Date of Degree

9-2022

Document Type

Dissertation

Degree Name

Ph.D.

Program

Mathematics

Advisor

Alf Dolich

Committee Members

Hans Schoutens

Philipp Rothmaler

Subject Categories

Discrete Mathematics and Combinatorics | Logic and Foundations

Keywords

Model Theory, Pseudofinite, Categoricity, Finite Structures

Abstract

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.

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