Both bar designs provide good retention and functional comfort. High stability appears to be an important factor for the patients' satisfaction and oral comfort. Rigid retention results in a higher force impact and appears to evoke the need for the retightening of occlusal screws, resulting in more maintenance service.

Abstract. The starting point of this article is an old question asked by Feferman in his paper on Hancock's conjecture [6] about the strength of ID*. This theory is obtained from the well-known theory ID] by restricting fixed point induction to formulas that contain fixed point constants only positively. The techniques used to perform the proof-theoretic analysis of ID* also permit to analyze its transfinitely iterated variants ID*. Thus, we eventually know that 11 D" | = 11D* |. §1. Introduction. The theories ID a of iterated inductive definitions formalize hierarchies of least (definable) fixed points. In the past years, these theories have been exhaustively studied and their proof-theoretic analysis has been carried out a long time ago, (cf. Buchholz et al. [3]). Also their metapredicative relatives ID a , that speak about hierarchies of (not necessary least) fixed points are well understood by now. The proof-theoretic ordinal of IDi is due to Aczel [1], who used a recursion theoretic argument, nowadays known as Aczel's trick, to embed IDi into 2}-AC. 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. It is natural to study theories, where fixed point induction is only restricted. Kreisel pointed out in [11], that "an inductive definition tells you what is in P^ not what is not in 9 s *". As mentioned in Feferman [6], this motivated to consider restricted versions of I Di such as IDj, a theory credited to H. Friedman where the scheme for proof by induction on fixed points is restricted to formulas that contain fixed point constants only positively. The question for a sharp upper bound is raised loc. cit. No answer to this question has yet been published, although partial results have been attained: If the fixed point axioms of ID, are restricted to so-called accessibility inductive definitions, then the resulting theory \D*{stf%W) can be embedded in S'-DC as sketched by Feferman in [6]. There, it is also stated that Friedman [8] introduced the theory IDj and showed that its ordinal is bounded by a\, where ao := so and a"+i := y>a"0. Further, upper bounds for

Let ≺ be a primitive recursive well-ordering on the natural numbers and assume that its order-type is greater than or equal to the proof-theoretic ordinal of the theory T. We show that the proof-theoretic strength of T is not increased if we add the negation of the statement which formalizes transfinite induction along ≺.

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