论文标题
一阶逻辑语义及其作为模型理论方法的应用的拓扑表示
A Topological Representation of Semantics of First-order Logic and Its Application as a Method in Model Theory
论文作者
论文摘要
数学逻辑研究通常涉及各种拓扑概念，几乎所有这些概念都可以视为从石材代表定理中发展。在石材代表定理中，布尔代数表示为石材空间的clopen套件的代数。基于这一点，在石材空间的结构和命题逻辑的语义之间建立了自然的联系。换句话说，命题理论的模型被表示为石头空间中的点。这使我们能够使用拓扑概念来描述逻辑中的许多事实。在本文中，我们为一阶逻辑做同样的事情。也就是说，我们将一阶逻辑语义的基本对象（例如理论，模型，基本嵌入等）组织为抽象定义的一种拓扑结构。确切地说，这种结构是一种丰富的流程空间，我们在本文中称为圆柱空间。此外，基于这一一阶逻辑语义的拓扑表示，我们系统地将点集拓扑的方法引入了模型理论的研究。我们通过一个示例证明了这种拓扑方法的巨大优势，并就其特征，优势和与类型空间的联系进行了一般讨论。
Various topological concepts are often involved in the research of mathematical logic, and almost all of these concepts can be regarded as developing from the Stone representation theorem. In the Stone representation theorem, a Boolean algebra is represented as the algebra of the clopen sets of a Stone space. And based on this, a natural connection is established between the structure of Stone space and the semantics of propositional logic. In other words, models of a propositional theory are represented as points in a Stone space. This enables us to use the concepts of topology to describe many facts in logic. In this paper, we do the same thing for the first-order logic. That is, we organize the basic objects of semantics of first-order logic, such as theories, models, elementary embeddings, and so on, into a kind of topological structure defined abstractly. To be precise, this kind of structure is a kind of enriched-topological space which we call cylindric space in this paper. Furthermore, based on this topological representation of semantics of first-order logic, we systematically introduce a method of point-set topology into the research of model theory. We demonstrate the great advantages of this topological method with an example and provide a general discussion of its features, advantages, and connection to the type space.
