On random models of finite power and monadic logic

Abstract

For any property θ of a model (or graph), let μ n( θ) be the fraction of models of power n which satisfy θ, and let μ( θ) = lim n →∞ μ n( θ) if this limit exists. For first-order properties θ, it is known that μ(θ) must be 0 or 1. We answer a question of K. Compton by proving in a strong way that this 0–1 law can fail if we allow monadic quantification (that is, quantification over sets) in defining the sentence θ. In fact, by producing a monadic sentence which codes arithmetic on n with probability μ = 1, we show that every recursive real is μ(θ) for some monadic θ.