Induction-recursion is a schema which formalizes the principles for introducing new sets in Martin-Löf's type theory. It states that we may inductively define a set while simultaneously defining a function from this set into an arbitrary type by structural recursion. This extends the notion of an inductively defined set substantially and allows us to introduce universes and higher order universes (but not a Mahlo universe). In this article we give a finite axiomatization of inductive-recursive definitions. We prove consistency by constructing a set-theoretic model which makes use of one Mahlo cardinal.

Abstract. This article provides the proof-theoretic analysis of the transfinitely iterated fixed point theories ID a and ID< a ; the exact proof-theoretic ordinals of these systems are presented. §1. Introduction. The transfinitely iterated fixed point theories ID« are relatives of the better known theories ID Q for iterated inductive definitions. These latter theories have been extensively studied during the last years (cf., e.g., Buchholz et al.[1]) and their proof-theoretic analysis has been carried through in all detail.The basic axioms of ID a provide hierarchies of least (definable) fixed points of a times iterated positive inductive definitions given by arithmetic operator forms. In the case of the fixed point theories ID Q , on the other hand, one confines oneself to hierarchies of arbitrary fixed points of the corresponding inductive definitions and drops the requirement for minimality.The finitely iterated fixed point theories ID" were first introduced in Feferman [5] in connection with his proof of Hancock's conjecture. Among other things, it is shown in this article that the proof-theoretic ordinal of ID" is a" for ao := e 0 and a n+ \ := ipa"0. Hence, the union of all ID" for n < co, i.e., the system ID co. It is a technical paper, which establishes the proof-theoretic ordinals of these systems. The relationship between transfinitely iterated fixed point theories and subsystems of second order arithmetic and the role of transfinitely iterated fixed point theories for metapredicativity in general are only briefly addressed in the conclusion.The plan of this paper is as follows. In Section 2 we discuss some ordinal-theoretic preliminaries; namely, we sketch an ordinal notation system which is based on nary

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