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On the induction schema for decidable predicates

  • Lev D. Beklemishev (a1)
Abstract

We study the fragment of Peano arithmetic formalizing the induction principle for the class of decidable predicates, IΔ1. We show that IΔ1 is independent from the set of all true arithmetical Π2-sentences. Moreover, we establish the connections between this theory and some classes of oracle computable functions with restrictions on the allowed number of queries. We also obtain some conservation and independence results for parameter free and inference rule forms of Δ1-induction.

An open problem formulated by J. Paris (see [4, 5]) is whether IΔ1 proves the corresponding least element principle for decidable predicates, LΔ1 (or, equivalently, the Σ1-collection principle BΣ1). We reduce this question to a purely computation-theoretic one.

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[1]Beklemishev, L. D., Induction rules, reflection principles, and provably recursive functions, Annals of Pure and Applied Logic, vol. 85 (1997), pp. 193242.
[2]Beklemishev, L. D., A proof-theoretic analysis of collection, Archive for Mathematical Logic, vol. 37 (1998), pp. 275296.
[3]Beklemishev, L. D., Open least element principle and bounded query computation, Computer science logic, 13th international workshop, CSL '99, Madrid, Spain, September 20–25, 1999, Proceedings (Flum, J. and Rodrigues-Artalejo, M., editors), Lecture Notes in Computer Science, vol. 1683, Springer-Verlag, Berlin, 1999, pp. 389404.
[4]Clote, P. and Krajíček, J., Open problems, Arithmetic, proof theory, and computational complexity (Clote, P. and Krajíček, J., editors), Oxford University Press, Oxford, 1993, pp. 119.
[5]Hájek, P. and Pudlák, P., Metamathematics of first order arithmetic, Springer-Verlag, Berlin, Heidelberg, New York, 1993.
[6]Kaye, R., Paris, J., and Dimitracopoulos, C., On parameter free induction schemas, this Journal, vol. 53 (1988), no. 4, pp. 10821097.
[7]Paris, J., A hierarchy of cuts in models of arithmetic, Model theory of algebra and arithmetic, Proceedings, Karapascz, Poland, 1979, Lecture Notes in Mathematics, vol. 834, Springer-Verlag, 1980, pp. 312337.
[8]Parsons, C., On a number-theoretic choice schema and its relation to induction, Intuitionism and proof theory (Kino, A., Myhill, J., and Vessley, R. E., editors), North-Holland, Amsterdam, 1970, pp. 459473.
[9]Parsons, C., On n-quantifier induction, this Journal, vol. 37 (1972), no. 3, pp. 466482.
[10]Rastsvetaev, A. and Beklemishev, L., On the query complexity of finding a local maximum point, Logic Group Preprint Series 206, University of Utrecht, 2000.
[11]Schwichtenberg, H., Some applications of cut-elimination, Handbook of mathematical logic (Barwise, J., editor), North-Holland, Amsterdam, 1977, pp. 867896.
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The Journal of Symbolic Logic
  • ISSN: 0022-4812
  • EISSN: 1943-5886
  • URL: /core/journals/journal-of-symbolic-logic
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