Generalized Inductive Definitions in Constructive Set Theory

The intent of this paper is to study generalized inductive definitions on the basis of Constructive Zermelo-Fraenkel Set Theory, CZF. In theories such as classical ZermeloFraenkel Set Theory, it can be shown that every inductive definition over a set gives rise to a least and a greatest fixed point, which are sets. This principle, notated GID, can also be deduced from CZF plus the full impredicative separation axiom or CZF augmented by the power set axiom. Full separation and a fortiori the power set axiom, however, are entirely unacceptable from a constructive point of view. It will be shown that while CZF + GID is stronger than CZF, the principle GID does not embody the strength of any of these axioms. CZF + GID can be interpreted in Feferman’s Explicit Mathematics with a least fixed point principle. The proof-theoretic strength of the latter theory is expressible by means of a fragment of second order arithmetic. MSC:03F50, 03F35