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arXiv · 1510.03780

Downward categoricity from a successor inside a good frame

Sebastien Vasey
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Abstract

We use orthogonality calculus to prove a downward transfer from categoricity in a successor in abstract elementary classes (AECs) that have a good frame (a forking-like notion for types of singletons) on an interval of cardinals: $\mathbf{Theorem}$ Let $K$ be an AEC and let $\text{LS} (K) \le λ< θ$ be cardinals. If $K$ has a type-full good $[λ, θ]$-frame and $K$ is categorical in both $λ$ and $θ^+$, then $K$ is categorical in all $λ' \in [λ, θ]$. We deduce improvements on the threshold of several categoricity transfers that do not mention frames. For example, the threshold in Shelah's transfer can be improved from $\beth_{\beth_{\left(2^{\text{LS} (K)}\right)^+}}$ to $\beth_{\left(2^{\text{LS} (K)}\right)^+}$ assuming that the AEC is $\text{LS} (K)$-tame. The successor hypothesis can also be removed from Shelah's result by assuming in addition either that the AEC has primes over sets of the form $M \cup \{a\}$ or (using an unpublished claim of Shelah) that the weak generalized continuum hypothesis holds.

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BibTeXRIS

Sebastien Vasey. 2016-10-03. Downward categoricity from a successor inside a good frame. https://doi.org/10.1016/j.apal.2016.10.003

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