#  1.2 超实数的诞生： ∗-变换

 1.2超实数的诞生： -变换

    在数学发展史上，超实数的诞生是一个里程碑事件。

   实际上，超实数系统是无穷小微积分的理论基础。

 1.2 Other-transforms; generating new numbers The-transform not only can be obtained for IR but also for IN, ZZ,Q, and in fact any set X of classical mathematics (and for much more, see Section 1.5). Their-transforms are indicated byIN,ZZ,Q, andX, respectively. Throwing all nonﬁnite numbers out ofIN andZZ we obtain again IN and ZZ, but something similar is not true forQ (forIR we know this already), simply becauseQ (just asIR) contains ﬁnite non-classical numbers. Yet there is a striking diﬀerence betweenQ andIR in this respect: the ‘standard part theorem’ discussed at the end of the preceding section does not hold forQ, that is to say, there are ﬁnite elements t ofQ that cannot be written as t = x + ε, with xQ, ε Q, ε 0. For let c be any irrational number, say c =2, and let (r1,r2,...) be some Cauchy sequence of rationals converging to c. Later on it will become clear that then the sequence (r1−c,r2−c,...) ‘generates’ an inﬁnitesimal δ inIR (because this sequenceconverges to zero). On the other hand (r1,r2,...) generates an element r Q IR, and r is ﬁnite (because the ri are rational, and this sequence converges), but it has no standard part in Q, for otherwise r = x + ε for some xQ and some ε Q, ε ' 0.But (r1 −c,r2 −c,...) also generates the ﬁnite number r −c IR, so that r −c = δ ' 0. It follows that x−c = δ −ε ' 0, hence x−c = 0 (as x−c is an ordinary real), which would mean that cQ, a contradiction. On the other hand, inIR we have that st(r) =c. (Carrying this argument further it turns out that there exists a 1−1 mapping between IR and the set of all ﬁnite elements ofQ modulo the set of all rational inﬁnitesimals, preserving addition and multiplication; i.e. the mapping is an isomorphism. Inother words, IR (notIR) can in a sense be produced byQ.)

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 There are various ways to introduce the new numbers. Below this will be done by means of inﬁnite sequences of classical numbers. In particular, the elements ofIR will be generated by means of inﬁnite sequences of reals, and it will be necessary to consider all such sequences. (Recall that the elements of IR can be generated by means of rather special inﬁnite sequences of rationals, i.e. the Cauchy sequences.) More generally, given any classical set X the elements of its-transformX will be generated by means of inﬁnite sequences of elements of X, and again all such sequences must be taken into account. Each such sequence ‘generates’ an element ofX, and in case X is a set of numbers (or n-tuples of numbers) special sequences generate the elements of X itself. For example, (1,2,3,...) generates a hyperlarge element ofIN, and (3/2,5/4,9/8,...) generates a ﬁnite element ofQ, that is equal to the sum of 1, generated by (1,1,1,...) andan inﬁnitesimal, generated by (1/2,1/4,1/8,...). Diﬀerent sequences may generate the same element ofX. In fact, given any x X there are many (uncountably many) diﬀerent sequences that generate x (if X contains at least two elements). For example, changing ﬁnitely many terms of a generating sequence has no eﬀect on the element generated. But there are many more variations on this theme. Wouldn’t it be possible to restrict ourselves to a suitable subset of all sequences? Unless we are satisﬁed with some sort of mutilated nonstandard analysis, most likely the answer is ‘no’. See Section 4.4.

 Anyway, the nuisance of having to use generating sequences is only temporary. Once the new numbers have been introduced (as well as new functions, etc.) in most cases it is not necessary at all to know that they came about by means of inﬁnite sequences. The situation is entirely analogous to that of introducing the real numbers: most of real analysis can be developed without the interference of Cauchy sequences. Most of the time an irrational such as2 is treated as just a number, not as a sequence. AlthoughIN,ZZ,Q andIR are extensions of IN, ZZ,Q and IR, respectively, in generalX is not always an extension of X. If, for example X = {IN}, thenX = {IN}, and since IN 6=IN, X is not contained inX.