On a question of Gaifman concerning invariant measures
In his 1964 paper “Concerning measures in first order calculi” Gaifman introduces the notion of a symmetric measure-model: a measure on the formulas of a first order calculus that is invariant under permutations of the elements instantiating the free variables of each formula, where these elements come from some fixed domain. To each symmetric measure-model there is associated a measure on sentences, which we can think of as a random (consistent) theory that the measure-model satisfies. Gaifman shows that every such random theory has a symmetric measure-model satisfying it. However, the symmetric measure-models that he constructs sometimes, necessarily, assign positive measure to instantiations of the formula “x=y” by unequal elements. Gaifman goes on to pose the question of characterizing those classical theories that admit symmetric measure-models without this pathology — those with so-called `strict equality’. In this talk I will show that when the instantiating domain is the set of natural numbers, a symmetric measure-model with strict equality is essentially a probability measure on a space of structures, with underlying set the natural numbers, that is invariant under the logic action. I will then give necessary and sufficient conditions for a classical theory to admit such an invariant measure, thereby providing an answer to the question posed by Gaifman. This is joint work with Nathanael Ackerman and Cameron Freer.