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Fragments of Heyting arithmetic

Published online by Cambridge University Press:  12 March 2014

Wolfgang Burr
Wolfgang Burr*
Affiliation:
Institut für Mathematische Logik und Grundlagenforschung, der Westfälischen Wilhelms-Universität Münster, Einsteinstr. 62, D-48149 Münster, Germany E-mail: Wolfgang.Burr@math.uni-muenster.de
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Abstract

We define classes Φn of formulae of first-order arithmetic with the following properties:

(i) Every φ ϵ Φn is classically equivalent to a Πn-formula (n ≠ 1, Φ1 := Σ1).

(ii)

(iii) IΠn and iΦn (i.e., Heyting arithmetic with induction schema restricted to Φn-formulae) prove the same Π2-formulae.

We further generalize a result by Visser and Wehmeier. namely that prenex induction within intuitionistic arithmetic is rather weak: After closing Φn both under existential and universal quantification (we call these classes Θn) the corresponding theories iΘn still prove the same Π2-formulae. In a second part we consider iΔ0 plus collection-principles. We show that both the provably recursive functions and the provably total functions of are polynomially bounded. Furthermore we show that the contrapositive of the collection-schema gives rise to instances of the law of excluded middle and hence .

Type
Research Article
Information
The Journal of Symbolic Logic , Volume 65 , Issue 3 , September 2000 , pp. 1223 - 1240
Copyright
Copyright © Association for Symbolic Logic 2000

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