# 1.9无穷小新生五十年

1.9无穷小新生五十年

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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 toIR by applying a rather considerable amount of mathematical logic, as indicated before. Another way to deﬁneIR 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 introduceIR 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 theoryIR 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 :yz : P(x,y)][x :sty : P(x,y)]. Standardization:stx :sty :stz : [zyzxP(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 variableis standard, because st is a label butis 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 : [zy⇔∗zxP(z)], where x, y and z are, of course, classical. Since alwayssS if and only if sS, it follows that, y = {zx : P(z)}, or equivalently,

y ={zx : 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)zINz 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)zINz < n, with nIN given such that n; then the results are as under 1). 3) P(z)zIRz ' 0; then y = {0} if 0x and y =if 0 6x. 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 nIN. 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 tof, 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.