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Archive for Mathematical Logic

, Volume 58, Issue 3–4, pp 347–352 | Cite as

A note on Gentzen’s ordinal assignment

  • Annika KanckosEmail author
Article
  • 79 Downloads

Abstract

Gentzen’s height measure of the 1938 consistency proof is a cumulative complexity measure for sequents that is measured bottom-up in a derivation. By a factorisation of the ordinal assignment a top-down ordinal assignment can be given that does not depend on information occurring below the sequent to which the ordinal is assigned. Furthermore, an ordinal collapsing function is defined in order to collapse the top-down ordinal to the one assigned by Gentzen’s own ordinal assignment. A direct definition of the factorised assignment follows as a corollary. This extraction of an ordinal collapsing function hopes to provide a formal or conceptual clarification of Gentzen’s ordinal assignment and its height-line argument.

Keywords

Relative consistency proof (03F25) Normalization (03F05) Ordinal notations (03F15) 

Mathematics Subject Classification

03F25 (Relative consistency and interpretations) 03F30 (First-order arithmetic and fragments) 03F05 (Cut-elimination and normal-form theorems) 03F15 (Recursive ordinals and ordinal notations) 

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References

  1. 1.
    Gentzen, G.: Neue Fassung des Widerspruchsfreiheitsbeweises für die reine Zahlentheorie. Forschungen zur Logik und zur Grundlegung der exakten Wissenschaften 4, 19–44 (1938)zbMATHGoogle Scholar
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    Mints, G.: Selected Papers in Proof Theory. Bibliopolis, Napoli (1992)zbMATHGoogle Scholar
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    Szabo, M.E. (ed.): The Collected Papers of Gerhard Gentzen. North-Holland, Amsterdam (1969)zbMATHGoogle Scholar
  4. 4.
    Takeuti, G.: Proof Theory: Second Edition. Dover Books on Mathematics. Dover Publications, New York (2013). First ed. 1987Google Scholar

Copyright information

© Springer-Verlag GmbH Germany, part of Springer Nature 2018

Authors and Affiliations

  1. 1.Department of PhilosophyUniversity of HelsinkiHelsinkiFinland

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