Pseudofinite Structures, Multidimensional Asymptotic Classes, and Generalised Measurable Structures

Abstract: 



I will discuss recent joint work with Anscombe, Steinhorn and Wolf, which takes its origin in the Chatzidakis-van den Dries-Macintyre results on approximate cardinalities of definable sets in finite fields. We develop a rather general framework for classes of finite structures with a certain uniformity on the (approximate or exact) cardinalities of definable sets in uniformly definable families. We also develop an analogous notion of `generalised measurability’  for infinite (not necessarily pseudofinite) structures, whereby definable sets are assigned values in an ordered semiring, with model-theoretic consequences.