Abstract
We use exact saturation to study the complexity of unstable theories, showing that a variant of this notion called pseudo-elementary class (PC)-exact saturation meaningfully reflects combinatorial dividing lines. We study PC-exact saturation for stable and simple theories. Among other results, we show that PC-exact saturation characterizes the stability cardinals of size at least continuum of a countable stable theory and, additionally, that simple unstable theories have PC-exact saturation at singular cardinals satisfying mild set-theoretic hypotheses. This had previously been open even for the random graph. We characterize supersimplicity of countable theories in terms of having PC-exact saturation at singular cardinals of countable cofinality. We also consider the local analog of PC-exact saturation, showing that local PC-exact saturation for singular cardinals of countable cofinality characterizes supershort theories.
Original language | English |
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Article number | 2250020 |
Pages (from-to) | 2250020:1-2250020:26 |
Number of pages | 26 |
Journal | Journal of Mathematical Logic |
Volume | 23 |
Issue number | 2 |
DOIs | |
State | Published - 1 Aug 2023 |
Bibliographical note
Publisher Copyright:© 2023 World Scientific Publishing Company.
Keywords
- Simple theories
- classification theory
- exact saturation
- stable theories