When Kleene extended his recursive realizability interpretation from intuitionistic arithmetic to analysis, he was forced to use more than recursive functions to interpret sequences and conditional constructions. In fact, he used what classically appears to be the full continuum. We describe here a generalization to higher type of Kleene's realizability, one case of which, -realizability, uses general recursive functions throughout, both to realize theorems and to interpret choice sequences. -realizability validates a version of the bar theorem and the usual continuity (...) principles, while also providing naturally, as Kleene's 1965 realizability does not, for versions of lawless sequence axioms, as well as of Church's Thesis. (shrink)

ZusammenfassungP. Bernays hat darauf hingewiesen, dass man, um die Widerspruchs freiheit der klassischen Zahlentheorie zu beweisen, den Hilbertschen flniter Standpunkt dadurch erweitern muss, dass man neben den auf Symbole sich beziehenden kombinatorischen Begriffen gewisse abstrakte Begriffe zulässt, Die abstrakten Begriffe, die bisher für diesen Zweck verwendet wurden, sinc die der konstruktiven Ordinalzahltheorie und die der intuitionistischer. Logik. Es wird gezeigt, dass man statt deesen den Begriff einer berechenbaren Funktion endlichen einfachen Typs über den natürlichen Zahler benutzen kann, wobei keine anderen (...) Konstruktionsverfahren für solche Funktionen nötig sind, als einfache Rekursion nach einer Zahlvariablen und Einsetzung von Funktionen ineinander .P. Bernays has pointed out that, in order to prove the consistency of classical number theory, it is necessary to extend Hilbert's finitary stand‐point by admitting certain abstract concepts in addition to the combinatorial concepts referring to symbols. The abstract concepts that so far have been used for this purpose are those of the constructive theory of ordinals and those of intuitionistic logic. It is shown that the concept of a computable function of finite simple type over the integers can be used instead, where no other procedures of constructing such functions are necessary except simple recursion by an integral variable and substitution of functions in each other. (shrink)

In this paper we provide a quantifier-free, constructive axiomatization of metric-Euclidean and of rectangular planes . The languages in which the axiom systems are expressed contain three individual constants and two ternary operations. We also provide an axiom system in algorithmic logic for finite Euclidean planes, and for several minimal metric-Euclidean planes. The axiom systems proposed will be used in a sequel to this paper to provide ‘the simplest possible’ axiom systems for several fragments of plane Euclidean geometry.

Elementary geometry can be axiomatized constructively by taking as primitive the concepts of the apartness of a point from a line and the convergence of two lines, instead of incidence and parallelism as in the classical axiomatizations. I first give the axioms of a general plane geometry of apartness and convergence. Constructive projective geometry is obtained by adding the principle that any two distinct lines converge, and affine geometry by adding a parallel line construction, etc. Constructive axiomatization allows solutions to (...) geometric problems to be effected as computer programs. I present a formalization of the axiomatization in type theory. This formalization works directly as a computer implementation of geometry. (shrink)

In this paper we provide a quantifier-free constructive axiomatization for Euclidean planes in a first-order language with only ternary operation symbols and three constant symbols . We also determine the algorithmic theories of some ‘naturally occurring’ plane geometries.