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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: woods@maths.uwa.edu.au
Ch. Cornaros
Affiliation:
Department of Mathematics, University of Aegean, GR-832 00 Karlovassi, Greece, E-mail: kornaros@aegean.gr
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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

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References

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