On a question of Gaifman concerning invariant measures

Model theory seminarFriday, April 15, 201612:30 pmGC 6417

Rehana Patel

On a question of Gaifman concerning invariant measures

Olin College

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.

Posted by on March 16th, 2016