August 12
Alex van Abel,
CUNY
Asymptotic Classes, Pseudofinite Cardinality and Dimension
We explore the consequences of various model-theoretic tameness conditions upon the behavior of pseudofinite cardinality and dimension. We show that for pseudofinite theories which are either Morley Rank 1 or uncountably categorical, pseudofinite cardinality in ultraproducts satisfying such theories is highly well-behaved. In the other direction, we construct simple and supersimple theories in which pseudofinite dimension is necessarily ill-behaved in all such ultraproducts. Additionally, we have novel results connecting various forms of asymptotic classes to each other and to their counting pairs.