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Remarks on Herbrand normal forms and Herbrand realizations

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Summary

LetA H be the Herbrand normal form ofA andA H,D a Herbrand realization ofA H. We show

  1. (i)

    There is an example of an (open) theory ℐ+ with function parameters such that for someA not containing function parameters

  2. (ii)

    Similar for first order theories ℐ+ if the index functions used in definingA H are permitted to occur in instances of non-logical axiom schemata of ℐ, i.e. for suitable ℐ,A

  3. (iii)

    In fact, in (1) we can take for ℐ+ the fragment (Σ 01 -IA)+ of second order arithmetic with induction restricted toΣ 01 -formulas, and in (2) we can take for ℐ the fragment (Σ 0,b1 -IA) of first order arithmetic with induction restricted to formulas VxA(x) whereA contains only bounded quantifiers.

  4. (iv)

    On the other hand,

    $$PA^2 \vdash A^H \Rightarrow PA \vdash A,$$

    wherePA 2 is the extension of first order arithmeticPA obtained by adding quantifiers for functions andA∈ℒ(PA). This generalizes to extensional arithmetic in the language of all finite types but not to sentencesA with positively occurring existential quantifiers for functions.

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Authors and Affiliations

  1. Fachbereich Mathematik, J.W. Goethe-Universität, Robert-Mayer-Strasse 6-10, W-6000, Frankfurt am Main, Federal Republic of Germany

    Ulrich Kohlenbach

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  1. Ulrich Kohlenbach
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I am grateful to the referee and to Prof. H. Luckhardt for many helpful suggestions which led to an improved presentation of our results

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Kohlenbach, U. Remarks on Herbrand normal forms and Herbrand realizations. Arch Math Logic 31, 305–317 (1992). https://doi.org/10.1007/BF01627504

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