Abstract As a continuation of ideas initiated in [19], we study bi-colored (generic) expansions of geometric theories in the style of the Fraïssé–Hrushovski construction method. Here we examine that the properties $NTP_{2}$ , strongness, $NSOP_{1}$ , and simplicity can be transferred to the expansions. As a consequence, while the corresponding bi-colored expansion of a red non-principal ultraproduct of p-adic fields is $NTP_{2}$ , the expansion of algebraically closed fields with generic automorphism is a simple theory. Furthermore, these theories are strong with $\operatorname {\mathrm {bdn}}(\text {"}x=x\text {"})=(\aleph _0)_{-}$ .

Abstract We define a class of computable functions over real numbers using functional schemes similar to the class of primitive and partial recursive functions defined by Gödel (1931, 1934) and Kleene (1936, Math. Ann., 112, 727–742). We show that this class of functions can also be characterized by MS-machines, which are Turing machine-like devices. The proof of the characterization gives a normal form theorem in the style of Kleene (1936, Math. Ann., 112, 727–742). Furthermore, this characterization is a natural combination of two most influential theories of computation over real numbers, namely the type-two theory of effectivity (see, e.g. Weihrauch (2000, Springer)) and the Blum–Shub–Smale (1989, Bull. Amer. Math. Soc. (N.S.), 21, 1–46) model of computation. Under this notion of computability, the recursive (or computable) subsets of real numbers are exactly effective $\varDelta ^0_2$ sets.

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.

This paper is a further investigation of a project carried out in Didehvar and Ghasemloo (2009) to study effective aspects of the metric logic. We prove an effective version of the omitting types theorem. We also present some concrete computable constructions showing that both the separable atomless probability algebra and the rational Urysohn space are computable metric structures.

Our aim in this article is twofold. First, for an arbitrary closed subset V of [0,1] containing both 0 and 1, the compactness theorem for the Gödel logic GV is investigated. Next, following Cintula and Navara (2004, Fuzzy sets and systems, 143, 59–73) and Tavana et al. (2012, Logic J. IGPL, 20, 254–265), for any subset K of [0,1], the notions of K-satisfiability and K-compactness are introduced for the standard first-order Gödel logic Gℝ. It is shown that whenever K is closed, the K-compactness fails for Gℝ if and only if K is infinitely countable and 1 ∉ K.