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

  • Alan R. Woods (a1) and Ch. Cornaros (a2)

Abstract

Over 25 years ago, the first author conjectured in [15] that the existence of arbitrarily large primes is provable from the axioms 0(π) + def(π), where π(x) is the number of primes not exceeding x, 0(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.

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[1]Berarducci, A. and Intrigila, B., Combinatorial principles in elementary number theory. Annals of Pure and Applied Logic, vol. 55 (1991), pp. 3550.
[2]Cornaros, Ch., On Grzegorzyk induction, Annals of Pure and Applied Logic, vol. 74 (1995), no. 1. pp. 121.
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[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. 237248.
[15]Woods, A. R., Some problems in logic and number theory, and their connections, Ph.D. thesis, University of Manchester, 1981.
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The Journal of Symbolic Logic
  • ISSN: 0022-4812
  • EISSN: 1943-5886
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