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Normalization proof for Peano Arithmetic

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Abstract

A proof of normalization for a classical system of Peano Arithmetic formulated in natural deduction is given. The classical rule of the system is the rule for indirect proof restricted to atomic formulas. This rule does not, due to the restriction, interfere with the standard detour conversions. The convertible detours, numerical inductions and instances of indirect proof concluding falsity are reduced in a way that decreases a vector assigned to the derivation. By interpreting the expressions of the vectors as ordinals each derivation is assigned an ordinal less than ɛ 0. The vector assignment, which proves termination of the procedure, originates in a normalization proof for Gödel’s T by Howard (Intuitionism and proof theory. North-Holland, Amsterdam, 1970).

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References

  1. Howard, W.A.: Assignment of ordinals to terms for primitive recursive functionals of finite type. In: Kino, J.M.A., Vesley, R. (eds.) Intuitionism and Proof Theory. North-Holland, Amsterdam (1970)

  2. Kanckos A.: Consistency of Heyting arithmetic in natural deduction. Math. Log. Q. 56(6), 611–624 (2010)

    Article  MathSciNet  MATH  Google Scholar 

  3. von Plato J.: Normal form and existence property for derivations in Heyting arithmetic. Acta Philos. Fennica 78, 159–163 (2006)

    Google Scholar 

  4. Prawitz D.: Natural deduction: a proof-theoretical study, 2nd edn. Dover Publications, UK (2006)

    Google Scholar 

  5. Prawitz, D.: A note on Gentzen’s 2nd consistency proof and normalization of natural deductions in 1st order arithmetic. In: Kahle, R., Rathjen, M. (eds.) Gentzen’s Centenary: The Quest for Consistency. Springer (2015)

  6. Siders A.: Gentzen’s consistency proof without heightlines. Arch. Math. Logic. 52(3–4), 449–468 (2013)

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  1. Department of Philosophy, University of Helsinki, Unioninkatu 40 A, P.O. Box 24, 00014, Helsinki, Finland

    Annika Siders

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  1. Annika Siders
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Correspondence to Annika Siders.

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Siders, A. Normalization proof for Peano Arithmetic. Arch. Math. Logic 54, 921–940 (2015). https://doi.org/10.1007/s00153-015-0450-y

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  • DOI: https://doi.org/10.1007/s00153-015-0450-y

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