Asymptotic Classes, Pseudofinite Cardinality and Dimension

Author

Alexander Van Abel, The Graduate Center, City University of New YorkFollow

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.

Recommended Citation

Van Abel, Alexander, "Asymptotic Classes, Pseudofinite Cardinality and Dimension" (2022). CUNY Academic Works.
https://academicworks.cuny.edu/gc_etds/5107