Model theory
Model theory
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模型理论
DOI：
10.1017/cbo9780511551574
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发表时间：
1993
期刊：
影响因子：
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通讯作者：
Wilfrid Hodges
中科院分区：
文献类型：
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作者：
Wilfrid Hodges
1954年，阿尔弗雷德·塔尔斯基(Alfred Tarski)[210]宣布，在模型理论的名义下出现了“元数学的一个新分支”。这个主题发展迅速：1987年欧米茄集团的模型理论书目长达617页[148]。到了20世纪80年代中期，已经有太多的模型理论方言，任何人都不可能精通其中的一小部分。例如，很少有模型理论家能够声称理解Zilber和Hrushovski在代数几何的边缘所做的工作，以及Immerman和Vardi对有限结构分类的研究。而且，这两个研究领域都没有太多接触英语哲学家和欧洲计算语言学家所称的“模型论”方法或概念。然而，所有这些模型理论的品牌都有共同的起源和重要的家族相似之处。其他一些被称为模特的东西肯定存在于家庭之外。例如，这一章没有提到“建模”，这意味着构建一个正式的理论来描述或解释一些现象。同样，在认知科学中，詹特纳和斯蒂芬斯[67]或约翰逊-莱尔德[100]的‘心理模型’超出了我们的主题。模型理论的所有特征都建立在一个基本概念上，那就是公式φ在解释I下为真的概念。经典的解释是塔斯基1933年的论文[202]。在本文中，塔尔斯基假设我们有一种具有精确定义的语法的语言L。忽略标点符号，L的符号有两种：常量和变量。常量有固定的含义；它们通常包括逻辑表达式，如‘and’和‘equals’。变量没有意义，但(稍微打断一下塔尔斯基非常仔细的表述)我们可以为每个变量分配一个对象，并询问当每个变量被视为其分配的对象的名称时，L的给定公式φ是否成立。变量的语法类别决定了可以分配给它们的宾语类型；例如，我们可以将个体分配给各个变量、类
In 1954 Alfred Tarski [210] announced that ‘a new branch of metamathematics’ had appeared under the name of the theory of models. The subject grew fast: the Omega Group bibliography of model theory in 1987 [148] ran to 617 pages. By the mid 1980s there were already too many dialects of model theory for anybody to be expert in more than a fraction. For example very few model theorists could claim to understand both the work of Zilber and Hrushovski at the edge of algebraic geometry, and the studies by Immerman and Vardi of classifications of finite structures. And neither of these lines of research had much contact with what English-speaking philosophers and European computational linguists had come to refer to as ‘model-theoretic’ methods or concepts. Nevertheless all these brands of model theory had common origins and important family resemblances. Some other things called models definitely lie outside the family. For example this chapter has nothing to say about ‘modelling’, which means constructing a formal theory to describe or explain some phenomena. Likewise in cognitive science the ‘mental models’ of Gentner and Stephens [67] or Johnson-Laird [100] lie outside our topic. All the flavours of model theory rest on one fundamental notion, and that is the notion of a formula φ being true under an interpretation I. The classic treatment is Tarski’s paper [202] from 1933. In this paper Tarski supposes that we have a language L with a precisely defined syntax. Ignoring punctuation, the symbols of L are of two kinds: constants and variables. The constants have fixed meanings; they will usually include logical expressions such as ‘and’ and ‘equals’. The variables have no meaning, but (to shortcircuit Tarski’s very careful formulation a little) we can assign an object to each variable, and ask whether a given formula φ of L becomes true when each variable is regarded as a name of its assigned object. The grammatical categories of the variables determine what kinds of object can be assigned to them; for example we can assign individuals to individual variables, classes
Lowenheim-Skolem-Tarski 的性质
DOI：
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发表时间：
2022
期刊：
数理解析研究所考究録
影响因子：
--
作者：
Egashira Kento;Yata Kazuyoshi;Aoshima Makoto;Toshimichi Usuba;只木孝太郎;Kenta Ozeki;野崎寛;薄葉季路
通讯作者：
薄葉季路