今日之无穷小微积分

今日之无穷小微积分
进入移动互联时代，死死搂着传统极限理论（古董），显得不合时宜。
为此，我们推荐DR. J. PONSTEIN教授的专著“非标准分析”，纷纷读者参考阅读。
本文附件1是该书的目录，附件2是该书的1.9节“20世纪的无穷小”。
袁萌  陈启清  7月8日
附件1：
NONSTANDARD ANALYSIS
DR. J. PONSTEIN

With love, love, love to those five women, who caressed me.

J. Ponstein former professor at the University of Groningen
A Naive Way to the Infinitesimals (an unorthodox treatment of Nonstandard Analysis)
ISBN: 90-367-1672-1
Contents
Prologue 11
Preface 15
1 Generalities 17
1.1 Infinitesimals and other nonstandard numbers: getting acquainted 17 
1.2 Other ∗-transforms; generating new numbers . . . . . . . . 19 
1.3 Bound and free variables; prenex normal form . . . . . . . . . . . 20
1.4 The purpose of nonstandard analysis . . . . . . . . . . . . . . . . 23 
1.5 More about the ∗-transform; transfer . . . . . . . . . . . . . . .25 
1.6 Standard, internal, and external constants . . . . . . . . . . . . . 28
1.7 Infinitesimals in Greek geometry? . . . . . . . . . . . . . . . . . . 29
1.8 Infinitesimals in the 17th to the 19th century . . . . . . . . . . . . 31
1.9 Infinitesimals in the 20th century . . . . . . . .. 34
1.10 Introducing infinitesimals by plausible reasoning; filters . . . . . . 37
1.11 Basic assumptions of formalism . . . . . . . . . . . . . . . . . . . 41
1.12 Basic assumptions of constructivism . . . . . . . . . . . . . . . . . 43
1.13 Selecting basic assumptions naively . . . . . . . . . . . . . . . . . 45
1.14 Basic definitions . . . . . . . . . . . . . . .. . . . . . 49
1.15 Filters . . . . . 51
1.16 About the nature of free ultrafilters . . . . . . 53
2 Basic theory  55
2.1 Reviewing the introduction of ZZ,Q and IR . . . . . . . . . . . . . 55
2.2 Introducing internal constants; definition of equality . . . . . . . . 57
2.3 Identification of internal constants . . . . . . . . . . . . . . . . . . 58
2.4 Standard constants; basic results for internal constants . . . . . . 63
2.5 External constants . . . . . . . . . . . . . . . . . . . . . . . . . . 67 
2.6 The ∗-transform of operations and expressions . . . . . . . . . . . 69 
2.7 The ∗-transform of relations and statements; L oˇs’ theorem; the internal definition principle . . . . . . . . . . . . . . . . . . . . . . 71
2.8 Transfer; the standard definition principle . . . . . . . . . . . . . 78 
2.9 The ∗-transform of attributes . . . . . . . . . . . . . . . . . . . . 81 
2.10 ∗IN, ∗ZZ, ∗Q, ∗IR: main definitions and properties . . . . . . . . . . 83
2.11 Overflow and underflow . . . . . . . . . . . . . . . . . . . . . . . . 85
2.12 ∗IN and ∗ZZ: more properties . . . . . . . . . . . . . . . . . . . . . 87 2.13 ∗Q and ∗IR: more properties; standard part . . . . . . . . . . . . . 91 
2.14 An alternative to introducing ∗ZZ, ∗Q and ∗IR . . . . . . . . . . . . 92
2.15 Getting away with generating sequences and H(si); summary . . 94
3 Some applications 99
3.1 Introduction and least upper bound theorem . . . . . . . . . . . . 99
3.2 Simplifying definitions and proofs of elementary calculus . . . . . 100
3.3 Continuity and limits for internal functions . . . . . . . . . . . . . 104
3.4 More nonstandard characterizations of classical notions . . . . . . 108
3.5 Inverse functions; bc . . . . . . . . . . . . . . . . . . . . . . . . . . 112
3.6 Differentiation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 114
3.7 Integration . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 118
3.8 Pitfalls in nonstandard analysis . . . . . . . . . . . . . . . . . . . 121
4 Some special topics  125
4.1 Principles of permanence . . . . . . . . . . . . . 125
4.2 The saturation principle . . . . . . . . . . . . . . . . . . . . . . . 130
4.3 Stirling’s formula . . . . . . . . . . . . . . . . . . . . . . . . . . . 132
4.4 Nonstandard mathematics without the axiom of choice? . . . . . . 134
Appendix 139
References 141
Index  143
附件2：
1.9 Infinitesimals in the 20th century
When in the 1870’s Weierstrass formulated the well-known ε−δ definitions of limit and continuity, definitions that completely ignore nonstandard numbers, the dispute regarding infinitesimals quickly settled in their disadvantage, but only temporarily, for in 1961 Robinson [6,7] presented a mathematically sound theory of the nonstandard numbers. These works embody the first fairly complete analysis of the nonstandard numbers. Not only are they based on work of forerunners, but also on an amount of mathematical logic that hitherto was unusual in mathematics. Only a few references should suffice here, see [8–12].
Robinson starts from the axioms of set theory due to Zermelo and Fraenkel, and the axiom of choice (called together the ZFC axioms), derives IR in a classical kind of way, and then extends IR to ∗IR by applying a rather considerable amount of mathematical logic, as indicated before. Another way to define ∗IR was already indicated by Hewitt [10] and worked out by Luxemburg [13]. Here the ZFC axioms are again the point of departure, but the more usual line of mathematical thinking is followed. (Except for the ZF axioms, this way is also followed in the next chapter.) Still another way to introduce ∗IR was found by Nelson [14]. Nelson adds three more axioms to the ZFC axioms, as well as a new symbol, st (for ‘standard’) that is used as a kind of label to distinguish standard constants from nonstandard constants. This leads directly to the set of all standard as well as all nonstandard constants, without the intermediary step of first introducing IR; consequently
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in internal set theory ∗IR is denoted by IR, and similarly, ∗IN is denoted by IN, etc. Actually, the point of view of internal set theory is that the IN of classical mathematics is the same as the IN of nonstandard analysis; and that all that happens is that unexpected elements of IN are discovered, elements that had always been there. In other words, according to this point of view, 0, 1, 2, etc. do not at all fill up IN (see Robert [15] and F. Diener et G. Reeb [16]). The additional axioms make sure that transfer is guaranteed (axiom of ‘transfer’), that nonstandard numbers exist (axiom of ‘idealization’), and that unique standard sets can be derived from given sets (axiom of ‘standardization’). Even though internal set theory uses relatively little of mathematical logic, the new axioms require some study, and do not seem to be as obvious as, for example, the axioms of Greek geometry: Transfer: ∀stt1 ...∀sttk : [∀stx : P(x,t1,...,tk) ⇒∀x : P(x,t1,...,tk)]. Idealization: [∀st finx : ∃x : ∀y ∈ z : P(x,y)] → [∃x : ∀sty : P(x,y)]. Standardization: ∀stx : ∃sty : ∀stz : [z ∈ y ⇔ z ∈ x∧P(z)]. Here stu means that the variable u must be standard, and similarly the label fin means that the corresponding variable must be finite (but beware, in internal set theory any hyperlarge natural number is finite, only the combination of standard and finite amounts to the classical notion of finiteness). Note that whereas stu means that the variable u is standard, ∗∧ means the variable ∗∧ is standard, because st is a label but∗is a mapping. P(...) denotes a given internal statement, except in the last axiom, where P(...) may even be external (see Section 1.6).
In naive nonstandard analysis these three additional axioms are not assumed but derived from the existence of the natural numbers and the axiom of choice. Transfer has already been discussed; and idealization is used to prove the existence of nonstandard elements in any internal set with an infinite number of elements. Perhaps standardization is the most intriguing of the three because it contains a statement P(z) that may be external. Reformulated naively it means that, ∀∗x : ∃∗y : ∀∗z : [∗z ∈∗y ⇔∗z ∈∗x∧P(∗z)], where x, y and z are, of course, classical. Since always ∗s ∈∗S if and only if s ∈ S, it follows that, y = {z ∈ x : P(∗z)}, or equivalently,
∗ y = ∗{z ∈ x : P(∗x)},
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which in internal set theory are illegal set formations. Here are a few examples, where x and y are still classical, but z need not be classical. 1) P(z) ≡ z ∈∗IN∧z is standard; then x = {1,2,3} gives y = x = 1,2,3}, x = IN gives y = x = IN, and x = IR also gives y = IN.
In fact IN is the largest y that is possible for variable x. 2) P(z) ≡ z ∈∗IN∧z < n, with n ∈∗IN given such that n ∼∞; then the results are as under 1). 3) P(z) ≡ z ∈∗IR∧z ' 0; then y = {0} if 0 ∈ x and y = ∅ if 0 6∈ x. For other details the reader should consult more adequate treatments of internal set theory.
In the mean time other versions of nonstandard analysis have been developed. In one of them external sets are ‘legalized’ by means of still other axioms, and another label, ext (for ‘external’).
By now many hundreds of publications have been devoted to nonstandard analysis: it is an established branch of mathematics.
No matter how infinitesimals are introduced, with or without the axioms of set theory, with or without extra axioms and new undefined symbols (st and ext), always the axiom of choice seems indispensible. If one tries to develop infinitesimal calculus without this axiom, it seems that one should be satisfied with a mutilated theory, as will be explained later on in Section 4.4. Here attempts by Chwistek [17,18] in this direction should be mentioned. In his 1926 paper Chwistek introduces new numbers by means of infinite sequences of classical numbers. These new numbers are called Progressionszahlen (‘sequence numbers’), and equality for them is defined as follows. Let Ni(αi) and Ni(βi) be two new numbers, then, Ni(αi) = Ni(βi) if and only if αi = βi for i > n for some n ∈ IN. Something similar is done to define inequality, and an operation like addition is defined by,
Ni(αi) + Ni(βi) = N(αi + βi).
A classical function f is extended by means of,
f(Ni(αi)) = Ni(f(αi)).
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The extended function happens to be quite similar to ∗f, the ∗-transform of f. Even so not much new calculus is developed. An extension of IR that includes all sequence numbers could be introduced, however.
In his 1948 book Chwistek spends less then ten pages on the subject, but nevertheless shows that he is well aware of the fact that ‘infinitely small’ numbers can be introduced, and he also introduces internal functions (called normal functions by him). Again there is no fully expanded calculus. Most likely, the deeper reason for this is that Chwistek defines (in)equality for his sequence numbers as indicated above. This definition has the advantage that the axiom of choice is not needed, but leads to rather serious problems, as will become clear in Section 4.4. It remains to remark that working with sequences is a technique used by Hewitt [10] and Luxemburg [13], and will be the technique of the next chapter, which is based on assumptions that from a naive, intuitive point of view are understandable, obvious, and acceptable, except perhaps the axiom of choice, and where everything that is not so obvious, such as transfer and all the rest, will be proved, rather than assumed.