New Models of the Real-Number Line

New Mo dels of the Real-Number Line

Recent developments in mathernatical logic reveal tltat tltere are a number of alternative wa)rs of defining the continttLlm, or connected nurnber system., to include all tlt,e rr:al, rtLLmbers

by Lynn Arthur Stccn

-f firtuallv all of mallrematics and tonic ideals-has beeu tl-re most enduring tempted to ans$,er this question by pos- \/ rrr,r"h'nf science is based on tlre and popular philosophy of mathernatics, tulatiug the a prioli existence in the hu- V abstlact concept of the leal-nrrrn- since it plovides mathematics with the man mind of a kernel of intuitive mathe- ber line: the continuum, or connected (scientiffc) discipline of confolming to matical and geometrical tluth. The con- number system, that includes all the real son-re kind of pelceived reality together siderable influence of Kant's philosophy nurlbels-zelo, positive and negative with the freedom to escape the bonds reinfolced the wideiy held view that the integels, rational numbers (flactions) of empilicism. axioms of mathematics should be self- and irrational numbers (such as r)-but N{ost wolking mathematicians, at evident truths. Euclidean geometry be- that excludes the "unreal," or complex, least those not directly engaged in prob- came the archetype of mathematics, numbers-explessions containing the Iems of mathematical logic, tend to be since it was a beautiful, useful theot'y imagirraly numbel y=1. tlre leal-num- Platonists in the sense that they talk and created by logical deduction from cer'- bel corrtinuum not only plovides the nat- act as if the abstract objects they study tain (nearly) self-evident axioms. In- ural setting for all the operations of have some kind of enduling ideal ex- deed, Kant hin-iself had asselted that the alithmetic and calculus but also selves istence. Standing in extieme opposition geometrical intuition of Euclidean space as our only intellectual model of time to the Platonists are the Formalists, rvho and of the time continuum r'vas one of and (one-dimensional) space. The plop- naintain that the essence of mathemat- the a priori characteristics of the human erties of the continuum were organized ics is not in its meaning but in its folm. mirrd. into a coherent axiomatic framework The problem of the natule of the real- Thus it came as a considerable shock duling the 19th century and have been number line is viewed quite difelently to the i.ntellectual community of the accepted and promulgated with great by adherents to these two schools of early 19th century to learn of the dis- conviction by most contempolary n'rath- thought. A Formalist is likely to say that covery of non-Euclidean geometries, in ematicians. the real-number iine is whatever rve de- r'vhich one of Euclid's axiorns (the paral- Yet in spite of the present unanimity ffne it to be; if we have several com- lel postulate) did not hold. One signifi- of opinion concerning the exact struc- peting defrnitions, then'"ve rvill have sev- cant consequence of this discovery rvas tule of the continuum several significant eral difielent real-number lines and that philosophels and mathematicians alternative systems have been discov- mathemp,tics will be enriched by their began to regard axioms not as self-evi- eled during the past 10 years. Although presence. A Platonist, on the other hand, dent truths but rather as arbitlary rules, none of these new models reflects any would be inclined to rvonder rvhich of subject only to the requiren-rent of con- Iogical flarv in the 19th-century theory, the different models replesented the sistency. For the axiorns to be great the their very existence shorvs quite clear'ly "real" real-number abstlaction of the theory delived from them had to be both that the epistemological foundation of space and time continuum, beautiful and useful, but to be rnathe- mathematical analysis is far from settled. Like beauty, mathematics exists in matics the theoly had only to be con- the eye of the beholder'. Unlike beauty, sistent. Ilthough mathcmatics is sometimes hor,vever, mathemdics enjoys an unpar- Yet in spite of the widespread 20th- I r called a science, it is usually distin- alleled r.rorldwide reputation for objec- century consensus on consistency as the guished from science by its relative in- tivity. If you labor to discover the plop- sole criterion of mathematical truth, the dependence flom empirical considera- erties of the real-number continuum and vast majolity of mathematicians and sci- tions. The intellectual models of science iI I do likewise, we shall leach the same entists maintain a Platonic vierv of the are judged by their ability to explain the conclusions. Science has a similar objec- continuum. Nearly everyone who stud- observed ploperties of the universe, tivity. But whereas the objectivity of ied calculus in the past 50 yeals was ex- rvhereas those of mathematics ale science is a rather plausible consequence pected either to have a clear intuitive judged (by mathematicians) according of its conformity with reality, the objec- (Platonic) image of the real-number line to their consistency and beauty and (by tivity of mathematics is harder to ex- or to believe that all its properties rvere scientists) according to their utility. Pla- plain. Why should the ideal mathemati- consequences of some 10 to 15 suPPos- to's description of .n"rathematics as the cirl universe in your mind be the same as edly self-evident truths that describe discovery of the properties of objects in the one in my mind? what is calleri formally a "complete or'- an ideal universe-the universe of Plir- Trvo centulies ago Immanuel Kant at- deled field" and informally the "real- number system" lsce illustration on tlis finitesimals played a key role in the de- niz had r.vrought, squirmed out of the pagel. velopment of the definitions and nota- dilemma by describing infinitesimals as A cynic might vierv this ploglam as tions of calculus. Indeed, during the "fictions, but useful fictions." N{ean- indoctrination in Platonism, since its l8th and l9th centuries the name for while Bishop Berkeley scorned Nerv- clear purpose is to convince students of what we now call simply "calculus" rvas ton's inffnitesimals (or fluxions, as Nerv- something their plofessors have accept- "calculus of inffnitesimals." What Nerv- ton called them) as "ghosts of departed ed, namely that thele is a unique real- ton and Leibniz did was to shorv horv quantities." number line with certain self-evident one could calculate with infinitesimals But mathematics flourisl.red in spite of properties, and th:rt mathematicians and obtain reliable lesults, results that the philosophers, arvakened by the cal- have succeeded, by listing the axioms of moreover could not be obtained by any culus of inffr.ritesimals as it had not been a complete oldeled field, in capturing other method. since the glories of Athens and Alexan- the essence of this line in a dozen or so The calculus of infinitesimals u'as re- dria. The Platonic image of the real- sentences. Thus mathematicians norv ceived with plofound skepticisrn by number line r.vas as yet only a vague talk about /he real-number line just as many philosophers rvho, follorving Aris- ideal, and calculus developed more as a mathematicians and philosophels of the totle, abhon'ed the absolute inffnite. In descriptive science than as a deductive 18th century talked about f/i.e geon.retry. his Plrysics Alistotle distinguished be- logical system. It rvas not until the 19th Of course, the existence of different trveen the potential infinite and the ab- century that mathematicians begau to geometries does not necessitate the ex- solute infinite, acceptirrg the former but echo the philosophers'skepticism; it rvas istence of different real-rrun'rber lines. rejecting the latter as untenable, or be- only then, as the axioms fol the real- Nonetheless, in I931 Kurt Gtidel shorvecl yond the firm glasp of the human mind. numbel system rvere distilled fi'om the that in any mathematical system suffi- (In taking this position Alistotle rvas great unolganized mass of mystical ciently large to contain arithmetic, there merely reflecting the r,videspread Greek properties, that it became clear that the will alrvays be ur.rdecidable sentences: mistlust of the infinite, most popularly existence of infinitesimals r.vas incon- statements about the system that can be illustrated by the paradores of Zeno.) sistent r'vith these axioms. neither proved nor disproved by Iogical The philosophel Leibniz, somervhat cha- This inconsistency is an immediate deduction fi'om the axioms lsee "Ciidel's grined at rvhat the mathematicinn Leib- eonsequence of one of the most char'- Proof," by Elnest Nagel and James R. Ner'vman; ScmNrrrrc AnrinrceN, lune, 1956]. Gridel's nor.v farnous "ur.rdecid- ability theorem" implies that in the Pla- 1. Addition and Multiplication tonic universe of ideal mathematical ob- lf xandyare real numbers,ihen so are x+ y and xy, jects thele are many-in fact, infinitely many-objects that satisfy the axioms for 2. Associativity the real-number line, since each unde- lf w, x and y are real numbers,then (w + x) + y = w + (x + y) and (wx)y = w(xy). cidable proposition about the leal-num- belline may be true in one ideal model 3. Commutativity ll x and y arereal numbers,thenx+y = y + x aad xy = yx. and false in another. In 1963, more than 30 yeals after 4. Distributivity Godel proved that diffelerrt models for lf w, x and y are real nr,rnrbers,then vt(x+ y) = wx + wy. the real-number axioms must exist, Paul J. Cohen of Stanfold University actuirlly 5. ldentities constlucted sorne models in r.vhich Georg Tlrere exist two special numbers z (or O) and u (or 1) ca1led the zero and the unit Cantor's fnmous "continuum hypothe- that sat sly x+ z = x and xu = xlor all real nunrbersx. sis" ."vas false Isee "Non-Ctrntorian Set 6. Additive lnverse Theory," by Paul J. Cohen and Reuben lf x is any real nr-rnrber, there is another real nr-tmber denoted by -x and called Hersh; ScrriNrrrrc Al.rnnrceN, Decem- the negatrve or aclclltive rrrverse of x that satisf ies x + -x = z, where z is the zero. ber, 1967]. Cohen's methods have been 7. Multiplicative lnverse r, applied extensively over the past eight lf x is any real nurnber except zero, tl-rere is another real number x called tlre rec procalor rlr-r trplicative irrverse of x, that sat sfies x x-1 Lt, where u is years to yield a lalge valiety of altema- = the unrt. 8. Trichotomy tive models in all areas of n'rathematics. lf xand yare real numbers, then eitherx< y or x = y a( x >y. Later in this alticle I shall shorv horv to constluct one of these models for the 9. Transitivity real-number line, but first I should like lfw z (where ziszero), Ihen xw < yw. attack on the Platonic ideal of the real- number line. 1'l . Completion Sr-rppose F rs a set of real numbers that has an upper bound, that is. srrppose p,ver since Isaac Nervtorr and Cottfried there is sonre real nurrber x such that y< x wlrenever y is a member of the set E, L Leibrriz laid the foundatiorrs of cal- Then E has a least r-rpper bound, that is, an upper bouncl x that is less than or culus in the late l7th century, mathema- equal to every other upper bourrd for F. ticians, philosophers and physicists have been quarreling about whether or not ELEVEN AXIOMS are customarily used to define the real-number line, rvhich is knorvn the real-number line contains infinitely formally as a complete ordered field, The first seven axioms define a field; the first l0 define small objects called infinitesimals. In- an ordered field. The symbol > means "greater than"; the symbol ( means "less than."

93 acteristic ploperties of inffnitesimals, tually all 20th-century mathematicians including the oldinary Platonic ideal of namely that any multiple of an infini- have adopted the deffnitions and con- the complete ordered field. Therefore in tesimal is still an infinitesimal. For in- cepts of Cauchy and Weierstrass. constructing our new models we shall stance, the infinitesimal dt used in cal- Within the past decade, hou'ever, feel free to use the old one. (This proc- culus is smaller than every ordinary Abraham Robinson of Yale University ess is quite analogous to the one fol- positive real number, and so is every developed a consistent mathematical Iowed in the construction of the non- multiple m(clx) lor any positive integer theory of infinitesimals. This new theory, Euclidean geometries, where the new m. Accordingly the set M of all multi- called "nonstandard analysis," resusci- models are defined within the standard ples of the infinitesimal dr has many up- tated the discredited ideas of the actual Euclidean space.) per bounds (any ordinary positive real inffnite and actual inffnitesimal and Let us now proceed to construct two number will do), but it has no least up- showed how much of modern mathe- speciffc models for the real-number sys- per bound, since for any given number matics could be consistently translated tem: one that will contain inffnitesimal b that is an uppel bound for M the into a language of infinities and infini- elements and another that will contain smaller number b - dx rvill also be an tesimals. In particular Robinson's new a set that violates Cantor's continuum upper bound for M. Thus M fails to sat- theory created a significant alternative hypothesis. Our method will be to con- isfy the final axiom for the real-number to the mathematician's real-number line struct a general model and then to ob- system, the "completeness" axiom. (alias the complete ordered field), an tain from it the two special models. The Hence at the same time that geome- alternative that contained infinitesimals symbol R will be used throughout the ters were being forced to change their and on which calculus could be done in discussion to stand for the ordinary real- criterion of truth from self-evidence to the spirit of Newton and Leibniz. number line; when we write r e R, we consistency, analysts were discovering Thus by the end of the 196O's there mean that r belongs to, or is a member that the supposedly self-evident axioms were available on two different fronts of, the set R, that is, r is a real number. for the calculus of infinitesimals were several pretenders to what can justiffably The objects in oul new models will be inconsistent. What was to be done? The be called the throne of mathematics: the real-valued functions I defined on some answer, developed principally by Au- real-number line. Although the techni- set S. In other words, f is a rule that as- gustin Cauchy and Karl Weierstrass, cal construction of each of these alterna- signs to each point s belonging to the rvas to abandon the infinitesimals but tives is long and complex, there is a com- set S a real number l(s) belonging to keep the calculus (whence our present paratively simple approach to both Co- the real-numbel set R. The symbolism abbreviated name for the subject). hen's and Robinson's models thlough the I : S + R expresses the fact that f causes Cauchy refolmulated the foundations of theory of probability; I shall norv out- each point of S to be assigned a value calculus by substituting the concept of a Iine this approach, emphasizing its spirit in R; the expression is usually read as "f limit for that of an infinitesimal; his more than its detail. maps S to R" or, more formally, as "l is method was a return to the Aristotelian a function from S to R." We shall use concept of the potential infinite as the pelore beginning this task it would be the symbol Rs to denote the set of all only secule basis fol reasoning. u well to consider a somewhat subtle functions from S to R. Weierstrass extended Cauchy's rvork philosophical issue. If we were attempt- In older for us to see how RB can be- by defining the concept of a limit in ing to prove the real-number line defec- gin to resemble the real-number system, terms of the more primitive concept of tive and to replace it with a better mod- we must understand horv to do addition real numbers. The Cauchy-Weielstrass el, we would have to take great care to and multiplication. Since the elements approach to calculus, now widely taught, avoid assuming the existence of the real- of RB are functions (that is, rules), one placed the epistemological foundation number line while constructing the al- must in effect define the addition and of calculus squarely on the shoulders of ternative. This, however, is not what we multiplication of rules. If f and g are the real-number line. Although many are trying to do. We are, rather, trying functions in Rs, we shall define f + g to users of calculus continue to prefer the to establish the existence of several dif- be the new function in Rs that assigns to intuitive language of infinitesimals, vir- ferent models of the real-number line, each point s belonging to the set S the real number l(s) + g(s); in pure symbols, ff + g)(r) : l(s) * g(s). Multiplication is similar: (lg)(r) : l(r)g(r). It is quite easy to show that the functions in Rs satisfy 1, Suppose S is the two-po rrt set {0,1}. the first six axioms of a complete ordered 2. Then each function /:S+R, which causes each point of S to be assigned a field, listed in the illustration on the pre- value n the real-number set fil can be defined by g v ng two real nurrbers, ceding page. A simple example, how- namely the va ue O and the va ue of aI 1. of lat f ever, suffices to demonstrate that Axiom the "multiplicative inverse" axi- 3 The identrty element u be onging to the trial set fl s given by u(O) = u (1) - 1, No. 7, and the zero eleme nt z be long ng to fl' rs grven by zQ) = z(1) = 0, om, is not satisfied by this set lsee illustration at leftl. Thus RB fails even to be a 4. Nowthefunction h belong ngto fl' def ned by h(O) Oand h(l) = 1 field and consequently falls far short of No, 7 is not yet h not have failsto satrsfy Axiom since h equal toz and does being a model for the real-number line. an rlverse. To correct this situation we shall en- 5. Therefore the set Rs rs rrot arr adequate modelof the real'nunrber ne, gage in a bit of mathematical gerryman- dering. Axiom No. 7 does not say that all elements of RB must have inverses; says does FAILURE 6p ps (the set of all functions from some arbitrary set S to the real-number set it only that if it is true that f R) to satisfy Axiom No. 7, the "multiplicative inverse" axiom, is demonstrated in this ex- not equal zero, then f must have an in- ample. For the purpose of the demonstration S is assumed to be the two.point set [0,1]. verse. Accordingly if we have some func- The conclusion is that Rs falls far short of being a model {or the real-number line. tions f (that is, elements of RB) that do

94 not Ilave inveLses, rve shall simply re- For our second example let us take for because I cannot be u'ritten as a se- defrne truth so that for them the state- the set O the unit interval I that consists quence. (The fact that the urrit interval ment "f equals zero" will be true. More of all real numbers betrveen 0 and l; in I is an uncountable set is actually a very precisely, we shall substitute fol the ab- symbols, I-{reR] 0 < r ( 1}. We famous theolem of Cantor's.) It follorvs solute notion of truth the more flexible build up a measure rit. on the subsets of I that R, rvhich contains I, is also uncount- concept of probable truth. In order to by ffrst assigning to each interval its able. show horv this can be carried out, I must length. Then for a subset A that colsists discuss some defi- are intervals u'e use the \Yf e noru retuln to oul oliginal task of digress for a rvhile to of pieces that \l nitions and examples from elementary maxim that the rvhole is equrrl to the l,uildi'rg nrodels for thl real-tturn- probability. surn of its parts to say tl-rat the rne:lsut'e bel line. We hLrd cor-rstructed the set Rs of A is the sum of the lengths of its of functions flom S to R and slrorved \ I athemrtical plobability is blsed on pieces. Continuing in tl.ris fashion, lve horv this set failed to satisfy Axiom No. *lI a special iunction that ussigrrs to can build a probability measure on I 7 for complete ordered fields. In older each subset A of a given set Q a positive tlrat is kno"vn as the Lebesgue neasure to correct this defect in Rs rve shall r-rotv real number that leplesents the proba- (named after the 2Oth-centuly Frer.rch assurne that S is equipped rvith a prob- bility that a point selected "at random" mathematician Henli Lebesgue). ability measure nl, so thlt rve shall be from the set O will actually be in A. This Since each point r belonging to the able to measure the size of subsets of S. function is called a "probability mea- set 1 is considered to be an intelval of The rnost plimitive sttrteurent that sure" on the set O, and rve shall denote length 0, the Lebesgue nteasure nr. of can be made concerning trvo fuuctions it by nl. The functiorr nr can be thought each point r is 0. According to the cou- in Rs is that they are equal (or unequal). "f of as a rule thnt measures the size of struction just desclibed, each finite set If I and g are given, then equals g" is sets, and the real uutnber rtr(A) can be r.vill also have measule 0, as u'ill each eithc:r true or false. It is true if ald or-rly thought of as the measure, or size, of tlte infinite set that cau be r'vlittetr as a se- if l(r) equals g(r) fol all points r belong- set A" Since the probability is I that a quence x1, xa, xr,. . . (since the nreasule ing to the real-numbel set R. In other point selected at random from O will be of the entire sequence is just the sum of rvolds, f c'quals g if and only if f and g in Q, the measure of Q must be l. In ad- the rneasules of each of its points). Since c\pl'ess the strme rule. If tl'rele is in R so dition lve require of nr only that it satis- tl-re set of rational numbers in 1 can be much as one poir.rt y rvhere l(y) does not ly the (self-evidcnt?) maxim that the listed in a sequence (I, I/2, l/3, 2/3, equal g(y), then rve ale forced to say rvhole is equ:rl to the sum of its parts: if l/4, 2/1, 3/1, ...) it too must have that the function f does not equal the the ,'vhole set A is bloken dorvn into measule 0, even though it appears to be function g-even if they aglee at every finitely or infinitely marry distinct parts a very large set. In othel rvords, the other point of R. Logicians call the terrns "false" Ar, A', .. ., An,. . ., then nt(A) : n(A1) -f chances are nil thtrt a numbel selected "tlue" and tluth values. In tra- rn(A1) f ... 1 nr(A,) * .... at random from the unit intelval rvill be ditional logic (rvhich rve have used Befole applying tliis tneasule to our rational. throughout this discussion) sentences set Rs I shall give trvo inpot'tarnt ex:tn- This minor paladox leads directly to hirve only trvo truth vllues: true or fa]se' ples of measules that rvill be used as the a major paradox concerniug the Le- We propose to change this tladition basis for our tu'o diffelent rnodels for besgue measure. If I select a nrlmber at by assigning to sentences not just one the real-number line. Filst, suppose O random from the unit interval, the prob- of two possible tluth virlues but a leal is the set N of positive integers: in syrn- ability is 0 that the number so chosen numbel that expresses the probtrbility of pleviously truth; this number may be 0 (if the sen- bols, N: { 1, 2, 3, ... } . We shall de- rvill equal some particular fine on N a very crucle measure by classi- selected number; intuitively there ru'e tence lus no chauce of being tlue) or I fying subsets of N as small if they are just too many nurnbers in the unit inter- (if the seltence is certain to be true) or finite and lalge if they ale cofinite, that varl to choose from. Yet the probability any nunrber in betu'eeu. We can accorn- is, if their complement (the set of posi- that I shall pick some number in the unit plislr ilris coup d'6tat ou Rs because rve tive integels not in them) is finite. Any interval is 1. Thus in this cnse it apperrls no."v hnve a [leans of measuling the size sn-rall set rvill be given measule 0 and that the r'vhole is not equal to the sun'r or probability of subsets of S. In par- any lalge set rvill be given meirsure 1. of its parts! This dilemma lias plagued ticular rve shall srry thnt the probtrbility Norv, the measul'e on N as just defined v:rrious philosophels throughout histoly; that "l equnls g" is tlue is the neasure fails to measure sets such as the set of horv can something of positive rveight be of tlie set of points in S rvhere it really even integers, since neither it nor its made up of parts that etrch have zet'o is true, that is, the measut'e of the set of complement is fir.rite. Tl.re ploblem is rveight? all points s belonging to S tltat satisfy thtrt the set of even firtegels is neither The tlouble is that there al'e too martv the lequirenrent tlrat l(s) equal g(s). If smirll rol large but soruehorv in be- points in the unit interval. Cantor called rve delote this plobability truth value t'"veen. We car colrect tl-ris deficiency in countable those inffnite sets that cau be of "f equals g" bV If - gl, rve hrrve :f = our measule rn by systematically cltrssi- rvlitten as a sequellce xb N., x:t, . . ., orte Sl : rn1{s e S I f(s) : g(s)}). {ying each intermediate set as eitl"rer poirt fol eirch integer'; othel infinite scts Of course, thele trre a lot of scltetrces sl'rall ol lalge on an albitrary bnsis, sub- are called uncount:rble. The plinciple about RN that are firr mole cterm has measure 0. Similar tion in truth v:rlues succeed is, begin- either 0 or I to every subset of N; lve reasoning does not allorv us to corrclude ning rvith the aton'ric sentences, to sys- shall call lr the cofinite rneasuLe on N. that the measure of the interval I is 0 ternatically rvork our rvay through the

95 entire catalogue of sentences about RB and determine how to assign a probability to each sentence in such a rvay that it represents the measure of the set of points in S r'vhele the sentence is true.

f he basic idea belrind tlris sclreme wls r orrtljned more than 100 years ago by the English mathematician George lEn.\ [nl Boole, who noticed that there was a considerable similality (in fact, a mathematical isomorphism) between the rules that govern the use in Iogic of the connectives "and," "or" and "not," and the operations on sets of, respectively, intersection, union and complementation [see illustration at left]. The abstract mathematical system that represents th'is structure is now called a Boolean algebra. To see how Boole's system can be used to solve our problem let us con- sidel a simple example. If ) is a sentence about Rs, we shall denote by [:] the set of all points in S rvhere ) is true, and by l:l the measure of [)]. Thus ]:l : m([>]) is the probability truth value of the sentence I >. If II is anothel sentence about Rs, trt {l:l- then one of the Boolean relationships \ is that and n] : fl In \ [) [)] [il]. \ rvords, the set of points in S where both \ ) and fI ale tlue is the intersection (denoted by 1-l) of the set where ) is true rvith the set whele II is true. Perhaps the only detail in our plan (to extend the probability truth values from atomic sentences to all sentences) that requires special comment is the treat- ment of the quantiffers V (meaning "for all") and 3 ("there exists"). The sentence "Vsfl(s)," which we read as "For all points s belonging to the set S, the sentence II(s) is true," means that II(s1) and II(sr) and ... (ad infinitum) are all true. In other words, V is an infinite repetition of "and." Thus it is appropri- ate that the set operation that corre- S*IT] sponds to V is the inffnite intersection, since the ordinary (finite) intersection corresponds to "and." Similarly, 3 rep- lesents an inffnite "or," so that its corre- sponding set opelation is the infinite union. In summary, then, although we have by no means discussed all the relevant detail, it is possible by means of the Boolean translation to determine for each sentence ) about RB a subset [)] govern CONSIDERABLE SIMILARITY exists between the rules that the use in logic of of S on which ) is true. The measure of ooor" oonot," the on sets of, respectively, inter- the connectives 'oand," and and operations this set is the probabiiity truth value of section (d), union (b) and complementation (c). This similarity (in fact, a mathematical ), denoted by A sentence I will be isomorphisrn) was discovered by the lgth-century English mathematician George Boole, l:1. considered valid and only if : 1; and the abstract system based on it is norv called Boolean algebra. The symbol stands if l>l [)] in other words, ) is valid if and only if for the set where the sentence > is true; IfI]l siSnifies the set where the sentence JI is true. The intersection of these two sets, symbolized n is the set where the sentence the measure of the set where ) fails to [>] In], "no ") anil II" is true; the union of these two sets, [>n U [II], is the set where ") or fl" is hold is 0, so that there is chance" of true. In the example of complementation S - [)] signifies the set where 'onot )," is true. failule. The examples of measures given v6 above reveal the essential distinction be- in question is cofinite. Thus in RS/m it Let 1 denote the unit interval (I : tween the concept of truth in the two- is valid to say that g is greater than n {renl0(r(r}), and Iet T be a valued logic and the concept of validity for all n,. This is just what we mean when set rvhose cardinal number is bigger' in the probability-valued logic; for ) to \\,e say that g is infinite. than c. Then we define S to be tlre set be true it must hold without exceptiotr We have now achieved the first of t'':\f :T-+I ]. The points of S ale for all points s belonging to the set S, our two goals: the description of a math- functions flom T to 1. The measure on but it remains valid even if it fails to ematically consistent model of the real S, rvhose details need not concern us, hold on a fairly large subset, provided numbers that contains infinitesimals. is a rather natulal extension to l?' of only that the measule of the exceptior.ral The other type of model, in rvhich Can- the Lebesgue measule on 1 desclibed set is 0. tor''s continuum hypothesis fails, re- above. Our ner'v model for the real-num- Let us denote by RB/m' the set RB quires some more wolk. First, hor'vever, ber line is RB /m' where S equals 1". rvith the new concept of vaiidity as de- I shall explain Cantor''s hypothesis. The fact that the points of S are real- termined by the measule m. We have ly functions can be the sout'ce of some seen that under the former notion of I antor developed his theoly of cardi- conceptual difficulty if lve confuse the truth Rs did not satisfy the axioms for v nal numbels by defining two scts as functions in S :1?' rvith those in Rs. a complete oldered field. But Rsz'rn will being of equal size (or of the same "caL- Nlathem:rticians avoid this confusion by always be an ordered field because, on dinality") if thele is some function, or emphasizing in their minds that S is a sel the basis of the concept of validity, it rule, that establishes a one-to-one cor- of functions rather than a set of functions. will always satisfy axioms No. I through respondence between them. All finite We put up with this double meaning be- No. 10. Furthermore, fol certain choices sets u'ith the same number of elements cause the entire key to Scott's model is of S and m it rvill also satisfy the com- have the same cardinality, and all in- a deliberate pun formed by interchang- pletion axiom, so that in these cases finite sequences have the same cardi- ing the role of functions and points. Rs,/rn will be a complete ordered field. nality as the set of integers. (The sets in To be precise, r've observe that each Suppose S is the set of positive irr- this latter category are those that r've point / belonging to the set T can be tcgers N the cofinite measul'e t?? have called countable.) Cantor's gleat thought of as a function in Rs tl'rat as- "vith defined above in our' first example of achievement '"vas to assign sizes (cardi- signs to each function f belonging to measures; nr assigns to finite sets the nal numbers) even to the uncountable 11 : S the real number l(t) belonging to value 0, to cofinite sets the value 1 and sets. In particular he identified a class 1. If rve call the function t by the name to irrtermediate sets eithel 0 or 1 accold- of sets of the same cardinality as the set i, then i(i) equals f(r) whenever f is a ing to some arbitrary but consistent pat- of real numbers R; he called the cardi- member of S. Norv our problem is to ffnd tern. In this case we can picture each nal number of sets in this class c, for a subset of Rs that is urrcountable, yet function I belonging to Rs as a sequcnce continuum. Sets of size c are those that not as large as RB. But just how Iarge is collespon- Rs? Since it contains a i lor each , be- f ,, f ,, f t,. . ., r.vhere for each integel i be- can be put into one-to-one longing to T, the cardinality of Rs must longing to N: S, 11 is the real number dence with the real numbers. that f assigns to i; f; is just rvhat we usu- Cantor's continuum hypothesis is be at least as large as that of T, but T ally call l(l). For example, the identity simply this: Every infinite set of real was selected so that its cardinality r'vas function zr is the sequence 1, l, 1, 1, . . ., numbels is either countable or of car- larger than c. Hence the cardinality of since a, - u(i) - 1 for each i belonging dinality c. There are no sets of interme- Rs, oul new real-number line, is gleater' to N: S. diate size. This hypothesis was formu- than c, the caldinality of the oid real- Given S and tn as defined above, lated (but not proved) by Cantor late in number lirre R. Ils/m contains both infirritesimals and the 19th century, and it rvas listed in Since the nerv real-number line is so infir'rite elements. The futrction I de- 1900 by David Hilbert as the first of vely much bigger than the old one, it fined by the sequence 1,1/2, L/3, l/1, his celebrated 23 problems for 20th-cen- rvill be much easier to find in the nerv . . . is an infinitesimal, since if rve multi- tury mathematics. We shall norv out- line an uncountable set that is not of the ply it by any integer r.r., the resulting se- line a version of Cohen's 1963 model same cardinality as the entire line. All we quence n, n/2, n/3, n/4, .. . is smaller that shorvs conclusively that Cantor's have to do is select a subset P of T r.vith tharr the identity element u except for continuum hypothesis could never be the ploperty that the caldinality of P '/' tlie finite set {n, n/2, n/3,..., n/(rt- proved from the oldinaly axioms of set is c. (If rve had been thinking of as the 7), n/nl, rvhich has measure 0. In other' theoly and leal numbers. porvel set of R, then we could take R u'olds, if n(n) is thc sentence "nf is less This nerv model, developed principal- itself for P.) Tlien the set P that consists functions fol t belonging to P than u," then ln(rz)l : I because{i c Nl ly by Dana S. Scott of Princeton Univer- of iill the i nf (i) < u(i)\ has measrrle 1. Thus sity, involves the use of a rather compli- is a subset of Rs of size c. Thus P is un- RB,/nz. it is not as large as RB. lvntt(n)l : 1, rvhich is to say that for cated set S in the general model countable, yet every n, rif is less than u. Thus f is an We first have to find a very large set T, This completes oul progt'arn of con- infinitesimal since only an infinitesimal a set rvhose cardinality is bigger than c, stlucting lelv models of the reai-number could have this ploperty. the cardinal number of R. The ploof line. To summarize bliefly, each model For a difierent pelspective let us look that such sets exist is anothel of Cantor''s consists of functions flon some set S to at the multiplicative inverse of f; f rnust significarrt contributions to mathen'ratics. the ordinaly real-number line R. The have an inverse since Rs,/rn is a field. An example of such a set is r'vhat is trvo-valued true-false logic of the old In reality the function g defined by the cnlled the power set of R: the set that system is replaced with a mole flexible sequence L, 2, 3,4, . . . is the invelse of consists of all subsets of R. Cantot''s probability-valued logic determined by f since, clear'ly, gf : u. Just as f is 1n proof that the cardinal number of this some measure on the set S. Validity in inffnitesimal, so g must be infirrite. In set is indeed bigger than c is essentially the ner'v model means truth rvith prob- fact, if n denotes the constant function the same as his proof that R, or equiva- ability 1, a definition that glants excep- n, n, n, ..., then lg> rrl:m(\i1gi) lently the unit interval 1, is strictly big- tions to sets of measure 0. The particu- nt\) - m(ltlt> nl): 1, since the set ger than the set of integers. lar models that contain infinitesimals or 97 counterexamples to the continuum hy- limited to real numbels. All the axioms quantum mechanics. The most common pothesis are then folmed by selecting of a complete ordered field are fir'st- delivation of quantum mechanics fi'om special sets S r'vith special probability order sentences except fol the comple- classical mechanics is accomplished by measul'es ?n. tion axiom (No. 1l), rvhich talks about a substituting in the classical theoly a wave ploperty of cll subsets. packet or plobability cloud for each clas- Jn ihe pleccding dcvelopment ol thc Norv, the models of nonstandard sicnl particle. In the quantum model the I tr,'n 'r'"ru models o[ the rcal-numlrer analysis (such as the fir'st of oul tr.vo ex- locrrtion of the palticle in the cloud is ljne I not only omitted all proofs and amples) have the property that every given bv a plobability distribution and technical detail but I also failed to nen- fir'st-oldcr sentence that is true about R the prilticle itself becomes a random tion a most irnpoltant distinction be- is true about them, and convelsely, In varirrble. tq'een thesc trl'o models: the Cohen- othel r.r'ords, it is impossible to distin- The nerv models for tl"re leal-number' Scott model (u4rere S equals 1'l ) is a guisl'r betrveen these nerv models and R line that u.ele dcscribed above ale cornplete ordered field, the by the use of first-oldel sentences. Since forn'red in just this rvay. Wc substituted "vl'rereas Robinson model (*'hcle S equals N) is a large part of calculus can be expresscd fol the oldinary leal numl>crs cet'tain just an oldeled field. The fact that Rob- in fir'st-ordel sentences, it is quite litelai- real-valued functions defined on a plob- insorr's n.rodel fails to be cornplete should ly true that for most practical purposes ability space. These functions are ple- come as no surprise, fot's'e did see that the nonstandard real-number line is the cisely u4rrrt statisticians and probribility any r-eal-number line that contained in- same as the oldinaly leal-number line. tl'reorists call random variables. The sub- finitesimals could not satisfy Axiorn No. The efiort required to shorv that the stitution of landon-r valiables for real 11. To complete the discussion of these completeness axiorn is satisfied by some numbers \\ras accompanied by a col're- models I should explain trvo things: Why palticulal model Rs/rn is decidedly sponding chauge from a trvo-valued sl-rould the tu'o r-r'rodels behave diffelent- more stLenuous than any of the ideas dis- logic to a more goneral scheme in u,hich ly u'ith r-cspect to Axiom No. 11, and by cussed above. Not only is tl-re axiom it- the truth value of a statement about the ',r'hat ligl'rt can we call ar.r object (such self mole difficult to verify, but also be- real nurnbels is expressed by a probabil- as Robirison's model) that fails to satisfy fore rve can even consider it rve must ity. This tlansition from trvo-valued to Axion'r No. 11 a leal-numbel line? wolk quite hard to extend the machinery probability-valucd iogic also occuls in The ansrvers to both questions de- of assigning plobability tluth values the derivation of action of rvliting dorvrr r9'' - The set oi a f Lrrct orrs f rorl 5 to R axiorns and theolems (r'vhich is to the m(,4) - The probab ty nreasL.re of the set A matl-rematician u4rat experimental ob- servation is to tl-re physicist) a limitation I A r' :rr R " on the amount of infoln'ration that can IIl - f he set ol all po nts n S where I s trlre be derived from a set of axion-rs (in tlie case) and a limitation on Thc truth va,l-re o{ the sertence I, that s the probab 1 ty measure of II] Giidel-Cohen a so written m(flI[) the capability of axiom systems to describe models uniquely (in Robinson's fl - Another scntcnce about t9s theory). The first of these tu'o limita- n : ntersect on of two sets tior.rs is the substance of Gcidel's unde- \, - Union of two sets cidability theorern, rvhereas the second is one version of a more general result V: "Fora ' knor.vn in mathematical logic as the I : "There ex sts" Lorvenhein-r-Skolem theolem. In differ'- c - Cartor's synrbol for the card nal nuinber of tlre cont nuum ent ways each limitation e\plesscs a basic uncertainty in mathematics.

t is not surplising that the intloduction GLOSSARY OF SYI-IROLS used in this article is provided in this il]ustration. The symbol rI c, used by Georg Cantor to signify continuum, denotes the cardinal number of sets in the of "unceltainty" results in mlthemat- class of infinite, uncountable sets of the same cardinality, or size, as the set of real numbers R. ics rvas accompanied by fundamental gR disagleement about their significance, just as Heisenbelg's unceltainty plinciple precipitnted arnong physicists a major dispute about its ultimate significirnce. \'Iost physicists tend to agree with Niels Bohr that the uncertninty in quantum mechanics is a fundamental larv of na- To preserve your copies ture. Some, however', stippolt tl-re position of Albelt Einstein, who argued thrrt the uncertainty principle melely express- SCIEI\TIFIC es a limitation on the plesent conceptr:al folrnulation of physics. UI^+ AMERIC-{N These two opposite poles in physics re{lect apploximtrtely the positions of tl're Formrrlist and the Platonist in n'rath- ernatics. The Forn'ralist believes tlrnt tT A choice of handsome and durable library files-or bind- mathematics is pule folm, and since the SCIENTIFIC AMERICAN. "ulcertainty" theolems of mathematics ers-for your copies of limit the scope and por.r,er of the formal- !l Both styles bound in dark green library fabric stamped isrn rvith which mathematics is con- gold leaf. cerned, then they must forever limit in mathematics itself. For a Foln'ralist the 'tT Files-and binders*prepacked for sale in pairs only. question of whethel the "real" real-number line satisfies the continuum hypothe- Indexfor entire year in December issue.* sis ol contains infinitesimals is as meaningless as is, for a physicist such as Bohr', the question of whether or not an electron "really" has an exact simultaneous FILES shown at lelt position and velocity. Robinson himself Hold 6 issues in each. reflects this position r'vhen he rvrites (in a paper titled "Folmalism 64") that "any Single copies easily accessible. mention of infinite totalities is literally Price: $4.00 for each pair meaningless." (U.S.A. only). In contrast, the Platonists, rv}ro count among their number even Gtjdel him- Address your order, enclosing self, believe (like Einsteilr) that the un- check or money order, decidabiiity in mathematics is a state- to: Department 6F ment about the inherent limitations of our present axiomatic mode of investiga- tion and not about the mathematical objects themselves. Gridel has algued that there is no leason to believe more in the objective existence of a physical object such as an electron than in the objective BINDERS shown at right eristence of a mathematical object such as the real-number line. And one who Hold 6 issues in each. believes that the real-number line has Copies open flat. an existence in a Platonic univelse can Irarldly avoid wondeling about rvhich of Price: $4.50 for each pair our several models is the more accurate (U.S.A. only). descliption of this Platonic continuum. Address your order, enclosing It is ironic that Robinson, the le-crea- check or money order, tol of infinitesimals, does not believe Department 68 they really exist, rvhereas Giidel, the to: plophet of undecid:rbility, believes in a Platoriic univelse in which the ploper- mathematical visible ties of objects are New York City residents please add rys sales tax for those rvho liave the eyes to see. Per- Other NYS residents please add 4/s state sales tax plus local tax harps these men are merely reflecting an *Supply of back copies of scrENTrFrc AMERTcAN is limited. To replace intense rnodesty about the significance missing copies, mail request with your order for files or binders. of their or.vn achievements. It seems unlikely, however, that rvithin the next few generations mathematicians rvill be SCIENTIFIC AMERICAN,4I5 Madison Ave., New York, N.Y. 10017 erble to aglee on whethel every mathematical statement thnt is tlue is also krrott irble.