Printed in The Netherlands Logic and Analysis On the constructive Dedekind reals

Abstract In order to build the collection of Cauchy reals as a set in constructive set theory, the only power set-like principle needed is exponentiation. In contrast, the proof that the Dedekind reals form a set has seemed to require more than that. The main purpose here is to show that exponentiation alone does not suffice for the latter, by furnishing a Kripke model of constructive set theory, Constructive Zermelo–Fraenkel set theory with subset collection replaced by exponentiation, in which the Cauchy reals form a set while the Dedekind reals constitute a proper class. Mathematics Subject Classification (2000) 03F50 ·03F55 ·03F60 ·03E35 ·03G30