Zero-One Laws, Random Graphs, and Fräıssé Limits

While at first glance the fields of logic and probability may appear as immiscible as oil and water, a closer look reveals that probabilistic questions about logic often have surprisingly elegant and powerful answers. For instance, one might wonder “What is the probability that a certain sentence in a certain logic holds for a randomly chosen structure?” Though in many cases this question will turn out to be hopelessly complex, it turns out that some logics in some situations are not powerful enough to define properties with any probabilities other than 0 or 1. Results of this form, known as zeroone laws, are of central importance in the study of this fascinating collision between logic and randomness. In this paper, we will begin by proving the zero-one law for first-order logic over graphs, using an ingenious construction known as the random graph. We will then use a related but much more general construction known as a Fräıssé limit in order to prove the zero-one law for first-order logic over the structures of any purely relational vocabulary.