Given a class ℱ oft otal functions in the set oft he natural numbers, one could study the real numbers that have arbitrarily close rational approximations explicitly expressible by means of functions from ℱ. We do this for classes ℱsatisfying certain closedness conditions. The conditions in question are satisfied for example by the class of all recursive functions, by the class of the primitive recursive ones, by any of the Grzegorczyk classes ℰnwith n ≥ 2, by the class of all (...) functions recursive in a given function and by the class of the functions primitive recursive in it, as well as by the class of all total functions in the set of the natural numbers. (shrink)

In order to avoid the use of individual variables in predicate calculus, several authors proposed language whose expressions can be interpreted, in general, as denotations of predicates . The present author also proposed a language of this kind [5]. The absence of individual variables makes all these languages rather different from the traditional language of predicate calculus and from the usual language of mathematics. The translation procedures from the ordinary predicate languages into the predicate languages without individual variables and vice (...) versa have a technical character. It is desirable to make possible carrying out such procedures by performing some transformations in a suitable new language which comprises both a sublanguage similar to the ordinary predicate languages and another one not using individual variables. We shall propose now a language of the desired kind. The expressions of this language could be interpreted, in general, as denoting predicates which depend on parameters on a given object domain – this predicate has one argument and depends on one parameter). The proposed language will have the modal features noted in [3] and [5]. It is worthy of mention that the language proposed at the end of [4] although containing expressions interpreted as predicates and expressions with parameters does not allow simultaneous availability of argument places and parameters. (shrink)

A theorem published by A. Grzegorczyk in 1955 states a certain kind of effective uniform continuity of computable functionals whose values are natural numbers and whose arguments range over the total functions in the set of the natural numbers and over the natural numbers. Namely, for any such functional a computable functional with one function-argument and the same number-arguments exists such that the values of the first of the functionals at functions dominated by a given one are completely determined by (...) their restrictions to numbers not exceeding the corresponding values of the second functional at the given function. We prove versions of this theorem for the class of the primitive recursive functionals and for the class of the elementary recursive ones. Analogous results can be proved for many other subrecursive classes of functionals. (shrink)

For any class of operators which transform unary total functions in the set of natural numbers into functions of the same kind, we define what it means for a real function to be uniformly computable or conditionally computable with respect to this class. These two computability notions are natural generalizations of certain notions introduced in a previous paper co-authored by Andreas Weiermann and in another previous paper by the same authors, respectively. Under certain weak assumptions about the class in question, (...) we show that conditional computability is preserved by substitution, that all conditionally computable real functions are locally uniformly computable, and that the ones with compact domains are uniformly computable. The introduced notions have some similarity with the uniform computability and its non-uniform extension considered by Katrin Tent and Martin Ziegler, however, there are also essential differences between the conditional computability and the non-uniform computability in question. (shrink)

The general framework of this paper is a reformulation of Hilbert’s program using the theory of locales, also known as formal or point-free topology [P.T. Johnstone, Stone Spaces, in: Cambridge Studies in Advanced Mathematics, vol. 3, 1982; Th. Coquand, G. Sambin, J. Smith, S. Valentini, Inductively generated formal topologies, Ann. Pure Appl. Logic 124 71–106; G. Sambin, Intuitionistic formal spaces–a first communication, in: D. Skordev , Mathematical Logic and its Applications, Plenum, New York, 1987, pp. 187–204]. Formal topology presents (...) a topological space, not as a set of points, but as a logical theory which describes the lattice of open sets. The application to Hilbert’s program is then the following. Hilbert’s ideal objects are represented by points of such a formal space. There are general methods to “eliminate” the use of points, close to the notion of forcing and to the “elimination of choice sequences” in intuitionist mathematics, which correspond to Hilbert’s required elimination of ideal objects. This paper illustrates further this general program on the notion of valuations. They were introduced by Dedekind and Weber [R. Dedekind, H. Weber, Theorie des algebraischen Funktionen einer Veränderlichen, J. de Crelle t. XCII 181–290] to give a rigorous presentation of Riemann surfaces. It can be argued that it is one of the first example in mathematics of point-free representation of spaces [N. Bourbaki, Eléments de Mathématique. Algèbre commutative, Hermann, Paris, 1965, Chapitre 7]. It is thus of historical and conceptual interest to be able to represent this notion in formal topology. (shrink)

The relation between least and diagonal fixed points is a well known and completely studied question for a large class of partially ordered models of the lambda calculus and combinatory logic. Here we consider this question in the context of algebraic recursion theory, whose close connection with combinatory logic recently become apparent. We find a comparatively simple and rather weak general condition which suffices to prove the equality of least fixed points with canonical (corresponding to those produced by the Curry (...) combinator in lambda calculus) diagonal fixed points in a class of partially ordered algebras which covers both combinatory spaces of Skordev and operative spaces of Ivanov. Especially, this yields an essential improvement of the axiomatization of recursion theory via combinatory spaces. (shrink)