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

Over 25 years ago, the first author conjectured in [15] that the existence of arbitrarily large primes is provable from the axioms IΔ0( ) + def( ), where (x) is the number of primes not exceeding x, IΔ0( ) denotes the theory of Δ0 induction for the language of arithmetic including the new function symbol , and def( ) is an axiom expressing the usual recursive definition of . We prove a modified version in which is replaced by a more general function that counts some of the primes below x (which primes depends on the values of parameters in ), and has the property that is provably Δ0( ) definable. §