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Proof theory for theories of ordinals—I: recursively Mahlo ordinals

Annals of Pure and Applied Logic 122 (1-3):1-85 (2003)

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Ordinal diagrams for recursively Mahlo universes.Toshiyasu Arai - 2000 - Archive for Mathematical Logic 39 (5):353-391.details In this paper we introduce a recursive notation system $O(\mu)$ of ordinals. An element of the notation system is called an ordinal diagram following G. Takeuti [25]. The system is designed for proof theoretic study of theories of recursively Mahlo universes. We show that for each $\alpha<\Omega$ in $O(\mu)$ KPM proves that the initial segment of $O(\mu)$ determined by $\alpha$ is a well ordering. Proof theoretic study for such theories will be reported in [9]. Areas of Mathematics in Philosophy of Mathematics Direct download (3 more) Export citation Bookmark 8 citations
A formalization of the theory of ordinal numbers.Gaisi Takeuti - 1965 - Journal of Symbolic Logic 30 (3):295-317.details Logic and Philosophy of Logic Logic and Philosophy of Logic, Miscellaneous in Logic and Philosophy of Logic Proof Theory in Logic and Philosophy of Logic Direct download (7 more) Export citation Bookmark 5 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) Proof Theory in Logic and Philosophy of Logic Direct download (4 more) Export citation Bookmark 42 citations
Subsystems of set theory and second order number theory.Wolfram Pohlers - 1998 - In Samuel R. Buss (ed.), Bulletin of Symbolic Logic. Elsevier. pp. 137--209.details Number Theory in Philosophy of Mathematics Direct download (2 more) Export citation Bookmark 22 citations
Zur Beweistheorie Der Kripke-Platek-Mengenlehre Über Den Natürlichen Zahlen.Gerhard Jäger - 1980 - Archive for Mathematical Logic 22 (3-4):121-139.details No categories Direct download Export citation Bookmark 24 citations

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