论文标题
无限逻辑和抽象小学课
Infinitary Logics and Abstract Elementary Classes
论文作者
论文摘要
我们证明，每个抽象的小学类（A.E.C.）具有LST数字$κ$和词汇$τ$的基质$ \ leqκ$可以在逻辑$ {\ mathbb l} _ {\ beth_2（κ2（κ）^{+++}，++}，κ^+} $的逻辑$ {\ mathbb l} _ {\ mathbb l} _ {\ mathbb l} _ {\ mathbb l} _ {\ mathbb l} _ {在此逻辑中因此，是EC类，而不仅仅是PC类。这构成了介绍定理先前给出的确定性水平的重大改进。作为证明的一部分，我们定义了A.E.C. $ \ MATHCAL K $。事实证明，这是班级的一个有趣的组合对象，超出了我们定理的目的。此外，我们研究了定义A.E.C.的句子之间的联系和相对较新的无限逻辑$ l^1_λ$。}
We prove that every abstract elementary class (a.e.c.) with LST number $κ$ and vocabulary $τ$ of cardinality $\leq κ$ can be axiomatized in the logic ${\mathbb L}_{\beth_2(κ)^{+++},κ^+}(τ)$. In this logic an a.e.c. is therefore an EC class rather than merely a PC class. This constitutes a major improvement on the level of definability previously given by the Presentation Theorem. As part of our proof, we define the \emph{canonical tree} $\mathcal S={\mathcal S}_{\mathcal K}$ of an a.e.c. $\mathcal K$. This turns out to be an interesting combinatorial object of the class, beyond the aim of our theorem. Furthermore, we study a connection between the sentences defining an a.e.c. and the relatively new infinitary logic $L^1_λ$.}
