A Walk on the Wild Side: Notions of maximality in first-order theories

TL;DR

This paper introduces a framework using patterns of consistency and inconsistency to analyze dividing lines in first-order theories, defining new notions of maximality and hierarchy, and providing examples to illustrate their properties.

Contribution

It develops a general framework for studying dividing lines in first-order theories using pattern-based notions of maximality and hierarchy, including new properties and examples.

Findings

$ ext{PM}^{(k+1)}$ theories are not $k$-dependent

Provided an example of a $ ext{PM}$ but $ ext{NSOP}_4$ theory

Constructed a hierarchy of theories with non-collapsing properties

Abstract

In the classification of complete first-order theories, many dividing lines have been defined in order to understand the complexity and the behavior of some classes of theories. In this paper, using the concept of patterns of consistency and inconsistency, we describe a general framework to study dividing lines and we introduce a notion of maximal complexity by requesting the presence of all the exhibitable patterns of definable sets. Weakening this notion, we define new properties (Positive Maximality and the $\mathrm{PM}^{(k)}$ hierarchy) and prove some results about them. In particular, we show that $\mathrm{PM}^{(k+1)}$ theories are not $k$-dependent. Moreover, we provide an example of a $\mathrm{PM}$ but $\mathrm{NSOP}_4$ theory (showing that $\mathrm{SOP}$ and the $\mathrm{SOP}_n$ hierarchy, for $n \geq 4$, can not be described by \emph{positive} patterns) and, for each $1<k<\omega$, an example of a $\mathrm{PM}^{(k)}$ but $\mathrm{NPM}^{(k+1)}$ theory (showing that the newly defined hierarchy does not collapse).