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A uniform approach for characterizing the provably total number-theoretic functions of KPM and its subsystems

Benjamin Blankertz & Andreas Weiermann

Studia Logica 62 (3):399-427 (1999)

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Proof theory of reflection.Michael Rathjen - 1994 - Annals of Pure and Applied Logic 68 (2):181-224.details The paper contains proof-theoretic investigation on extensions of Kripke-Platek set theory, KP, which accommodate first-order reflection. Ordinal analyses for such theories are obtained by devising cut elimination procedures for infinitary calculi of ramified set theory with Пn reflection rules. This leads to consistency proofs for the theories KP+Пn reflection using a small amount of arithmetic and the well-foundedness of a certain ordinal system with respect to primitive decending sequences. Regarding future work, we intend to avail ourselves of these new cut (...) Download Export citation Bookmark 37 citations
Proof-theoretic analysis of KPM.Michael Rathjen - 1991 - Archive for Mathematical Logic 30 (5-6):377-403.details KPM is a subsystem of set theory designed to formalize a recursively Mahlo universe of sets. In this paper we show that a certain ordinal notation system is sufficient to measure the proof-theoretic strength ofKPM. This involves a detour through an infinitary calculus RS(M), for which we prove several cutelimination theorems. Full cut-elimination is available for derivations of $\Sigma (L_{\omega _1^c } )$ sentences, whereω 1 c denotes the least nonrecursive ordinal. This paper is self-contained, at least from a technical (...) point of view. (shrink) Download Export citation Bookmark 42 citations
Ordinal notations based on a weakly Mahlo cardinal.Michael Rathjen - 1990 - Archive for Mathematical Logic 29 (4):249-263.details Download Export citation Bookmark 18 citations
Collapsing functions based on recursively large ordinals: A well-ordering proof for KPM. [REVIEW]Michael Rathjen - 1994 - Archive for Mathematical Logic 33 (1):35-55.details It is shown how the strong ordinal notation systems that figure in proof theory and have been previously defined by employing large cardinals, can be developed directly on the basis of their recursively large counterparts. Thereby we provide a completely new approach to well-ordering proofs as will be exemplified by determining the proof-theoretic ordinal of the systemKPM of [R91]. Download Export citation Bookmark 22 citations
Proof theory and ordinal analysis.W. Pohlers - 1991 - Archive for Mathematical Logic 30 (5-6):311-376.details In the first part we show why ordinals and ordinal notations are naturally connected with proof theoretical research. We introduce the program of ordinal analysis. The second part gives examples of applications of ordinal analysis. Download Export citation Bookmark 20 citations
A Uniform Approach to Fundamental Sequences and Hierarchies.Wilfried Buchholz, Adam Cichon & Andreas Weiermann - 1994 - Mathematical Logic Quarterly 40 (2):273-286.details In this article we give a unifying approach to the theory of fundamental sequences and their related Hardy hierarchies of number-theoretic functions and we show the equivalence of the new approach with the classical one. Download Export citation Bookmark 23 citations

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