is 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 the first-order rational G\"odel logic, the ultrametric logic is introduced. This serves as a suitable framework for analyzing structures which carry an ultrametric function $d$ 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.