J.M. HENLE Non-nonstandard Analysis: Real Infinitcsimals

~n athematics has had a troubled relationship with , a relationship that stretches back thousands of years. On the one ,hand, infinitesimals make intuitive sense. They're easy to deal with algebraically. They make a lot of fun. On the other hand, they seem impossible to nail down. They're hard to deal "standard." Nor is it "nonstandard," however, as this term with intellectually. They can mask a fundamental lack of now has a well-def'med meaning. In what follows, we ma- understanding of analysis. nipulate sequences of real . We treat them (mostly) There was hope, when developed as numbers. We add them, subtract them, and put them nonstandard analysis [R], that intuition and rigor had at last into functions. They aren't numbers, however. Trichotomy joined hands. His work indeed gave infinitesimals a foun- fails, for example. dation as members of the set of hyperreai numbers. But it The central construction in this article is a rediscovery. was an awkward foundation, dependent on the of Its first discoverer probably was D. Laugwitz. More on this Choice. Unlike standard systems, there is no later. canonical set of hyperreal numbers. There is a way, however, of constructing infinitesimals Notation. We denote a sequence {an}nE ~ by a boldface naturally. Ironically, the seeds can be found in any calcu- a. We permit as to be undefined for fmitely many n. The lus book of sufficient age. At the turn of the century, it was key idea throughout is that of "from some point on." An typical of texts to define an as a "variable equation or inequality involving sequences will be in- whose is zero" [C]. That is the inspiration for the terpreted as being true or false, depending on whether present approach to calculus. Its infmitesimals are se- the associated equation or inequality involving the terms quences tending to 0. of the sequences is true or false from some point on. I call the system "non-nonstandard analysis" to draw at- For example, tention to its misfit nature. Having infinitesimals, it is not a>2

@ 1999 SPRINGERWERLAG NEW YORK, VOLUME 21, NUMBER 1, 1999 67 simply means "an > 2 from some point on," that is, that Proposition 1. Suppose a, b ~ 0. Then for some k, n -> k implies that an > 2. We use ordinary (1) a + b ~ 0. letters for real numbers. Note that a r can (2) a - b ~ 0. be viewed as the sequence a, with an = r for all n. (3) If e is finite, then ac ~ 0. (4) If Icl < lal, then e ~ O. Equations and inequalities involving sequences are inter- preted in the same way; that is, a + b = c means an § bn = PROOF. Given any positive d, we know that lal < d/2 and Cn from some point on, and sin(a) = b means sin(an) = bn Ibt < d/2. Then, la + b I -< lal + Ibl < d. Thisproves Part (1). from some point on. Note that forf(a) to be defined, it is Part (2) is proved similarly. only necessary that f(an) be defined from some point on. For Part (3), because Icl < r for some real r, and lal < The sequence b: 0, 0, 0, 1, 2, 3, 4,..., for example, does d/r for all positive d, we have lacl < d. have a reciprocal, I/b, because, as noted, finitely many For Part (4), observe that because lel < lal < d, Icl terms of a sequence may be undefined.

a+4=3b Definition. For sequences a and b, we say that a and b are infinitely close, or a ~ b, iff a - b ~ 0. and

c <45. Proposition 2. If a ~- r and b ~- s, then (1) a+b~r+s. Then, (2) a- b ~ r- s. (3) ab ~ rs. an + 4 = 3bn, from some point on, (4) If a -< s, then r -< s. and (5) a + b ~ r/s, if s r O. Cn < 45, from some point on. PROOF. Parts (1) and (2) follow easily from Proposition 1. Because the two are true together from some point on, For Part (3), note that (a-r)s, (b-s)r, and we can add them to get (a - r)(b - s) are all ~ 0. Adding these gives us ab - rs ~ O. an + 4 + Cn < 3bn + 45, from some point on. For Part (4), if r > s, then we would have both an -< s from some pont on and lan- r I < (r- s)/2 from some In other words, point on, which is impossible. a+4+c<3b+45. In view of Part (3), we need only show lfo ~ 1/s to establish Part (5). From s ~ b, we get Is~21 ~ Ib/21 < Ibl, so What we are doing, with this construction, is taking the by Part (4), Is~21 -< Ibl. Then I1/b - 1/s I = I(s - b)/bs I -< equivalence classes of sequences under the relation "equal- (1/Is/21)(1/Isb]s- b I. By Proposition 1, this is infinitesi- ity from some point on." To present it in that way, how- mal. 9 ever, would entail excessive formalism. One of my pur- This is all we need to get started. poses is to demonstrate that there is a fairly simple approach to infmitesimals, one that can reasonably be pre- Elementary Calculus sented to ordinary calculus students.

Definition 1. A function f is continuous at x = r, r real, Infinitely Small and Infinitely Close iff

Definition. A sequence a is infinitely small if lal < d a~rimpliesf(a)~f(r). for all positive real numbers d. For infinitely small a, we write a ~ 0. If lal < r for some real r, we say a is finite Equivalently, f is continuous at r if Ax ~ 0 implies or bounded. A sequence a is infinitesimal if a r 0 and f(r + Ax) -f(r) ~ O. a~0. We can also discuss continuity at a sequence a, as op- These definitions hide quantifiers. When we say that a r posed to a real r, but continuity on a set X corresponds to 0, we are actually saying that there is a k such that an r 0 continuity at all real numbers in X. As in nonstandard analy- for all n --> k. Similarly, this defmition of a ~ 0 is, in real- sis, continuity at all points (real numbers and sequences) ity, the more familiar and complicated Ve > 0 3k, n ~ k in a set X is equivalent to uniform continuity. I will prove lanl < e. this later.

(}8 THE MATHEMATICAL INTELLIGENCER Proposition 3. The sum, difference, product, and quo- Definition 3. a is a subsequence of b (written a C b) if tient (when the divisor is nonzero) of functions contin- every term of a is a term of b; more precisely, if there is uous at x = r are continuous at x = r. an increasing function, k : N ~ N such that a n = bk(n).

This follows easily from Proposition 2. Proposition 6. Let a C b be given. (1) Ifb is infinitesimal, then so is a. (2) If b ~ r, then a ~ r. Definition 2. For a functionfand a real r, we sayf'(r) = d iff for all infinitesimal Ax, (3) Ifb satisfies a given equation or inequality, then so does a. f(r + Ax) -f(r) ~ d. Ax The proof of this is routine. Here is what the computation looks like of the standard Proposition 7. (The Chain Rule). If y =fix) and z = first example: f(x) = x 2. g(y), then dz/dx = (dz/dy) 9(dy/dx). (x + Ax) 2 - x 2 x 2 - 2xAx + Ax 2 - x 2 Ax Ax PROOF. For Ax infinitesimal, dy/dx ~ Ay/Ax, where Ay = 2xAx + AX 2 f(x + Ax)-fix). By Proposition 4, Ay ~0, so seemingly Ax Az/Ay --~ dz/dy, where Az = g(y + Ay) -- g(y). But there is = 2x + Ax another way to write Az, since g(y + Ay) - g(y) = g(f(x) + ~2X. Ay) - g(f(x)) = g(f(x + Ax)) - g(f(x)), so dz/dx ~ Az/h,x.

Proposition 4. If f is differentiable at r, it is continuous Putting this together, at r. dz Az Az Ay ~ dz dy PROOF. Just multiply dx Ax Ay Ax dy dx" The only difficulty with this is that Ay, while infinitely close f(r + Ax) - f(r) ~ d and Ax --~ O to 0, may not be infinitesimal because it may equal 0 infi- Ax nitely often. In that case, we can't claim that dz/dy .~ to get Az/Ay, because the definition of requires Aye 0. f(r + Ax) -f(r) ~ 0. 9 But then, let Ax* C Ax be the subsequence such that The proofs of the differentiation rules are simple. The the corresponding Ay* is a sequence entirely composed of is typical: O's. Then, dy/dx ~ Ay*/Ax* = 0. And, because Ay* = 0, the corresponding Az* = g(y + Ay*)- g(y) is also 0, so Proposition 5. If the function h is the product of differ- dz/dx ~ Az*/Ax* = 0. Thus, once again, entiable functions f and g, then h is differentiable and d z = d z. d y. 9 h' =f'g + g'f. dx dy dx

PROOF. Writing Af forf(x + Ax) -f(x) and similarly for g For integration, we can use sequences of step functions. and h, we have f(x + Ax) = f(x) + Af, so

= h(x + Ax) - h(x) Definition 4. A function s is a step function on an in- = f(x + Ax)g(x + Ax) -f(x)g(x) terval, [a, b], if it is defined on the interval and changes = (f(x) + Af)(g(x) + Ag) -f(x)g(x). value only a finite number of times over the interval.

Thus, Step functions are simply functions that are piece-wise con- Ah if(x) + Af)(g(x) + Ag) -f(x)g(x) stant. A sequence of step functions, s, is a sequence where Ax Ax Sn is a step function from some point on. _ f(x)Ag + g(x)Af + AfAg Ax Definition 5. If s is a step function on [a0, an] with s(x) = (x) Af + AfAg ci on each subinterval, (ai, ai+l), i = 0,..., n - 1, then = f(x) Ax + g Ax Ax n 1 --~ f(x) 9g'(x) + g(x) . f'(x) + 0 9g'(x) f? s(x)dx ~. ci(ai+l - ai). 0 i=0 =f(x) 9g'(x) + g(x) .f'(x). 9 Definition 6. For a function f and an interval [p, q], The proof of the Chain Rule, omitted or banished to the appendices in virtually all texts today, is easy, but we need to discuss subsequences. f; f c~ = r

VOLUME 21, NUMBER 1, 1999 69 iff there are sequences of step functions d -

ddx ~ r ,~. udx. sequence c C a, and a real number, r = k.dld2dad4.. 9and s; s; by construction,

As usual, d -< f-< u simply means that dn -< f-< u,, on Icl - r I < 1, [p, q] from some point on, and fpq d dx is the sequence Ic2 - r] < 0.1 {fq dn dx}. Ic3 - rI < 0.01 The integral, r, is unique, for if di, ui, and ri satisfy the conditions above for i = 1, 2, then

Thus, for any q > 0, Icn - r I < q from some point on. That rl ~ dl dx -< u2 dx ~ r2 ~ d2 dx ~ u 1 dx ~ rl. l; s; s; s; means Ic - r I ~ O, or c ~ r. 9 The basic theorems on integrals are easily proved. The fol- Proposition 10 (Intermediate Value Theorem). Iff is lowing is an example: continuous on [p, q] and f(p) -< s -< f(q), then for some r E [p, q],f(r) = s. Proposition 8. If f is integrable over [a, b] and [b, c], then it is integrable over [a, c] and PROOF: I begin by constructing two sequences a and bin the interval [p, q]. For n = 1, let al = P and bl = q. For n f dx= f dx + f dx. ;a f; in general, divide the interval [p, q] by points

PROOF. Let dab , Uab , dbc, Ubc be the step-functions witness- p =X0 ~Xl ~X2 ~ "'" ~Xn = q, ing the integrability off over [a, b] and [b, c]. Define d and equally spaced, a distance of (q - p)/n apart. As f(x0) -< u on [a, c] by gluing together the respective lower and up- s <-f(Xn), there must be two adjacent points, Xk and Xk+l per functions. Then, we certainly have such that f(xk) -< s -

70 THE MATHEMATICALINTELLIGENCER :IGURE

on each subinterval to be the minimum value off (guar- How Did We Do It?. anteed to exist by Proposition 11). Similarly, define Un as How did I avoid the Axiom of Choice? I simply asked the maximum value off. We have that I -<-f-< u. less of our sequences than one asks of numbers. They aren't Because f is bounded, the sequences L = fq 1 dx and totally ordered, for example. The sequences 0, 1, 0, t, ... U = fpq u dx are bounded, so there is a subsequence l* C and 1, 0, 1, 0,... are incomparable. There are also zero di- 1 and real r such that the corresponding L* = f~ l* dx ~ r. visors. This doesn't cause any problems. Let u* be the subsequence of u correpsonding to l*, and Why didn't I need the ? The let, for each n, Ax~ be the length of the corresponding Transfer Principle is a powerful schema of nonstandard subinterval. I claim that fq u* dx ~ r and so fq f dx = r. analysis that says that any statement (in a particularly rich, Proof of claim. For any n, we can find the greatest differ- well-defmed language) that is true about the real number ence, Rn, between In(x) and Un(X) on [p, q], and we have system is true about the system and vice that versa. We have here a weak form of this, namely that all equations and inequalities true about reals are true about Un dx -- In dx <- Rn(q - p). sequences. We also have conjunctions of these (but not dis- s;* s;" junctions). The difference, Rn, can be represented as If(Sn) - f(tn)l, for In elementary calculus and analysis, the Transfer some Sn and tn with ISn - tn] <-- AXn. Principle is used chiefly to prove that the nonstandard de- fmitions are equivalent to the standard definitions. But, here, these equivalences are easy. Here's an example: Then,

Definition 7. f is uniformly continuous on C iff u* dx - l* dx - S(q - p) = ~f(s) -f(t)l(q - P). s; s; (Standard) Ve > 0 38 > 0, Vx, y E C We would like to say that as is - tl -< Ax ~ O, then, by con- Ix - Yl < ~ ~ If(x) - f(Y)t < ~. tinuity, If(s) -f(t)l ~ O, so (Non-nonstandard) Va, b E C, a = b ~f(a) ~f(b).

u*dx- l*dx~O and l*dx~r. Proposition 13. The standard and the non-nonstandard P s; s; definitions of uniform continuity are equivalent. But continuity requires that one of s and t be real. The property s ~ t ~f(s) ~f(t) is actually equivalent to uni- PROOF. Suppose the non-nonstandard definition holds, and form continuity. We can easily work around this, however, suppose we are given an e for which there is no suitable & by finding a subsequence s** C s and real r, with s** -~ r Then, for each natural number n, choose an and bn such and using the corresponding subsequences u**, 1"*, and that la.n - bnI < 1/n, but If(an) - f(bn)I >- ~. Then, we have t** ~ r ~ s** to finish the proof. 9 a -~ b, but f(a) r f(b), a contradiction.

:IGURE ;

VOLUME 21, NUMBER 1, 1999 71 Suppose, now, that the standard definition holds, and and the power to find ever smaller numbers. In a weak we are given a ~ b and d, a positive real. By the standard sense, it is the difference between inf'mitesimals and lim- definition, there is a 6 such that Vx, y Ix - y] < 8 ~ If(x) - its. Archimedes used infinitesimals for intuition [A1], then f(y)] < d. Then, because ]a~ - b~] < 8 from some point on, verified his results by proving them with (what we would If(an) -f(b,~)] < d from some point on, and so f(a) ~ f(b). call today) limits [A2]. Calculus, as formulated in the seventeenth century, was first expressed in terms of infinitesimals. As infinitesimals What happened to the Axiom of Completeness? I themselves were not well understood, there were criti- did use the completeness of the real line, but in a most in- cisms and misunderstandings. On the whole, however, the nocuous and comprehensible form: I simply assumed that doubts were overshadowed by the outstanding success of every infinite decimal corresponds to a real number. 1 the theory as developed in the eighteenth century. What happened to all the quantifiers? In the case In the nineteenth century, Cauchy, Weierstrass, and oth- of uniform continuity, for example, I went from "re 3 8 Vx, ers made infinitesimals unnecessary. Absolute y..." (logicians call this a II3 statement) to "Va, b,..." (a were replaced by limits, now rigorously defined. But lin- II1 statement). Some of the quantifiers are buried. The guistic habits didn't change. Mathematicians and physicists statement "a ~ b" is, in reality, something like "Vr > 0 continued to talk in terms of infinitesimals. Infinitesimals lan - bnl < r from some point on," or "Vr > 0 3k Vn > k didn't disappear from calculus texts for over 100 years. .... " Essentially, I simplify definitions by coding up the Infinitesimals began to reappear in the twentieth cen- most difficult part. tury. The idea of sequences as infinitesimals appears in a One can also ask, What is next? There are theorems remarkable book, The Limits of Science, Outline of Logic of analysis where nonstandard proofs are awkward or do and Methodology of Science by Leon Chwistek, painter, not exist. In the former category is the theorem that the philosopher, and mathematician [Ch]. Published in 1935, uniform limit of continuous functions is continuous. This the book is not well known today (it is in Polish). is proved in [HK] by taking a nonstandard model of a non- Chwistek's definitions are similar to those presented here, standard model. It can be done in the present system by although there are differences and limitations. His work taking sequences of sequences. Indeed, by closing under foreshadows not only non-nonstandard analysis, but non- the operation of "taking sequences," a great deal more standard analysis. analysis can be handled. In [He], this is pursued to prove In [L1] and [L2], Laugwitz formulated the system of se- the Baire Category Theorem, for which no simple non- quences described in this article, but with different nota- standard proof exists, and to develop measure theory. tion. Laugwitz used his "~-Zahlen" to investigate distribu- tions and operators. An earlier paper by Schmieden and Pedagogy Laugwitz [SL] used a more primitive system with the idea The non-nonconstructive infmitesimals presented here of justifying the infmitesimals of Leibniz. Laugwitz's work could improve the teaching of calculus. In both "standard" was not carried further, possibly because the discovery of and "reform" calculus courses, rigor has been almost en- nonstandard analysis made real infinitesimals unglam- tirely omitted. Consequently, students are not asked to orous. prove theorems until they have a fairly strong intuition for In 1960, Abraham Robinson constructed nonstandard the subject and have met infinite sequences. This back- models of the real number system using mathematical logic ground makes non-nonstandard analysis very attractive [R]. Robinson credits the papers of Laugwitz and Schmieden for, say, an Advanced Calculus course, or even Calculus with some inspiration for his work. Nonstandard analysis III. requires a substantial investment (mathematical logic and For those wishing to remain as "standard" as possible, the Axiom of Choice) but pays great dividends. Non- one can still use sequences. The chief adjustment is to re- standard analysis has been used to discover new theorems place a ~ 0 with a --~ 0. The definitions and proofs in this of analysis. It has been fruitfully applied to measure the- article are easily converted. This approach is being used ory, Brownian motion, and economic analysis, to name just now in [CH]. a few areas. Attempts to reform calculus instruction along infinitesimal lines, however, did not have much success Some History [HK], [K]. Robinson's book contains an excellent history of The Greeks explored the ideas of infmity and infinitesi- infinitesimals. mals. On the whole, they rejected them. To Aristotle, there There are other systems of standard infmitesimals, was no absolute . Completed infinite sets such as Conway's surreal numbers, for example [Co]. There are other the natural numbers {1, 2, 3,...}, did not exist. There could systems for avoiding e's and fi's (see [Hi] for a recent exam- only be potential infinities; for example, for every number, ple). There may also be other rediscoveries of this system. there is another number that is larger, and so on. The dis- The intended contribution of the present article is to place tinction is similar to that between an infinitesimal number the structure in an algebra suitable for students of calculus.

1This unremarkable statement is accepted easily by students. It is not equivalent to Completeness, but is stronger, and yields the Archimedean Principle as well.

72 THE MATHEMATICALINTELLIGENCER REFERENCES [A1] Archimedes, The Method of Archimedes, recently discovered by Heiberg, edited by Sir Thomas L. Heath, Cambridge, Cambridge University Press (1912). [A2] Archimedes, "The quadrature of the parabola, in The Works of Archimedes, edited by Sir Thomas L. Heath, Cambridge: Cambridge University Press (1897). [C] Chandler, G. H., Elements of the Infinitesimal Calculus, 3rd ed., New York: John Wiley & Sons (1907). [Ch] Chwistek, L., The Limits of Science, Outline of Logic and Methodology of Science, Lwow-Warszawa: Ksiaznica-Atlas (1935). [CH] Cohen, D., and Henle, J. M., Conversational Calculus, Reading, MA: Addison-Wesley/Longman (in press). [Co] Conway, J. H., On Numbers and Games, New York: Academic Press (1976). [He] Henle, J. M., Higher non-nonstandard analysis: Category and mea- sure, preprint. [HK] Henle, J. M., and Kleinberg, E. M., Infinitesimal Calculus, Cambridge, MA: M.I.T. Press, (1979). [Hi] Hijab, O., Introduction to Calculus and Classical Analysis, New York: Springer-Verlag, (1997). [K] Keis~er, H. J., Elementary Calculus, An infinitesimal Approach, 2rid ed., Boston: Prindle, Weber & Schmidt (1986). ILl] Laugwitz, D., Anwendungen unendlich kleiner Zahlen I, J. Reine Angew. Math. 207 (1960), 53-60. [L2] Laugwitz, D. Anwendungen unendlich kleiner Zahlen II, J. Reine Angew. Math. 208 (1961), 22-34. Acknowledgments [R] Robinson, A., Nonstandard Analysis, Amsterdam: North-Holland, I would like to thank Michael Henle, Ward Henson, Roman (1966). Kossak, Dan Velleman, and several anonymous reviewers [SL] Schmieden, C., and Laugwitz, D., Eine Erweiterung der for careful reading and helpful remarks. Infinitesimalrechnung, Math. Zeitschr. 69 (1958), 1-39.