Lefschetz's principle
Lefschetz's principle
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莱夫谢茨原理
DOI：
10.1016/0021-8693(69)90117-3
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发表时间：
1969
期刊：
影响因子：
--
通讯作者：
P. Eklof
中科院分区：
文献类型：
--
作者：
J. Barwise;P. Eklof
本文描述了一种语言，用它可以方便地表达代数几何的陈述，并证明莱夫谢茨原理。我们在第1节中详细描述了该语言并证明了mctamathematical结果。第1节的结果适用于代数几何在第2节。在第3节中，WC做了一些简短的历史评论。我们要感谢亚伯拉罕·罗宾逊教授在这项工作期间进行了许多令人感兴趣的讨论。什么WC所谓的Lcfschetz的原则'已被韦伊如下所述（[II]，第306页）：“f或ag ivcn值的characteristicp，每一个结果，只涉及有限数目的点和品种，这已被证明为一些选择的普遍域仍然有效，没有限制;只有一个代数几何的特征p的每一个值的p，而不是一个代数几何的每一个选择的普遍域。”韦伊说，这一原则的正式证明需要“一个正式的'元数学'表征的类型的命题”，它适用于;“这将不得不依赖于'元数学'，即逻辑分析我们所有的定义。在本文中，我们尝试执行这个程序。因此，在对比以前metamathcmatical制定的莱夫谢茨原则所产生的一般逻辑考虑（见第3节），我们的出发点一直是分析的定义代数几何。其结果是一种语言，从逻辑的角度来看是有点复杂（这是一个高阶，无限语言），但这是密切类似于自然语言的代数几何。因此，例如，在语言的模型中，代表理想的变量在多项式集合上的范围，代表仿射n空间范围内的变量在n元组集合上的范围。
In this paper we describe a language in which the statements of algebraic geometry can readily be cxpresscd and for which Lefschetz’s principle can be proved. We describe the language in detail and prove the mctamathematical results in Section 1. The results of Section 1 are applied to algebraic geometry in Section 2. In Section 3 WC make some brief historical remarks. We wish to express our appreciation to Professor Abraham Robinson for many interesting discussions during the course of this work. What WC call Lcfschetz’s principle’has been stated by Weil as follows ([II], p. 306):“f or ag ivcn value of the characteristicp, every result, involving only a finite number of points and of varieties, which has been proved for some choice of the universal domain remains valid without restriction; there is but one algebraic geometry of characteristic p for each value of p, not one algebraic geometry for each choice of the universal domain.” Weil says that a formal proof of this principle would require “a formal ‘metamathematical’characterization of the type of proposition” to which it applies;“this would have to depend upon the ‘metamathematical’ie logical analysis of all our definitions.” In this paper we attempt to carry out this program. Thus in contrast to previous metamathcmatical formulations of the Lefschetz principle which arose from general logical considerations (see Section 3) our starting point has been an analysis of the definitions of algebraic geometry. The result is a language which from the logical point of view is somewhat complex (it is a higher-order, infinitary language), but which is closely akin to the natural language of algebraic geometry. Thus, for example, in models of the language, variables standing for ideals range over sets of polynomials, and variables standing for varieties in affine n-space range over sets of n-tuples.