A simplified version of local predicativity

Buchholz, Wilfried

doi:10.1017/cbo9780511896262.006

Cited by 59 publications

(89 citation statements)

References 12 publications

(8 reference statements)

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“…The set of ordinals C Ω (ε Ω+1 , 0) gives rise to an elementary computable ordinal representation system (cf. [14,8,23,26]). In what follows, C Ω (ε Ω+1 , 0) will sometimes be denoted by T (Ω).…”

Section: Logical Axiomsmentioning

confidence: 99%

“…For the presentation of infinitary proofs we draw on [8]. Buchholz developed a very elegant and flexible setting for describing uniformity in infinitary proofs, called operator controlled derivations.…”

Section: H-controlled Derivationsmentioning

confidence: 99%

“…Standing in sharp contrast to the ordinal analysis of KP (cf. [14,8]), however, the terms s i may and often will contain subterms that the operator H does not control, that is, subterms t with | t | ∈ H(∅).…”

Section: H-controlled Derivationsmentioning

confidence: 99%

“…Also note that we cannot eliminate cuts with ∆ P 0 -formulae since we don't have predicative cut elimination [8,Theorem 3.16] as in the case KP. Corollary 7.7.…”

Section: ω+1mentioning

confidence: 99%

“…Note that in contrast to the infinitary system used for the ordinal of KP (see [14,8]) the terms of RS P Ω may contain free variables. This will be crucial in proving the Soundness Theorem 8.1.…”

mentioning

confidence: 99%

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Relativized ordinal analysis: The case of Power Kripke–Platek set theory

Rathjen

2014

Annals of Pure and Applied Logic

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The paper relativizes the method of ordinal analysis developed for KripkePlatek set theory to theories which have the power set axiom. We show that it is possible to use this technique to extract information about Power Kripke-Platek set theory, KP(P).As an application it is shown that whenever KP(P) + AC proves a Π P 2 statement then it holds true in the segment V τ of the von Neumann hierarchy, where τ stands for the Bachmann-Howard ordinal.

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Section: Logical Axiomsmentioning

confidence: 99%

Section: H-controlled Derivationsmentioning

confidence: 99%

Section: H-controlled Derivationsmentioning

confidence: 99%

“…Also note that we cannot eliminate cuts with ∆ P 0 -formulae since we don't have predicative cut elimination [8,Theorem 3.16] as in the case KP. Corollary 7.7.…”

Section: ω+1mentioning

confidence: 99%

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confidence: 99%

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Relativized ordinal analysis: The case of Power Kripke–Platek set theory

Rathjen

2014

Annals of Pure and Applied Logic

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Finitary Treatment of Operator Controlled Derivations

Buchholz

2001

MLQ - Math. Log. Quart.

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By combining the methods of two former papers of ours ([6] and [7]) we develop a finitary ordinal analysis of the axiom system KPi of Kripke-Platek set theory with an inaccessible universe. As a main result we obtain an upper bound (in the fast-growing hierarchy) for the provably recursive functions of KPi. Mathematics Subject Classification: 03F05, 03F15.In this paper we continue our work on finitary representations of infinitary derivations which we have started in [6] and resumed in [8]. On the basis of [7] we develop a finitary ordinal analysis of the axiom system KPi of Kripke-Platek set theory with an inaccessible universe. As a main result we obtain the following T h e o r e m . If KPi ∀z ("z = HF" → ∀x∈z ∃y∈z φ(x, y)) with φ(x, y) ∈ ∆ 0 , then there exists a <·-primitive recursive function F : ω −→ ω (in the sense of [13, p. 117]) with <· < ψ Ω (ε I+1 ) such that ∀n∈ω ∀x∈L n ∃y∈L F (n) φ(x, y) holds. Here "z = HF" is a set-theoretic formula saying "z is the set of all hereditarily finite sets".One goal of this line of work (from which we still are very far away) is to develop some kind of "semantics" for Arai's purely combinatorial proof-theoretic analysis of very strong impredicative theories (cf. [1, 2, 3]) and to find connections to Rathjen's work [10,11,12].In Section 1 we introduce the language of L RS of ramified set theory and the infinitary proof system RS ∞ , essentially as in [7]. One technical modification is the following: For RS-terms a with lev(a) < β we set aThis has the advantage that nowAt the end of Section 1 we introduce the main concept of this paper namely the notion of a notation system for RS ∞ -derivations and we define what it means that such a notation system is controlled by an operator H.In Section 2 we introduce a notation system RS 0 which besides others contains notations for RS ∞ -derivations of all KPi-axioms (relativized to L I ). In Section 3 we extend RS 0 to a notation system RS + which in addition contains notations for 1)

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On Relating Theories: Proof-Theoretical Reduction

Rathjen

Toppel

2019

Mathesis Universalis, Computability and Proof

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The notion of proof-theoretical or finitistic reduction of one theory to another has a long tradition. Feferman and Sieg [13, Chap. 1] and Feferman in [22] made first steps to delineate it in more formal terms. The first goal of this paper is to corroborate their view that this notion has the greatest explanatory reach and is superior to others, especially in the context of foundational theories, i.e., theories devised for the purpose of formalizing and presenting various chunks of mathematics.A second goal is to address a certain puzzlement that was expressed in Feferman's title of his Clermont-Ferrand lectures at the Logic Colloquium 1994: "How is it that finitary proof theory became infinitary?" Hilbert's aim was to use proof theory as a tool in his finitary consistency program to eliminate the actual infinite in mathematics from proofs of real statements. Beginning in the 1950s, however, proof theory began to employ infinitary methods. Infinitary rules and concepts, such as ordinals, entered the stage.In general, the more that such infinitary methods were employed, the farther did proof theory depart from its initial aims and methods, and the closer did it come instead to ongoing developments in recursion theory, particularly as generalized to admissible sets; in both one makes use of analogues of regular cardinals, as well as "large" cardinals (inaccessible, Mahlo, etc.). ([19]).The current paper aims to explain how these infinitary tools, despite appearances to the contrary, can be formalized in an intuitionistic theory that is finitistically reducible to (actually Π 0 2 -conservative over) intuitionistic first order arithmetic, also known as Heyting arithmetic. Thus we have a beautiful example of Hilbert's program at work, exemplifying the Hilbertian goal of moving from the ideal to the real by eliminating ideal elements.

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A simplified version of local predicativity

Cited by 59 publications

References 12 publications

Relativized ordinal analysis: The case of Power Kripke–Platek set theory

Relativized ordinal analysis: The case of Power Kripke–Platek set theory

Finitary Treatment of Operator Controlled Derivations

On Relating Theories: Proof-Theoretical Reduction

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