On the spectrum of limit models
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arXiv
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| Natura: | Preprint |
| Pubblicazione: |
2025
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| author | Beard, Jeremy Mazari-Armida, Marcos |
| author_facet | Beard, Jeremy Mazari-Armida, Marcos |
| contents | We study the spectrum of limit models assuming the existence of a nicely behaved independence notion. Under reasonable assumptions, we show that all `long' limit models are isomorphic, and all `short' limit models are non-isomorphic.
$\textbf{Theorem.}$ Let $\mathbf{K}$ be a $\aleph_0$-tame abstract elementary class stable in $λ\geq \operatorname{LS}(\mathbf{K})$ with amalgamation, joint embedding and no maximal models. Suppose there is an independence relation on the models of size $λ$ that satisfies uniqueness, extension, non-forking amalgamation, universal continuity, and $(\geq κ)$-local character in a minimal regular $κ< λ^+$.
Suppose $δ_1, δ_2 < λ^+$ with $\operatorname{cf}(δ_1) < \operatorname{cf}(δ_2)$. Then for any $N_1, N_2, M \in \mathbf{K}_λ$ where $N_l$ is a $(λ, δ_l)$-limit model over $M$ for $l = 1, 2$,
\[N_1 \text{ is isomorphic to } N_2 \text{ over } M \iff \operatorname{cf}(δ_1) \geq κ\]
Both implications in the conclusion have improvements. High cofinality limits are isomorphic without the $\aleph_0$-tameness assumption and assuming the independence relation is defined only on high cofinality limit models. Low cofinality limits are non-isomorphic without assuming non-forking amalgamation.
We show how our results can be used to study limit models in both abstract settings and in natural examples of abstract elementary classes. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2503_11605 |
| institution | arXiv |
| publishDate | 2025 |
| record_format | arxiv |
| spellingShingle | On the spectrum of limit models Beard, Jeremy Mazari-Armida, Marcos Logic Primary: 03C48. Secondary: 03C45 We study the spectrum of limit models assuming the existence of a nicely behaved independence notion. Under reasonable assumptions, we show that all `long' limit models are isomorphic, and all `short' limit models are non-isomorphic. $\textbf{Theorem.}$ Let $\mathbf{K}$ be a $\aleph_0$-tame abstract elementary class stable in $λ\geq \operatorname{LS}(\mathbf{K})$ with amalgamation, joint embedding and no maximal models. Suppose there is an independence relation on the models of size $λ$ that satisfies uniqueness, extension, non-forking amalgamation, universal continuity, and $(\geq κ)$-local character in a minimal regular $κ< λ^+$. Suppose $δ_1, δ_2 < λ^+$ with $\operatorname{cf}(δ_1) < \operatorname{cf}(δ_2)$. Then for any $N_1, N_2, M \in \mathbf{K}_λ$ where $N_l$ is a $(λ, δ_l)$-limit model over $M$ for $l = 1, 2$, \[N_1 \text{ is isomorphic to } N_2 \text{ over } M \iff \operatorname{cf}(δ_1) \geq κ\] Both implications in the conclusion have improvements. High cofinality limits are isomorphic without the $\aleph_0$-tameness assumption and assuming the independence relation is defined only on high cofinality limit models. Low cofinality limits are non-isomorphic without assuming non-forking amalgamation. We show how our results can be used to study limit models in both abstract settings and in natural examples of abstract elementary classes. |
| title | On the spectrum of limit models |
| topic | Logic Primary: 03C48. Secondary: 03C45 |
| url | https://arxiv.org/abs/2503.11605 |