On the relationship between fixed points and iteration in admissible set theory without foundation

Probst, Dieter

doi:10.1007/s00153-004-0251-1

Cited by 3 publications

(4 citation statements)

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“…Using this method, the application of pseudo-hierarchy arguments is no longer limited to second order analysis and can be applied in the context of explicit mathematics and admissible set theory as well; cf. [13,14]. In the sequel, Since S 2 G D Q and Wo^(ft), it has a <-least element.…”

Section: Ra a \^^-Hr(ss)mentioning

confidence: 99%

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

Probst¹

2006

J. symb. log.

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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

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Section: Ra a \^^-Hr(ss)mentioning

confidence: 99%

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

Probst¹

2006

J. symb. log.

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“…(ii) In view of a result in Avigad (1996) we thus know that KP + (Π 1 -Red) proves that every 𝑋-positive arithmetical formula 𝜑[𝑥, 𝑋 + ] has a fixed point (which is a set). (iii) On the other hand, we also know by results due to Gregoriades (2019) and Probst (2005) that there are positive arithmetical formulas that do not have hyperarithmetical fixed points. (iv) So we conclude that 𝐿 𝜔 𝐶𝐾 1 is not a model of KP + (Π 1 -Red), implying that KP does not prove (Π 1 -Red).…”

Section: Comments and Questionsmentioning

confidence: 96%

Some Set-Theoretic Reduction Principles

Bärtschi,

Jäger

2024

Outstanding Contributions to Logic

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In this article we study several reduction principles in the context of Simpson's set theory ATR 𝑆 0 and Kripke-Platek set theory KP (with infinity). Since ATR 𝑆 0 is the set-theoretic version of ATR 0 there is a direct link to second order arithmetic and the results for reductions over ATR 𝑆 0 are as expected and more or less straightforward. However, over KP we obtain several interesting new results and are lead to some open questions.

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“…Example 9 We consider two subsystems of second order arithmetic and one theory of (iterated) admissible sets.…”

Section: Theorem 8 Suppose That T Is a Theory Which Is Formulated In mentioning

confidence: 99%

“…Recent work on metapredicative theories for iterated admissible sets and explicit mathematics with comparatively weak induction principles -see Probst [8] and some papers in preparation [5,9] for all relevant detailsmakes it desirable to apply similar strategies, but the use of pseudo-hierarchies in subsystems of set theory and explicit mathematics seems to be a different matter.…”

Section: Introductionmentioning

confidence: 99%