Minimum Classical Extensions of Constructive Theories

Abstract

Reverse constructive mathematics, based on the pioneering work of Kleene, Vesley, Kreisel, Troelstra, Bishop, Bridges and Ishihara, is currently under development. Bishop constructivists tend to emulate the classical reverse mathematics of Friedman and Simpson. Veldman’s reverse intuitionistic analysis and descriptive set theory split notions in the style of Brouwer. Kohlenbach’s proof mining uses interpretations and translations to extract computational information from classical proofs. We identify the classical content of a constructive mathematical theory with the Gentzen negative interpretation of its classically correct part. In this sense HA and PA have the same classical content but intuitionistic and classical two-sorted recursive arithmetic with quantifier-free countable choice do not; $\varSigma ^0_1$ numerical double negation shift expresses the precise difference. Other double negation shift and weak comprehension principles clarify the classical content of stronger constructive theories. Any consistent axiomatic theory S based on intuitionistic logic has a minimum classical extension S$^{+g}$, obtained by adding to S the negative interpretations of its classically correct consequences. Subsystems of Kleene’s intuitionistic analysis and supersystems of Bishop’s constructive analysis provide interesting examples, with the help of constructive decomposition theorems.