Skip to main content
SpringerLink
Log in

Remark on Kreisel's conjecture

  • Published:
Journal of Soviet Mathematics Aims and scope Submit manuscript

Abstract

At the end of the 60's, Kreisel conjectured that formal arithmetics admit infinite induction rules if the lengths of proofs of their premises are uniformly bounded by the same number. By the length of a proof we mean the number of applications of axioms and rules of inference. In this article we construct a theory R * with a finite number of specific axioms. The language of R * contains a constant 0, a unary function symbol, equality, and ternary predicates for addition and multiplication. It is proved that for any consistent axiomatizable extension

R* it is possible to find a formula A(a) that satisfies the following conditions: a) ∀xA(x) cannot be derived in

; b) for any n the length of the proof of the formula A(0(n)) is no greater than c1[log2 (n+1)]+c2, where the constants c1 and c2 are independent of n. Here the expression 0(n) indicates 0 with n primes.

This is a preview of subscription content, log in via an institution to check access.

Access this article

Log in via an institution

Price excludes VAT (USA)
Tax calculation will be finalised during checkout.

Instant access to the full article PDF.

Institutional subscriptions

Similar content being viewed by others

Literature cited

  1. S. K. Kleene, Introduction to Metamathematics, Van Nostrand, New York (1952).

    Google Scholar 

  2. S. K. Kleene, Mathematical Logic, Wiley, New York (1967).

    Google Scholar 

  3. V. P. Orevkov, “Theorems with very short proofs can be strengthened,” Semiotika i Informatika, No. 12, 27–38 (1979).

    Google Scholar 

  4. V. P. Orevkov, “Elimination of cross sections for estimating proof lengths,” Dokl. Akad. Nauk SSSR,296, No. 3, 539–542 (1987).

    Google Scholar 

  5. M. Davis, Yu. Matijasevič, and J. Robinson, “Hilbert's tenth problem. Diophantine equations: positive aspects of a negative solution,” in: Proc. Symp. Pure Math.,28, 323–378 (1976).

    Google Scholar 

  6. H. Friedman, “One hundred and two problems in mathematical logic,” J. Symb. Logic,40, No. 2, 113–129 (1975).

    Google Scholar 

  7. J. Krajíček, “Generalization of proofs,” in: Proc. Fifth Easter Conf. on Model Th., East Germany (1987).

  8. R. J. Parikh, “Some results on the length of proofs,” Trans. Am. Math. Soc.,177, 29–36 (1973).

    Google Scholar 

  9. D. Richardson, “Sets of theorems with short proofs,” J. Symb. Logic,39, No. 2, 235–242 (1974).

    Google Scholar 

  10. T. Yukami, “A note on formalized arithmetic with function symbols' and +,” Tsukuba J. Math.,2, 69–73 (1978).

    Google Scholar 

  11. T. Yukami, “Some results on speed-up,” Ann. Jpn. Assoc. Philos. Sci.,6, No. 4, 195–205 (1984).

    Google Scholar 

Download references

Authors
  1. V. P. Orevkov
    View author publications

    You can also search for this author in PubMed Google Scholar

Additional information

Translated from Zapiski Nauchnykh Seminarov Leningradskogo Otdeleniya Matematicheskogo Institua im. V. A. Steklova Akademii Nauk SSSR, Vol. 176, pp. 118–126, 1989.

Rights and permissions

Reprints and permissions

About this article

Cite this article

Orevkov, V.P. Remark on Kreisel's conjecture. J Math Sci 59, 850–855 (1992). https://doi.org/10.1007/BF01104108

Download citation

  • Issue Date:

  • DOI: https://doi.org/10.1007/BF01104108

Keywords

Search

Navigation