Tame pseudofinite theories with wild pseudofinite dimensions

TL;DR

This paper constructs two pseudofinite theories that are tame in neostability but exhibit pathological pseudofinite dimensions, challenging existing assumptions about the relationship between pseudofinite dimension and tameness.

Contribution

It provides counterexamples showing that pseudofinite dimension does not necessarily imply tameness in pseudofinite theories, and introduces a new method for proving quantifier elimination using pseudofinite cardinality.

Findings

Pseudofinite theories can be tame yet have wild pseudofinite dimensions.

Pseudofinite cardinality remains well-behaved with respect to definability.

A novel approach to proving quantifier elimination via pseudofinite cardinality.

Abstract

We construct two pseudofinite theories which are tame from a neostability perspective, yet have pathological fine pseudofinite dimension in all models. These theories serve as counterexamples to potential converses of results by Garcia, Macpherson and Steinhorn relating pseudofinite dimension to tameness. We demonstrate that pseudofinite cardinality in these theories is well behaved with regards to definability, and provide a novel method of proving quantifier elimination using pseudofinite cardinality.