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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Abstract. In this article we introduce systems for metapredicative Mahlo in explicit mathematics and admissible set theory. The exact upper proof-theoretic bounds of these systems are established. §1. Introduction. In classical set theory an ordinal K is called a Mahlo ordinal if it is a regular cardinal and if, for every normal function / from K to K, there exists a regular cardinal fi less than K so that {/(£) : £ < ju} C [i. The statement that there exists a Mahlo ordinal is a powerful set existence axiom going beyond theories like ZFC. It also outgrows the existence of inaccessible cardinals, hyper inaccessibles, hyperhyperinaccessible and the like.There is also an obvious recursive analogue of Mahlo ordinal. Typically, an ordinal a is baptized recursively Mahlo, if it is admissible and reflects every TI2 sentence on a smaller admissible ordinal. The corresponding formal theory KPM has been proof-theoretically analyzed by Rathjen [14, 15]. KPM is a highly impredicative theory, and its proof-theoretic strength is significantly beyond that of KPi, the second order theory (A 2 -CA) + (Bl) and Feferman's theory To, which are all proof-theoretically equivalent (cf. [3,6,10]).This article can be seen as a further contribution to the general program of metapredicativity. We have studied other metapredicative theories in Jager, Kahle, Setzer and Strahm [8], Jager and Strahm [11], and Strahm [21, 20]; there also some further background material can be found.One aim here is to look at metapredicative Mahlo in admissible set theory. The corresponding theory, named KPm°, is admissible set theory above the natural numbers as urelements plus n2 reflection on the admissibles. As induction principles we have complete induction on the natural numbers for sets, but do not include e induction.A further aim of this paper is to introduce the concept of Mahloness into explicit mathematics and to analyze the proof-theoretic strength of its metapredicative version. An extension of Feferman's theory T 0 by Mahlo axioms is studied in Jager and Studer [12]. Setzer [18] presents a related formulation in the framework of Martin-L6f type theory.For the formalization of Mahlo in explicit mathematics we work over the basic theory EETJ which comprises the axioms of applicative theories and has type

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This paper studies systems of explicit mathematics as introduced by Feferman [9, 11]. In particular, we propose weak explicit type systems with a restricted form of elementary comprehension whose provably terminating operations coincide with the functions on binary words that are computable in polynomial time. The systems considered are natural extensions of the first-order applicative theories introduced in Strahm [19,20].

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