  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.
