论文标题
连续逻辑中的最小设置和分类性
Strongly Minimal Sets and Categoricity in Continuous Logic
论文作者
论文摘要
众所周知，经典的鲍德温 - 拉克兰经典表征在连续逻辑上失败了，因为并非每个密不可分的分类理论都具有很小的集合。在这里，我们通过发展连续逻辑中强烈最小的集合以及检查Banach空间的不可分割扩展的理论来研究这些问题。 为此，我们介绍并表征了“词典理论”，其中可定义的集合足够普遍，以至于可以执行许多熟悉离散逻辑的结构。在Banach理论的背景下，我们还介绍了“难以分辨的子空间”的概念，我们用来改善Shelah和Usvyatsov的结果。这两个概念都适用于在不可分割的分类理论背景之外的连续逻辑。 最后，我们构建或提出了许多反例，包括$ω$稳定的理论，没有vaughtian Pairs，而没有密不可分的分类和一个密不可分的分类理论，其家庭中仅在其家庭中占有很小的设置。
The classical Baldwin-Lachlan characterization of uncountably categorical theories is known to fail in continuous logic in that not every inseparably categorical theory has a strongly minimal set. Here we investigate these issues by developing the theory of strongly minimal sets in continuous logic and by examining inseparably categorical expansions of Banach space. To this end we introduce and characterize 'dictionaric theories,' theories in which definable sets are prevalent enough that many constructions familiar in discrete logic can be carried out. We also introduce, in the context of Banach theories, the notion of an 'indiscernible subspace,' which we use to improve a result of Shelah and Usvyatsov. Both of these notions are applicable to continuous logic outside of the context of inseparably categorical theories. Finally, we construct or present a slew of counterexamples, including an $ω$-stable theory with no Vaughtian pairs which fails to be inseparably categorical and an inseparably categorical theory with strongly minimal sets in its home sort only over models of sufficiently high dimension.
