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Intuitions of Infinity INTUITIONS OF INFINITY by David Tall, Mathematics Education Research Centre, University of Warwick What is infinity? It is an extrapolation of our finite experience. But for many this leaves a bitter taste in the mouth. Some· As such our intuitions of infinity depend very much on the thing has gone wrong somewhere. How can "nought point nine kind of experience which is extrapolated. For instance, if we recurring" equal one? Surely when we subtract it from one we consider the "samesize" property of sets which have a bijection* get a single digit left over, way off in the "infinitieth ):>lace": between them then we get the kind of infinity called a cardinal 1-0.999 ... 999 ... =0.000 ... 000 ...... 1.· number. This is the most widely considered notion of infinity amongst. mathematicians ,yet it has properties which usually These problems are usually explained away by quoting the seem most unintuitive to the uninitiated. If we look at the current mathematical dogma relating to the appropriate .situ· experiences of pupils in school, they rarely concern the "same· ation. In handling the limit of 1/( 1 �x) as x tends to 1 we might size" aspect of sets so this lack of intuitive appeal is hardly say that the limit from the left is + oo and from the right is � oo surprising. Secondary pupils have different ·experiences of and 1/(1-x) is "not defined at x = 1". The problem as to infinity which give them a totally different feel for the idea. whether 2oo3 + 3 is a bigger infinity than oo2 + 4 is responded For instance, in drawing the graph of to by a rap over the knuckles "don't do arithmetic with infinity, oo/oo /(x)""'1/(1-x) it leads to contradictions,, "a symbol like is meaningless" and so on.The problem of the "infinitieth place" argument also they find that as x approaches 1 from the left,/(x) gets very large has a slick response. "Let and positive but as x approaches 1 from the right, f(x) gets very sn= 0. 999... 9 (to n decimal places), large and negative. The expression 2 3 so that s1 =0.9,s =0.99, and so on. Then (x + 4)/(2x + 3) 2 1-sn= 1/1011 approaches 0 as x grows large, so although x2 + 4 and 2x3 + 3 3 both grow without limit as x gets large, 2x + 3 grows even and taking the limit as n tends to infinity gives faster than x2 + 4: 1-lims11=0, How can one reconcile these two examples? Id the first case n-+Oll f(x) tends to a "plus infinity" from one side and a "minus so infinity" from the other. Is "plus infinity" the same as "minus 2 1 -0.999... 999 ...= 1- lim S11= 0 . infinity"? In the second case if x + 4 tends to "plus infinity" ,.-.eo 3 as x grows large, how can 2x + 3 tend to something even larger? and there is no digit left over in the inflnitieth place." · Can w.e have something that is even bigger than infinity? We often learn riot to "understand" these arguments but we . One thing is certain. Talking about cardinal infinity, the do "get used to them". So we become enlightened in the true "samesize" of sets, will not resolve this problem in ':i" natural wRys of mo!1ern mathematics. way.These ideas of "growing large", "tending to infinity" and However, there are two fundamental problems with this so on have absolutely nothing to do with comparing the size of eventual acceptance of the wisdom of our elders and betters. sets. The first is why do we so persistently obtain these early As another example, consider that hoary character the decimal intuitions of infinity as a direct product of school experience? expansion The second is a more fundamental one. If everyone seems to 0.999 ... 999... get such wild ideas, in what sense is the accepted mathematical definition so utuch better? "nought point nine recurring''. Does it equal one, or is it just When we look back at the history of mathematics we find a less? Ask school pupils this question and the vast majority will new twist. Three hundred years ago when the calculus was say "less". I suspect that the majority of teachers would say invented the accepted mathematical notion of infinity was quite the same. Mathematicians assert that this decimal expansion different from what it is now. In fact it is closer to the notions equals one. For them the decimal expansion means the actual of infinity given by the limiting processes which are discussed real number which is the limit' of the decimal approximations. in school.The joke is that many of the intuitions that we sense The decimal expansion nowadays would have been perfectly acceptable in the theories 0.333 ; ..333 ... of three centuries ago. Moreover a moderfi invention called ''non·standard analysis" permits the use of these old ideas of equals a third. (On that matter most pupils and teachers might infinity and it is possible to conceive of such ideas in a muc:h agree.) Multiplying this expression by three gives the proverbial simpler context, as I tried to demonstrate in a recent article in "nought point nine recurring" which therefore should equal the Matltematital Ga�mel, one. My purpose in discussing these matters is not to suggest to * A b1j'tction is a one-one, onto function. readers that they must now learn a revolutionary new piece of 30 Mathematic& in School, May 1981 mathematics so that they may achieve mathemati<;al salvation - cardinal number n and has no element in common with IN, so far from it. What I would like to do is to give an explanation K0 + n is the cardinal number of the set as to why these intuitions of infinity are produced in school­ INU ( - 1, -2, ... , -n I work. Then I would like to demonstrate that these intuitions are not as stupid as the accepted mathematical defiriTtions make But the map f: IN ......U IN ( - 1, - 2, ... , -n I given by . out. The formal mathematical definition is perfectly alright in f(r)= -r for r�n a context where number means a comparison of size of sets and f(r)=r-n for r>n cardinal number gives a theoretical extension to the counting concept.However, in other contexts, such as limiting processes, is a bijection, so the cardinal concept is singularly inappropriate to explain K0+n= K0 (2) intuitions of infinity which arise. To demonstrate this I shall introduce a simple number system which contains infinite for any finite cardinal n. elements and infinitesimally small elements as well as the usual Equations (I) and (2) tell us that it is hopeless to try to define real numbers. I hope the average reader won't be frightened a-P for cardinals a, P; subtracting X0 from both sides of(l) by my intention to do this. I guarantee that(s)he will have met gives K 0 = 0 and subtracting K0 from both sides of (2) gives the number system before but will just have called it by a n = 0 for any natural number n! different name.We shall see that it is possible to have a number Analogous definitions and difficulties hold for the product of system with infinities of different sizes which can be added, two cardinal numbers a, p. Here we choose sets X, Y with multiplied, subtracted and divided in a sensible way, and where cardinal numbers a, P respectively and define the product the multiplicative inverse of an infinite element is an infinitesi­ cardinal ap to be the cardinal of the cartesian product X X Y. mal. This is quite different from the case of cardinal numbers For instance if a= 2, P= X0, we can take X= ( 0, 1 I, Y =IN which can only be added and multiplied. Subtraction and divi­ then X X Y is the set of ordered pairs of the form (0, n) or sion of infinite cardinal numbers cannot be defined, a fact (1, n) for n a natural number. The function which Cantor took as "proof" that infinitesimals could not exist. f:Xx Y--+IN I shall deal with these properties of cardinal infinity first since they arc instructive in showing how different kinds of infinity given by have diiierent properties. Then I shall go on to describe the /((0, n)) = 2n- I simple system which allows a full arithmetic of infinite elements /((1, n))=2n ' of diiierent sizes. After that I shall look at the way intuitions is a bijection, so of infinity arise in school and consider which mathematical 2 X0= X o· theory is more appropriate for the understanding of these This immediately tells us that we cannot define division in intuitions. The problem of "nought point nine recurring" general because dividing both sides by X 0 would give 2 = 1. requires an allusion to non-standard analysis which I shall briefly In recognising the problems of subtracting and dividing mention. Once again we shall see that the intuition, though cardinals Cantor decreed that infinitesimals (found by dividing incompatible with the modern theory of cardinal infinity does a finite number by an infinite one) could not exist2• His con­ fit very well with an alternative mathematical theory. Finally clusion is incorrect. It should have been that infinitesimals do I shall consider the consequences of what I have had to say for not exist within the context of cardinal numbers. In a similar way teachers. It will not mean the learning of new mathematics, solutions of the problems 3 + 4 or 5-8 do not exist in the rather a greater respect for the mathematics that is used in context of natural numbers IN.
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