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- W2232652977 endingPage "1642" @default.
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- W2232652977 abstract "We prove: $mathbf{Theorem}$ Let $K$ be a universal class. If $K$ is categorical in cardinals of arbitrarily high cofinality, then $K$ is categorical on a tail of cardinals. The proof stems from ideas of Adi Jarden and Will Boney, and also relies on a deep result of Shelah. As opposed to previous works, the argument is in ZFC and does not use the assumption of categoricity in a successor cardinal. The argument generalizes to abstract elementary classes (AECs) that satisfy a locality property and where certain prime models exist. Moreover assuming amalgamation we can give an explicit bound on the Hanf number and get rid of the cofinality restrictions: $mathbf{Theorem}$ Let $K$ be an AEC with amalgamation. Assume that $K$ is fully $operatorname{LS} (K)$-tame and short and has primes over sets of the form $M cup {a}$. Write $H_2 := beth_{left(2^{beth_{left(2^{operatorname{LS} (K)}right)^+}}right)^+}$. If $K$ is categorical in a $lambda > H_2$, then $K$ is categorical in all $lambda' ge H_2$." @default.
- W2232652977 created "2016-06-24" @default.
- W2232652977 creator A5045065882 @default.
- W2232652977 date "2017-09-01" @default.
- W2232652977 modified "2023-09-25" @default.
- W2232652977 title "Shelah's eventual categoricity conjecture in universal classes: Part I" @default.
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- W2232652977 doi "https://doi.org/10.1016/j.apal.2017.03.003" @default.
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