Classical consequences of constructive systems

Abstract

This is a survey of formal axiomatic systems for the three main varieties of constructive analysis, in a common language and with intuitionistic logic, which are as nearly as possible compatible with classical analysis and with one another. Classically sound consequences of principles of intuitionistic mathematics are emphasized. Compatibility with classical analysis is of two kinds. On the one hand, Bishop’s constructive mathematics and a very substantial part of intuitionistic analysis are classically correct, sharing with constructive recursive mathematics a neutral subsystem adequate for recursive function theory and elementary real analysis. On the other hand, each constructive system considered here is separately consistent with the negative interpretation of each of its classically sound subsystems, establishing internal compatibility with the classical context it is intended to refine. A possibly new criterion for the transposability of unique existential quantifiers and a recent theorem of Vafeiadou on the internal compatibility of classical and intuitionistic analysis are included. This article is part of the theme issue ‘Modern perspectives in Proof Theory’.