Abstract. We introduce a new type of generalized Turing machines (GTMs), which are intended as a tool for the mathematician who studies computability in Analysis. In a single tape cell a GTM can store a symbol, a real number, a continuous real function or a probability measure, for example. The model is based on TTE, the representation approach for computable analysis. As a main result we prove that the functions that are computable via given representations are closed under GTM programming. This generalizes the well known fact that these functions are closed under composition. The theorem allows to speak about objects themselves instead of names in algorithms and proofs. By using GTMs for specifying algorithms, many proofs become more rigorous and also simpler and more transparent since the GTM model is very simple and allows to apply well-known techniques from Turing machine theory. We also show how finite or infinite sequences as names can be replaced by sets (generalized representations) on which computability is already defined via representations. This allows further simplification of proofs. All of this is done for multi-functions, which are essential in Computable Analysis, and multirepresentations, which often allow more elegant formulations. As a byproduct we show that the computable functions on finite and infinite sequences of symbols are closed under programming with GTMs. We conclude with examples of application.

This paper is devoted to systematic studies of some extensions of firstorder Gödel logic. The first extension is the 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 the first-order rational Gödel 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 Gödel 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.

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.

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