Date of Degree
Discrete Mathematics and Combinatorics | Logic and Foundations
Model Theory, Pseudofinite, Categoricity, Finite Structures
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. On the other hand, it has been shown that pseudofinite dimension is not necessarily well-behaved in all ultraproducts of theories which are simple or supersimple; we extend such an observation by constructing 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.
Van Abel, Alexander, "Asymptotic Classes, Pseudofinite Cardinality and Dimension" (2022). CUNY Academic Works.