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On n-quantifier induction

Published online by Cambridge University Press:  12 March 2014

Charles Parsons
Charles Parsons*
Affiliation:
Columbia University, New York, New York 10027
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Extract

In this paper we discuss subsystems of number theory based on restrictions on induction in terms of quantifiers, and we show that all the natural formulations of ‘n-quantifier induction’ are reducible to one of two (for n ≠ 0) nonequivalent normal forms: the axiom of induction restricted to (or, equivalently, ) formulae and the rule of induction restricted to formulae.

Let Z0 be classical elementary number theory with a symbol and defining equations for each Kalmar elementary function, and the rule of induction

restricted to quantifier-free formulae. Given the schema

let IAn be the restriction of IA to formulae of Z0 with ≤n nested quantifiers, IAn′ to formulae with ≤n nested quantifiers, disregarding bounded quantifiers, the restriction to formulae, the restriction to , formulae. IRn, IRn′, , are analogous.

Then, we show that, for every n, , , IAn, and IAn′, are all equivalent modulo Z0. The corresponding statement does not hold for IR. We show that, if n ≠ 0, is reducible to ; evidently IRn is reducible to . On the other hand, IRn′ is obviously equivalent to IAn′ [10, Lemma 2].

Type
Research Article
Information
The Journal of Symbolic Logic , Volume 37 , Issue 3 , September 1972 , pp. 466 - 482
Copyright
Copyright © Association for Symbolic Logic 1972

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References

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