On the Strength of the Uniform Fixed Point Principle in Intuitionistic Explicit Mathematics

Abstract

The paper is concerned with a line of research that plumbs the scope of constructive theories. The object of investigation here is Feferman’s intuitionistic theory of explicit mathematics augmented by the monotone fixed point principle which asserts that every monotone operation on classifications (Feferman’s notion of set) possesses a least fixed point. To be more precise, the new axiom not merely postulates the existence of a least solution, but, by adjoining a newfunctional constant to the language, it is ensured that a fixed point is uniformly presentable as a function of the monotone operation. The strength of the classical non-uniform version, MID, was investigated in [6] whereas that of the uniform version was determined in [16, 17] and shown to be that of subsystems of second order arithmetic based on $\varPi_{2}^{1}$-comprehension. This involved a rendering of $\varPi_{2}^{1}$-comprehension in terms of fixed points of non-monotonic $\varPi_{1}^{1}$-operators and a proof-theoretic interpretation of the latter in specific operator theories that can be interpreted in explicit mathematics with the uniform monotone fixed point principle. The intent of the current paper is to show that the same strength obtains when the underlying logic is taken to be intuitionistic logic.