论文标题
短AEC的分类转移与集合合并
Categoricity transfer for short AECs with amalgamation over sets
论文作者
论文摘要
令$ {\ bf k} $为$ \ mathrm {ls}（{\ bf k}）$ - 简短的抽象基本类，并假设比存在怪物模型的存在更多（集合和任意大型模型的合并）。假设$ {\ bf k} $在某些$μ> \ mathrm {ls}（{\ bf k}）$中是分类的，那么它在所有$μ'\geqμ$中都是分类的。我们的结果消除了Grossberg-Vandieren制造的$μ$的后继需求，以使用短暂而不是驯服。并使用合并对集合而不是模型。它还消除了Vasey的素数要求，该需求假设模型的驯服和合并。作为推论，我们获得了Morley和Shelah的一阶理论的向上分类转移的替代证明。在我们的构建中，我们简化了Vasey的结果来构建一个弱成功的框架。这使我们可以使用Shelah-Vasey的论点来获得足够饱和模型的素数。如果我们用$ \ mathrm {ls}（{\ bf k}）$替换分类假设，那么$ {\ bf k} $对于足够饱和的模型已经很棒。这阐明了对ZFC中不可数的一阶理论的主要差距定理的研究。
Let ${\bf K}$ be an $\mathrm{LS}({\bf K})$-short abstract elementary class and assume more than the existence of a monster model (amalgamation over sets and arbitrarily large models). Suppose ${\bf K}$ is categorical in some $μ>\mathrm{LS}({\bf K})$, then it is categorical in all $μ'\geqμ$. Our result removes the successor requirement of $μ$ made by Grossberg-VanDieren, at the cost of using shortness instead of tameness; and of using amalgamation over sets instead of over models. It also removes the primes requirement by Vasey which assumes tameness and amalgamation over models. As a corollary, we obtain an alternative proof of the upward categoricity transfer for first-order theories by Morley and Shelah. In our construction, we simplify Vasey's results to build a weakly successful frame. This allows us to use Shelah-Vasey's argument to obtain primes for sufficiently saturated models. If we replace the categoricity assumption by $\mathrm{LS}({\bf K})$-superstability, ${\bf K}$ is already excellent for sufficiently saturated models. This sheds light on the investigation of the main gap theorem for uncountable first-order theories within ZFC.
