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A note on theΠ 02 -induction rule

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Abstract

It is well-known (due to C. Parsons) that the extension of primitive recursive arithmeticPRA by first-order predicate logic and the rule ofΠ 02 -inductionΠ 02 -IR isΠ 02 -conservative overPRA. We show that this is no longer true in the presence of function quantifiers and quantifier-free choice for numbersAC 0,0-qf. More precisely we show that ℐ :=PRA 2 +Π 02 -IR+AC 0,0-qf proves the totality of the Ackermann function, wherePRA 2 is the extension ofPRA by number and function quantifiers andΠ 02 -IR may contain function parameters.

This is true even forPRA 2 +∑ 01 -IR+Π 02 -IR −+AC 0,0-qf, whereΠ 02 -IR − is the restriction ofΠ 02 -IR without function parameters.

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Authors and Affiliations

  1. Fachbereich Mathematik, J.W. Goethe-Universität, Robert-Mayer-Strasse 6-10, D-60054, Frankfurt, Germany

    Ulrich Kohlenbach

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  1. Ulrich Kohlenbach
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I am grateful to an anonymous referee whose suggestions led to an improved discussion of our results

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Kohlenbach, U. A note on theΠ 02 -induction rule. Arch Math Logic 34, 279–283 (1995). https://doi.org/10.1007/BF01469385

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