超乘积构造方法

    进入本世纪，2010年6月55日，数学家J.Keisler发表论文，对模型论基础构造方法超乘积（Ultraproducts）的历史发展进行综述，并且列出了54篇相关论文，很有参考价值。

简而言之，所谓“超乘积”∏ Ai（i是指标集合U的元素，U一般是可数无限结合）。

利用不同的滤子，结合给定的超乘积，可以构造出各种数学模型，比如：超实数系统。

有兴趣的读者可参阅本文附件。

袁萌  陈启清  元月11日

附件：以下是J.Keisler于2010年6月5日，发表的“超乘积构造方法”的原文：

THE ULTRAPRODUCT CONSTRUCTION

H. JEROME KEISLER

Abstract. This is a brief survey of the ultraproduct construction, which is meant to provide background material for the readers of this volume.

1. Introduction

The ultraproduct construction is a uniform method of building models of rst order theories which has applications in many areas of mathematics. It is attractive because it is algebraic in nature, but preserves all properties expressible in rst order logic. The idea goes back to the construction of nonstandard models of arithmetic by Skolem [51] in 1934. In 1948, Hewitt [16] studied ultraproducts of elds. For rst order structures in general, the ultraproduct construction was dened by L o´s [37] in 1955. The subject developed rapidly beginning in 1958 with a series of abstracts by Frayne, Morel, Scott, and Tarski, which led to the 1962 paper [14]. Other early papers are [31] by Kochen, and [18] by the author. The groundwork for the application of ultraproducts to mathematics was laid in the late 1950’s through the 1960’s. The purpose of this article is to give a survey of the classical results on ultraproducts of rst order structures in order to provide some background for the papers in this volume. Over the years, many generalizations of the ultraproduct construction, as well as applications of ultraproducts to non-rst order structures, have appeared in the literature. To keep this paper of reasonable length, we will not include such generalizations in this survey. For earlier surveys of ultraproducts see [7], [12], [24]. For much more about ultraproducts see the books [9], [10], [49], and [54]. We assume familiarity with a few basic concepts from model theory. For the convenience of the reader we give a crash course here. The cardinality of a set X is denoted by |X|. The cardinality of N is denoted by ω. The set of all subsets of a set I is denoted by P(I), and the set of nite subsets of I by Pω(I). Given mappings f : X → Y and g : Y → Z, the composition g f : X → Z is the mapping x 7→ g(f(x)). A rst order vocabulary L consists of a set of nitary relation symbols, function symbols, and constant symbols. We use A,B,... to denote L-structures with universe sets A,B,.... By the cardinality of A we mean the cardinality of its universe set A. The notationA|= φ(a1,...,an) means that the formula φ(x1,...,xn) is true in A when each xi is interpreted by the corresponding ai. The notation h : A→B means that h is a homomorphism ofAintoB, that is, h maps A into B and each atomic formula which is true for a tuple inAis true for the h-image of the tuple in B. The notation h : A⊆B means that h is an (isomorphic) embedding ofAintoB, that is, h maps A into B and each quantier-free formula of

Date: June 1, 2010. 2000 Mathematics Subject Classication. Primary 03C20, Secondary 03H05, 54D80 . 1（全文参阅“无穷小微积分”网站）