Let φ be a monadic second order sentence about a finite structure from a class K which is closed under disjoint unions and has components. Compton has conjectured that if the number of n element structures has appropriate asymptotics, then unlabelled (labelled) asymptotic probabilities ν(φ) (μ(φ) respectively) for φ always exist. By applying generating series methods to count finite models, and a tailor made Tauberian lemma, this conjecture is proved under a mild additional condition on the asymptotics of the number (...) of single component K-structures. Prominent among examples covered, are structures consisting of a single unary function (or partial function) and a fixed number of unary predicates. (shrink)

Over 25 years ago, the first author conjectured in [15] that the existence of arbitrarily large primes is provable from the axioms I Δ₀(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 de f(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. (shrink)