FOUNDATIONSOFNONSTANDARDANALYSIS
(2018-09-26 17:40:16)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
whomwehavewrittenareexpe
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.) Our experience shows that it is often helpful at the beginning to have such explicit nonstandard extensions at hand. As noted above, however, we think it is limiting to become dependent on such a construction and we encourage readers to adopt the more exible axiomatic approach as quickly as possible. In writing this exposition we have benetted greatly from conversations with Lou van den Dries about the best ways to present ideas from model theory to the general mathematical public. His ideas are presented in [5] and, with Chris Miller, in [6], and our treatment obviously depends heavily on that work. We have also freely used many ideas from other expositions of nonstandard analysis (listed among the References) and from the other Chapters in this book. To all these authors we express our sincere appreciation, and we recommend their writings to the reader who nishes this exposition with a desire to learn more about how nonstandard methods can be applied.
2. Nonstandard Extensions(省略)