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June 2009 On bounded arithmetic augmented by the ability to count certain sets of primes
Ch. Cornaros, Alan R. Woods
J. Symbolic Logic 74(2): 455-473 (June 2009). DOI: 10.2178/jsl/1243948322

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.

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Ch. Cornaros. Alan R. Woods. "On bounded arithmetic augmented by the ability to count certain sets of primes." J. Symbolic Logic 74 (2) 455 - 473, June 2009. https://doi.org/10.2178/jsl/1243948322

Information

Published: June 2009
First available in Project Euclid: 2 June 2009

zbMATH: 1174.03026
MathSciNet: MR2518806
Digital Object Identifier: 10.2178/jsl/1243948322

Rights: Copyright © 2009 Association for Symbolic Logic

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Vol.74 • No. 2 • June 2009
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