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

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.