Explicit mathematics with the monotone fixed point principle. II: Models

Abstract

AbstractThis paper continues investigations of the monotone fixed point principle in the context of Feferman's explicit mathematics begun in [14]. Explicit mathematics is a versatile formal framework for representing Bishop-style constructive mathematics and generalized recursion theory. The object of investigation here is the theory of explicit mathematics augmented by the monotone fixed point principle, which asserts that any 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 new constant to the language, it is ensured that a fixed point is uniformly presentable as a function of the monotone operation. Let T0 + UMID denote this extension of explicit mathematics. [14] gave lower bounds for the strength of two subtheories of To + UMID in relating them to fragments of second order arithmetic based on comprehension. [14] showed that To ↾ + UMID and To ↾ + INDk + UMID have at least the strength of ( – CA), and ↾ and – CA), respectively.Here we are concerned with the exact reversals. Let UMIDN be the monotone fixed-point principle for subclassifications of the natural numbers. Among other results, it is shown that To ↾ + UMIDN and To ↾ + INDN + UMIDN have the same strength as ( – CA) ↾ and ( – CA), respectively.The results are achieved by constructing set-theoretic models for the aforementioned systems of explicit mathematics in certain extensions of Kripke-Platek set theory and subsequently relating these set theories to subsystems of second arithmetic.