TY - JOUR

T1 - On bounded arithmetic augmented by the ability to count certain sets of primes

AU - Woods, Alan

AU - Cornaros, C.

PY - 2009

Y1 - 2009

N2 - 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.

AB - 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.

U2 - 10.2178/jsl/1243948322

DO - 10.2178/jsl/1243948322

M3 - Article

SN - 0022-4812

VL - 74

SP - 455

EP - 473

JO - The Journal of Symbolic Logic

JF - The Journal of Symbolic Logic

ER -