初等类的一个注记  

A Note on Elementary Class

作　　者：裘江杰[1] Jiangjie Qiu(School of Philosophy,Renmin University of China)

机构地区：[1]中国人民大学哲学院

出　　处：《逻辑学研究》2022年第4期47-56,共10页Studies in Logic

摘　　要：初等类是模型论的一个核心概念。对初等类有多个刻画定理,其中一个使用“对初等等价封闭”与“对超积封闭”。《初等模型论》是新近出版的一部优秀的模型论作品,在其中根据上述刻画定理,给出了可数无穷步得到一个结构类的初等类闭包的“操作性”方法。这一方法颇为直观,可以加深我们对初等类这一概念的理解,然而由于疏忽了“共尾”现象,这一方法或有漏洞。本文引入“取κ超积闭”这一概念,首先“优化”了前述刻画定理,然后依据这一“优化”后的结果修补了上述“操作性”方法。本文也据此初步讨论了结构类上几个性质之间的关系。Elementary class is a core concept of model theory.There are several characterization theorems for elementary class,one of which uses“closed under elementary equivalence”and“closed under ultraproducts”.Elementary model theory is an excellent newly published textbook,in which,based on the above characterization theorem,provide a method,which could be used to form the elementary closure of a given class of structures in countably infinite steps.This method is intuitive,can deepen our understanding of the concept of elementary class.Due to the neglect of the“cofinality”problem,however,this method may have loopholes.In this paper,we introduced the concept of“closed underκ­ultraproducts”firstly,and improved the above characterization theorem,and perfected the above method finally.This paper also discusses the relationship among some properties of classes of structures.

关 键 词：初等类 初等等价 超积 

分 类 号：B81[哲学宗教—逻辑学]