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On n-quantifier induction

  • Charles Parsons (a1)

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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].

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[1]Diller, Justus and Schütte, Kurt, Simultane Rekursionen in der Theorie der Funktionale endlicher Typen, Archiv für mathematische Logik und Grundlagenforschung, vol. 14 (1971), pp. 6974.
[2]Dreben, Burton and Denton, John, Herbrand-style consistency proofs, in Intuitionism and proof theory, North-Holland, Amsterdam, 1970, pp. 419434.
[3]Howard, W. A., Functional interpretation of bar induction by bar recursion, Compositio Mathematica, vol. 20 (1968), pp. 107124.
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[6]Kreisel, G., Interpretation of analysis by constructive functionals of finite types, Constructivity in mathematics (Heyting, A., ed.), North-Holland, Amsterdam, 1959, pp. 101129.
[7]Kreisel, G., Inessential extensions of Heyting's arithmetic by means of functionals of finite type, this Journal, vol. 24 (1959), p. 284 (Abstract).
[8]Kreisel, G., A survey of proof theory, this Journal, vol. 33 (1968), pp. 321384.
[9]Kreisel, G., Lacombe, D. and Shoenfield, J. R., Effective operations and partial recursive functionals, Constructivity in mathematics (Heyting, A., ed.), North-Holland, Amsterdam, 1959, pp. 290297.
[10]Parsons, Charles, On a number-theoretic choice schema and its relation to induction, Intuitionism and proof theory, North-Holland, Amsterdam, 1970, pp. 459473.
[11]Spector, Clifford, Provably recursive functionals of analysis, Proceedings of Symposia in Pure Mathematics, Vol. 5, 1962, pp. 127.
[12]Tait, W. W., Nested recursion, Mathematische Annalen, vol. 143 (1961), pp. 236250.
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[14]Tait, W. W., Constructive reasoning, Logic, methodology, and philosophy of science. III (Rootselaar, B. van and Staal, J. F., Editors), North-Holland, Amsterdam, 1968, pp. 185198.
[15]Parsons, Charles, Proof-theoretic analysis of restricted induction schemata, this Journal vol. 36 (1971), p. 361 (Abstract).
[16]Parsons, Charles, On a number-theoretic choice schema. II, this Journal, vol. 36 (1971), p. 587 (Abstract).

On n-quantifier induction

  • Charles Parsons (a1)

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