There are two main results concerning the nature of constructive extensions of the countable atomless Boolean algebra, say 28 a, which appear in [1]. These results are (I) if <& is a constructive extension of 23a, then each element of # is recursive open and regular, and (II) the characterization of the simple constructive extensions 28JJJ) as those for which U is recursive open and regular. We...

Euclidean geometry, as presented by Euclid, consists of straightedge-and-compass constructions and rigorous reasoning about the results of those constructions. A consideration of the relation of the Euclidean “constructions” to “constructive mathematics” leads to the development of a first-order theory ECG of the “Euclidean Constructive Geometry”, which can serve as an axiomatization of Euclid ...

We define a constructive topos to be a locally cartesian closed pretopos. The terminology is supported by the fact that constructive toposes enjoy a relationship with constructive set theory similar to the relationship between elementary toposes and (impredicative) intuitionistic set theory. This paper elaborates upon one aspect of the relationship between constructive toposes and constructive ...

Circumscription is one of the most powerful nonmonotonic reasoning formalisms. Its classical modeltheoretic semantics serves as an effective mechanism to ensure minimality in commonsense reasoning. However, as the lack of proof-theoretic feature, it is incapable of expressing derivation order that sometimes plays a crucial role in various reasoning tasks. On the other hand, the general theory o...

Partially ordered sets are investigated from the point of view of Bishop’s constructive mathematics, which can be viewed as the constructive core of mathematics and whose theorems can be translated into many formal systems of computable mathematics. The relationship between two classically equivalent notions of supremum is examined in detail. Whereas the classical least upper bound is based on ...

James Ladyman has argued that constructive empiricism entails modal realism, and that this renders constructive empiricism untenable. We maintain that constructive empiricism is compatible with modal nominalism. Although the central term ‘observable’ has been analyzed in terms of counterfactuals, and in general counterfactuals do not have objective truth conditions, the property of being observ...