Constructing Quasiminimal Structures

TL;DR

This paper investigates methods for constructing quasiminimal models in first-order theories, providing new constructions and bounds that enhance understanding of their existence and properties in non-elementary categoricity.

Contribution

It introduces several new constructions for quasiminimal models with varying levels of control, and establishes an upper bound on the Hanf number for their existence.

Findings

Multiple constructions of quasiminimal models with different properties

An upper bound on the Hanf number for large quasiminimal models

Enhanced understanding of quasiminimal structures in non-elementary categoricity

Abstract

Quasiminimal structures play an important role in non-elementary categoricity. In this paper we explore possibilities of constructing quasiminimal models of a given first-order theory. We present several constructions with increasing control of the properties of the outcome using increasingly stronger assumptions on the theory. We also establish an upper bound on the Hanf number of the existence of arbitrarily large quasiminimal models.