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| Main Author: | |
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| Format: | Preprint |
| Published: |
2024
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| Subjects: | |
| Online Access: | https://arxiv.org/abs/2409.19236 |
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Table of Contents:
- 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 $1<k<ω$, an example of a $\mathrm{PM}^{(k)}$ but $\mathrm{NPM}^{(k+1)}$ theory (showing that the newly defined hierarchy does not collapse).