The proof-theoretic analysis of transfinitely iterated fixed point theories

Abstract

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 <to , has proof-theoretic ordinal To, thus providing a framework for predicative mathematics.In this article we make the step to transfinite iterations of fixed points and are interested in the theories ID Q for a > 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 <p functions. In Section 3 we introduce the theories ID a and we state the Main Theorem about the proof-theoretic ordinal of ID a . Section 4 is devoted to the definition of transfinitely iterated theories for self-reflecting truth SRT". These systems are contained in ID Q , and greatly facilitate the wellordering proofs for the latter, which we will present in Section 5. A full cut elimination procedure for ID a is provided in Section 6. Finally, we discuss some related topics in Section 7.

The proof-theoretic analysis of transfinitely iterated quasi least fixed points

“…The theories ID" of n-times iterated inductive definitions have been analyzed by Feferman in connection with Handcock's conjecture in [6]. The proof-theoretic analysis of ID tt has been carried out in all details by Jager, Kahle, Setzer and Strahm [9]. _ Some problems however, have remained unsolved: In the theories ID a , induction on fixed points is dropped completely.…”

Upper bounds for metapredicative Mahlo in explicit mathematics and admissible set theory

“…By combining (10) and (11) and applying persistency, we can finally derive The following main result of this paragraph is the natural generalization of Main Lemma II in Jager, Kahle, Setzer and Strahm [8] PROOF. This theorem is proved by induction on n < a>.…”

Theories of Proof-Theoretic Strength Ψ (ΓΩ +1)

“…• n , for all natural numbers n ≥ 1, and to reduce FP • n are the analogues of the well-known fixed point theories ID n , see Feferman [12] or Jäger, Kahle, Setzer, and Strahm [18], but with ID 1 rather than PA as the base theory. The languages L…”