Henson构建超实数直接法(全文),何处寻?
(2018-09-28 12:50:04)Henson构建超实数直接法(全文),何处寻?
一般而言,高校教授超实数微积分,中国与美国情况都一一样,师生惧怕形式逻辑与公理集合论。怎么办?
美国知名数学家C.W.Henson教授发明了于一种构建超实数的直接法,可以避开上述实际问题。
袁萌
附录:
FOUNDATIONS OF NONSTANDARD ANALYSIS
A Gentle Introduction to Nonstandard Extensions
BY C. WARD HENSON
1. Introduction There are many introductions to nonstandard analysis, (some of which are listed in the References) so why write another one? All of the existing introductions have one or more of the following features:
(A) heavy use of logical formalism right from the start;
(B) early introduction of set theoretic apparatus inexcess of what is needed formost applications;
(C)
dependence on an explicit construction of the nonstandard model,
usually by means of the ultrapower construction. All three of these
features have negative consequences. The early use of logical
formalism or set theoretic structures is often uncomfortable for
mathematicians who do not have a background in logic, and this can
effectively deter them from using nonstandard methods. The explicit
use of a particular nonstandard model makes the foundations too
specic and inexible, and often inhibits the free use of the ideas
of nonstandard analysis. In this exposition we intend to avoid
these disadvantages. The readers for who
mathematics encounter, and we carefully show how it can be used without diculty to obtain useful facts about nonstandard extensions. In Section 4 we extend the concept of nonstandard extension to mathematical settings in which there may be several basic sets (such as the vector space setting, where there is a eld F and a vector space V ). In Sections 5 and 6 we show how these ideas can be used to introduce nonstandard extensions in which sets and other objects of higher type can be handled, as is certainly necessary for applications of nonstandard methods in such areas as abstract analysis and topology. However, we do this in stages; in particular, in Section 5 we indicate how to deal with nonstandard extensions in a simple setting where a limited amount of set theoretic apparatus has been introduced. Such limited frameworks for nonstandard analysis are nonetheless adequate for essentially all applications. Section 6 treats the full superstructure apparatus which has become one of the standard ways of formulating nonstandard analysis and which is frequently used in the literature. In Section 7 we briey discuss saturation properties of nonstandard extensions. In several places we introduce specic nonstandard extensions using the ultraproduct construction, and we explore the meaning of certain basic concepts (such as internal set) in these concrete settings. (See the last parts of Sections 2, 4, 5, and 7.) (本文栏目容量有限,以下省略)