Hostname: page-component-586b7cd67f-t7czq Total loading time: 0 Render date: 2024-11-25T18:16:59.476Z Has data issue: false hasContentIssue false
  • English
  • Français

On bounded arithmetic augmented by the ability to count certain sets of primes

Published online by Cambridge University Press:  12 March 2014

Alan R. Woods  and
Ch. Cornaros
Alan R. Woods
Affiliation:
School of Mathematics and Statistics, University of Western Australia, Crawley, W.A. 6009., Australia, E-mail: [email protected]
Ch. Cornaros
Affiliation:
Department of Mathematics, University of Aegean, GR-832 00 Karlovassi, Greece, E-mail: [email protected]
Get access Rights & Permissions [Opens in a new window]

Abstract

Over 25 years ago, the first author conjectured in [15] that the existence of arbitrarily large primes is provable from the axioms IΔ0(π) + def(π), where π(x) is the number of primes not exceeding x, IΔ0(π) denotes the theory of Δ0 induction for the language of arithmetic including the new function symbol π, and def(π) is an axiom expressing the usual recursive definition of π. We prove a modified version in which π is replaced by a more general function ξ that counts some of the primes below x (which primes depends on the values of parameters in ξ), and has the property that π is provably Δ0(ξ) definable.

Type
Research Article
Information
The Journal of Symbolic Logic , Volume 74 , Issue 2 , June 2009 , pp. 455 - 473
Copyright
Copyright © Association for Symbolic Logic 2009

Access options

Get access to the full version of this content by using one of the access options below. (Log in options will check for institutional or personal access. Content may require purchase if you do not have access.)

References

REFERENCES

[1]Berarducci, A. and Intrigila, B., Combinatorial principles in elementary number theory. Annals of Pure and Applied Logic, vol. 55 (1991), pp. 35–50.CrossRefGoogle Scholar
[2]Cornaros, Ch., On Grzegorzyk induction, Annals of Pure and Applied Logic, vol. 74 (1995), no. 1. pp. 1–21.CrossRefGoogle Scholar
[3]D'Aquino, P., Local behaviour of the Chebyshev theorem in models of IΔ0, this Journal, vol. 57 (1992), no. 1, pp. 12–27.Google Scholar
[4]Grzeoorczyk, A., Some classes of recursive functions, Rozprawy Matematycine, vol. IV (1953). pp. 1–45.Google Scholar
[5]Nguyen, P., Bounded reverse mathematics, Ph.D. thesis, Graduate Department of Computer Science, University of Toronto, 2008.Google Scholar
[6]Nguyen, P., Proving infinitude of prime numbers using binomial coefficients, preprint, University of Toronto, 2008, pp. 1–20.Google Scholar
[7]Paris, J. B. and Wilkie, A. J., Counting problems in bounded arithmetic. Methods in Mathematical Logic: Proceedings of the 6th Latin American Symposium on Mathematical Logic 1983, Lecture Notes in Mathematics, vol. 1130, 1985, pp. 317–340.CrossRefGoogle Scholar
[8]Paris, J. B., Wilkie, A. J., and Woods, A. R., Provability of the pigeonhole principle and the existence of infinitely many primes, this Journal, vol. 53 (1988), pp. 1235–1244.Google Scholar
[9]Ramanujan, S., A proof of Bertrand's Postulate, Journal of the Indian Mathematical Society, vol. 11 (1919), pp. 181–182.Google Scholar
[10]Shapiro, H. N., Introduction to number theory, Wiley-Interscience, New York, 1983.Google Scholar
[11]Sylvester, J. J., On Tchebycheff's theory of the totality of the prime numbers comprised within given limits, American Journal of Mathematics, vol. 4 (1881), pp. 230–247, reprinted in The Collected Mathematical Papers of James Joseph Sylvester, vol. 3, 530-545, Chelsea, New York, 1973.CrossRefGoogle Scholar
[12]Sylvester, J. J., On arithmetical series. Messenger of Mathematics, vol. 21 (1892), pp. 1-19 and 87–120, reprinted in The Collected Mathematical Papers of James Joseph Sylvester, vol. 4, 687–731, Chelsea, New York, 1973.Google Scholar
[13]Tchebychef, P. L., Mémoire sur les nombres premiers, Journal de mathématiques pares et appliquées, I série, vol. 17 (1852), pp. 366–390, reprinted in Oeuvres de P. L. Tchebychef, Tome 1, 51–70, Chelsea, New York. A translation into English of part of this work is included in D. E. Smith, A Source Book in Mathematics, 127–148, Dover, New York, 1959.Google Scholar
[14]Wilkie, A. J., Some results and problems on weak systems of arithmetic, Logic Colloquium '77 (Macintyre, A.. Pacholski, L., and Paris, J., editors), North-Holland, 1978, pp. 237–248.Google Scholar
[15]Woods, A. R., Some problems in logic and number theory, and their connections, Ph.D. thesis, University of Manchester, 1981.Google Scholar