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A Note on How to Extend Gentzen’s Second Consistency Proof to a Proof of Normalization for First Order Arithmetic

In: Gentzen's Centenary

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  • Dag Prawitz

    (Stockholms universitet, Filosfiska institutionen)

Abstract

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.

Suggested Citation

  • Dag Prawitz, 2015. "A Note on How to Extend Gentzen’s Second Consistency Proof to a Proof of Normalization for First Order Arithmetic," Springer Books, in: Reinhard Kahle & Michael Rathjen (ed.), Gentzen's Centenary, edition 1, pages 131-176, Springer.
    • Handle: RePEc:spr:sprchp:978-3-319-10103-3_6
    DOI: 10.1007/978-3-319-10103-3_6
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