# 实数系统为何还需要保序扩张？

这个问题是荷兰格罗宁根大学知名数学家J.Ponstein教授在其专著“非标准分析”（1996年发表的PDF电子版）中明确提出的，并且做出了自己的=

Preface

An inﬁnitesimal is a ‘number’ that is smaller then each positive real number and is larger than each negative real number, so that in the real number system there is just one inﬁnitesimal, i.e. zero. But most of the time only nonzero inﬁnitesimals are of interest. This is related to the fact that when in the usual limit deﬁnition x is tending to c, most of the time only the values of x that are diﬀerent from c are of interest. Hence the real number system has to be extended in some way or other in order to include all inﬁnitesimals.

This book is concerned with an attempt to introduce the inﬁnitesimals and the other ‘nonstandard’ numbers in a naive, simpleminded way. Nevertheless, the resulting theory is hoped to be mathematically sound, and to be complete within obvious limits. Very likely, however, even if ‘nonstandard analysis’ is presented naively, we cannot do without the axiom of choice (there is a restricted version of nonstandard analysis, less elegant and less powerful, that does not need it). This is a pity, because this axiom is not obvious to every mathematician, and is even rejected by constructivistic mathematicians, which is not unreasonable as it does not tell us how the relevant choice could be made (except in simple cases, but then the axiom is not needed).

The remaining basic assumptions that will be made would seem to be acceptable to many mathematicians, although they will be taken partly from formalistic mathematics – i.e. the usual logical principles, in particular the principle of the excluded third – as well as from constructivistic mathematics – i.e. that at the start of all of mathematics the natural numbers (in the classical sense of the term) are given to us. Not only the natural number, but also the set and the pair will be taken as primitive notions. The net eﬀect of this is a version of mathematics that, except for truly nonstandard results, would seem to produce the same theorems as produced by classical mathematics.

One of the consequences of combining ideas from the two main schools of mathematical thinking is that the usual axioms of set theory, notably those due to Zermelo and Fraenkel, will be ignored. First of all, there will be elements that are

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not sets, the natural numbers to begin with, only then sets will be formed from them in stages (or day by day), whereas when starting from the Zermelo-Fraenkel axioms each mathematical entity, in particular each natural number, is some set. From a formal point of view the latter has the advantage that there is just one primitive notion, but from a naive point of view it is not so obvious why numbers should be sets (in formalistic mathematics after the natural numbers come to life in the form of sets, this fact is concealed as soon as possible). Moreover, aren’t we presupposing at least the order of the natural numbers already when writing down axioms by means of suitable symbols?

To a certain extent nonstandard analysis is superﬂuous! For if a theorem of classical mathematics has a nonstandard proof, it also has a classical proof (this follows from what in nonstandard analysis is known as the ‘transfer’ theorem). Often the nonstandard proof is intuitively more attractive, simpler and shorter, which is one of the reasons to be interested in nonstandard analysis at all. Another reason is that totally new mathematical models for all kinds of problems can be (and in the mean time have been) formulated when inﬁnitesimals or other nonstandard numbers occur in such models. A trivial example is a problem involving a heap of sand containing very many grains of sand, but where the number of grains of sand must not be inﬁnite. Then taking the inverse of some positive inﬁnitesimal and rounding the result up or down produces a so-called inﬁnitely large ‘natural number’ that is larger than each ordinary natural number, but is smaller than inﬁnity. It can be manipulated in much the same way as the ordinary numbers, which cannot, of course, be said of inﬁnity. As a consequence the mathematics of inﬁnitely large sets is essentially simpler than that of inﬁnite sets. A peculiarity, however, is that the ‘selected’ inﬁnitesimal and hence the inﬁnitely large natural number are not speciﬁed the way the number of elements of a set of, say, 25 elements is speciﬁed. On the other hand, if ω is that inﬁnitely large natural number, it makes sense to consider another heap of sand with ω2 grains of sand, that can be thought of as the result of combining ω heaps of sand each containing w grains of sand. But in what follows the analysis of practical models containing nonstandard numbers will not be stressed.

Chapter 1

Generalities