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A Note on How to Extend Gentzen’s Second Consistency Proof to a Proof of Normalization for First Order Arithmetic
Stockholm University, Faculty of Humanities, Department of Philosophy.
2015 (English)In: Gentzen's Centenary: The Quest for Consistency / [ed] Reinhard Kahle, Michael Rathjen, Springer, 2015, 131-176 p.Chapter in book (Refereed)
Abstract [en]

The purpose of this note is to show that the normalization theorem can be proved for first order Peano arithmetic by adapting to natural deduction the method used in Gentzen’s second consistency proof. Gentzen explained the intuitive idea behind his proof by informally arguing for the possibility of a normalization theorem of natural deduction, but what he actually proved was a special case of the Hauptsatz for a sequent calculus formalization of arithmetic. To transfer Gentzen’s method to natural deduction, I shall assign his ordinals to notations for natural deductions that use an explicit operation of substitution. The idea is first worked out for predicate logic. The main problems reside there and consist in finding a normalization strategy that harmonizes with the ordinal assignment. The result for predicate logic is then extended to arithmetic without effort, and thereby full normalization of natural deductions in first order arithmetic is achieved.

Place, publisher, year, edition, pages
Springer, 2015. 131-176 p.
National Category
Philosophy
Research subject
Theoretical Philosophy
Identifiers
URN: urn:nbn:se:su:diva-125728DOI: 10.1007/978-3-319-10103-3_6ISBN: 978-3-319-10102-6 (print)ISBN: 978-3-319-10103-3 (electronic)OAI: oai:DiVA.org:su-125728DiVA: diva2:894915
Available from: 2016-01-17 Created: 2016-01-17 Last updated: 2017-03-23Bibliographically approved

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