A Note on the History of the Cantor Set and Cantor Function
Author(s): Julian F. Fleron
Source: Mathematics Magazine, Vol. 67, No. 2 (Apr., 1994), pp. 136-140
Published by: Mathematical Association of America
Stable URL: http://www.jstor.org/stable/2690689 .
Accessed: 14/10/2013 11:42
Your use of the JSTOR archive indicates your acceptance of the Terms & Conditions of Use, available at .
http://www.jstor.org/page/info/about/policies/terms.jsp
.
JSTOR is a not-for-profit service that helps scholars, researchers, and students discover, use, and build upon a wide range of
content in a trusted digital archive. We use information technology and tools to increase productivity and facilitate new forms
of scholarship. For more information about JSTOR, please contact support@jstor.org.
.
Mathematical Association of America is collaborating with JSTOR to digitize, preserve and extend access to
Mathematics Magazine.
http://www.jstor.org
This content downloaded from 132.178.2.64 on Mon, 14 Oct 2013 11:42:52 AM
All use subject to JSTOR Terms and Conditions
MATHEMATICS MAGAZINE
136
A Noteon theHistory
oftheCantorSet
and CantorFunction
JULIAN F. FLERON
SUNY at Albany
Albany,New York12222
A searchthroughthe primaryand secondaryliteratureon Cantoryieldslittleabout
the historyofthe Cantorset and Cantorfunction.
In thisnote,we wouldlike to give
some of thathistory,a sketchof the ideas underconsiderationat the timeof their
discovery,and a hypothesisregardinghow Cantorcame upon them.In particular,
Cantorwas not the firstto discover"Cantorsets." Moreover,althoughthe original
discoveryof Cantorsets had a decidedlygeometricflavor,Cantor'sdiscoveryof the
Cantorset and Cantorfunction
was neithermotivatedby geometry
nordid it involve
geometry,
even thoughthisis how theseobjectsare oftenintroduced(see e.g. [1]). In
fact,Cantormayhave come upon themthrougha purelyarithmetic
program.
The systematic
studyofpointset topologyon the real line arose duringthe period
1870-1885 as mathematicians
investigated
twoproblems:
could be integrated,
and
1) conditionsunderwhicha function
2) uniquenessof trigonometric
series.
It was withinthe framework
of these investigationsthat the two apparently
independentdiscoveriesofthe Cantorset were made; each discoverylinkedto one of
theseproblems.
BernhardRiemann(1826-1866) spentconsiderabletimeon the firstquestion,and
suggestedconditionshe thoughtmightprovidean answer.Althoughwe will not
discuss the two formshis conditionstook(see [2, pp. 17-18]), we note thatone of
these conditionsis importantas it eventuallyled to the developmentof measure
theoreticintegration
[2, p. 28]. An important
step in thisdirectionwas the workof
HermannHankel (1839-1873) duringthe early 1870s. Hankel showed,withinthe
framework
of Riemann,thatthe integrability
of a functiondepends on the natureof
certainsets of pointsrelatedto the function.In particular,"a function,
he [Hankel]
wouldbe Riemann-integrable
thought,
if,and onlyif,it were pointwisediscontinuous
thatforeverya > 0 theset ofpointsx at
[2, p. 30],"meaning,in modernterminology,
which the functionoscillatedby more than a- in every neighborhoodof x was
nowheredense. Basic to Hankel's reasoningwas his belief that sets of the form
forall nowheredense subsetsofthereal line.Workingunder
{1/2'} were prototypes
thisassumptionHankelclaimedthatall nowheredense subsetsofthe real line could
be enclosedin intervalsof arbitrarily
smalltotallength(i.e. had zero outercontent)
[2, p. 30]. As we shall see, thisis notthe case. (See also [3].)
intothenatureofcertainpointsets wouldbecome
AlthoughHankel'sinvestigation
extremelyimportant,"as was the case with Dirichlet and Lipschitz,it was the
of the possibilitiesof infinitesets-in particular,
inadequacyof his understanding
nowheredense sets-that led him astray.It was not until it was discoveredthat
nowheredense sets can have positiveoutercontentthatthe importanceofnegligible
sets in the measure-theoretic
sense was recognized[2, p. 32]." The discoveryof such
sets, nowheredense sets withpositiveoutercontent,was made by H. J. S. Smith
(1826-1883), SavilianProfessorof Geometryat Oxford,in a paper [4] of 1875. After
an expositionofthe integration
ofdiscontinuous
Smithpresenteda method
functions,
forconstructing
nowheredense sets thatwere muchmore"substantial"thanthe set
This content downloaded from 132.178.2.64 on Mon, 14 Oct 2013 11:42:52 AM
All use subject to JSTOR Terms and Conditions
137
VOL. 67, NO. 2, APRIL 1994
he observedthe following:
{1/2'}. Specifically,
Let mnbe any givenintegralnumbergreaterthan2. Divide the intervalfrom0 to 1 into mnequal
parts;and exemptthe last segmentfromany subsequentdivision.Divide each of the remaining
in - 1 segmentsintom equal parts;and exemptthe last segmentsfromanysubsequentsubdivision.
we shallobtainan infinite
numberofpointsof division
If thisoperationbe continuedad infinitum,
P upon the line from0 to 1. These pointslie in loose order... [4, p. 147].
Smith's'loose order'is whatwe referto as nowheredense.
In modernterminology
discussionis the assumptionthatthe exemptedintervals
Implicitin Smith'sfurther
are open, so the resultingset is closed. Today thisset would be knownas a general
Cantorset,and thisseems to be the firstpublishedrecordof such a set.
Later in the same paper, Smithshows thatby dividingthe intervalsremaining
beforethe nth step into Mn equal partsand exemptingthe last segmentfromeach
subdivisionwe obtaina nowheredense set ofpositiveoutercontent.Smithwas well
as he states,"the resultobtainedin thelast
ofthisdiscovery,
awareoftheimportance
example deserves attention,because it is opposed to a theoryof discontinuous
whichhas receivedthe sanctionof an eminentgeometer,Dr. Hermann
functions,
in the contemporary
Hankel [4, p. 149]." He continuesby explainingthe difficulties
examples
illuminate.
that
his
theoriesof integration
to note thatan editor'sremarkat the conclusionof Smith'spaper
It is interesting
states"thispaper,thoughit was notread, was offeredto the societyand acceptedin
the usual manner."(Emphasis added.)1 In fact,this paper went largelyunnoticed
Smith'scrucial
on the Europeancontinentand unfortunately
amongmathematicians
almosta decade later,of similar
discoverieslay unknown.It tookthe rediscovery,
of contemporary
theoriesof integration
ideas by Cantorto illuminatethe difficulties
integration.
and to begin the evolutionofmeasure-theoretic
GeorgCantor(1845-1918) came to the studyofpointset topologyaftercompleting
a thesison numbertheoryin Berlinin 1867. He began workingwithEduard Heine
(1821-1881) at the Universityof Halle on the question of the uniqueness of
series.This questioncan be posed as follows:
trigonometric
If forall x exceptthosein someset P we have
00
2ao+
E
(an cos(nx) + bnsin(nx))= 0
n1
mustall thecoefficients
an and bnbe zero?
"when the convergencewas uniform
Heine answeredthe questionin the affirmative
in general with respectto the set P, which is thus takento be finite[2, p. 23],"
thatdid
on anysubinterval
thatthe convergencewas uniform
meaning,by definition,
notcontainanypointsof the finiteset P.
withthis problem.In papers [5, 6] of 1870 and
Cantorproceeded much further
in general"and
1871,he removedtheassumptionthattheconvergencewas "uniform
set. In doingso he beganto lookat
beganto considerthecase when P was an infinite
whatwe now considerthe fundamental
pointset topologyofthe real line. In a paper
[8] of 1872, Cantorintroducedthe notionof a limitpointof a set thathe definedas
we do today,callingthe limitpointsofa set P the derivedset,whichhe denotedby
P'. Then P" was thederivedset of P', and so on. Cantorshowedthatiftheset P was
IIt is possible that "not read" simplymeant thatthe paper was not presentedat a meetingof the
of thisnote,one mustconsiderthat
London MathematicalSociety.However,in weighingthe significance
in vols. 3-10 of the Proceedingsof the London MathematicalSociety(1871-1879), and perhaps even
no otherpaperwas similarlynoted.
further,
This content downloaded from 132.178.2.64 on Mon, 14 Oct 2013 11:42:52 AM
All use subject to JSTOR Terms and Conditions
MATHEMATICS MAGAZINE
138
series Ia0 +
such that p(n) = 0 for some integer n and the trigonometric
=
+
on
then
all
of
the
coefficients
P,
except
possibly
0,
sin(nx))
cos(nx)
bn
Eon=(an
had to be zero. Cantor'sworkon thisproblemwas "decisive"[9, p. 49], and doubly
role in muchofhis upcoming
as his derivedsets would play an important
important
work.
In theyears1879-1884 Cantorwrotea seriesofpapersentitled"Uber unendliche,
[10-15]," thatcontainedthe firstsystematictreatlinearePunktmannichfaltigkeiten
ofthreetermsin
mentofthepointset topologyofthereal line.2It is theintroduction
of this series Cantor
this series thatconcernsus mosthere. In the firstinstallment
dense (literally"uiberalldicht"),a
defineswhatit meantfora set to be everywhere
termwhose usage is still current.He gives a few examples,includingthe set of
numbersoftheform22n?+1/211 wheren and m are integers,and continuesby noting
the relationshipbetween everywheredense sets and theirderived sets. Namely,
dense in (a, ,B) if[and onlyif] P' = (a, ,3) [10, pp. 2-3]. In
P c (a, ,3) is everywhere
the fifthinstallmentof this series Cantordiscusses the partitionof a set into two
of a
componentsthathe termsreducibleand perfect[14, p. 575]. His definitionperfectset is also stillcurrent:A set P is perfectprovidedthat P = P'.
Cantor states that
Afterintroducingthe term perfectin the fifthinstallment,
perfectsets need not be everywheredense [14, p. 575]. In the footnoteto this
statementCantorintroducesthe set thathas become knownas the Cantor(ternary)
set: The set ofreal numbersof the form
Cv
Cl
x=3 + * + cv
31) +*
perfectset
wherec1 is 0 or 2 foreach integerv. Cantornotesthatthisis an infinite,
withthe propertythatit is not everywheredense in any interval,regardlessof how
smalltheintervalis takento be. We are givenno indicationofhowCantorcame upon
thisset.
papers,
During the time Cantorwas workingon the 'Punktmannichfaltigkeiten'
otherswere workingon extensionsof the FundamentalTheorem of Calculus to
functions.
Cantoraddressedthisissue in a letter[18] dated November
discontinuous
1883, in whichhe definesthe Cantorset,just as it was definedin the paper [14] of
1883 (whichhad actuallybeen writtenin Octoberof 1882). However,in theletterhe
It
the firstknownappearanceof thisfunction.
goes on to definethe Cantorfunction,
whose values
is firstdefinedon the complementofthe Cantorset to be the function
are
I (c2
2
1
foranynumberbetween
6 3 3+ *
+ 3_l-+
3/1-1
31L
and b=
3-l + .+
31`1
2
+ 3/i'
whereeach c1V
is 0 or 2. Cantorthenconcludesthissectionoftheletterby notingthat
on [0,1].
can be extendednaturallyto a continuousincreasingfunction
thisfunction
to Harnack'sextensionofthe FundamentalTheorem
Thatservesas a counterexample
which was in vogue at the time (see e.g.
of Calculus to discontinuousfunctions,
[2, p. 60]). We are givenno indicationof how Cantorcame upon thisfunction.
There are two othertopicsthatinterestedCantorthatwe would like to mention
and it is
witharithmetic
constructions
because theyare indicativeof Cantor'sfacility
(see [16, 17]),
2in addition,thesepaperscontainedmanyothertopicsthathad farreachingimplications
of higherorder derived sets thatmarkedthe "beginningsof Cantor's
includingCantor'sinvestigation
numbers[2, p. 72]."
theoryoftransfinite
This content downloaded from 132.178.2.64 on Mon, 14 Oct 2013 11:42:52 AM
All use subject to JSTOR Terms and Conditions
139
VOL. 67, NO. 2, APRIL 1994
possiblywithinthis settingthat Cantor came upon the Cantor set and Cantor
function.First,Cantorspent some time in the mid 1870s consideringthe possible
betweena line and a plane,a questionmostof
existenceofa bijectivecorrespondence
In 1877,in a letterto RichardDedekind
as
absurd.
dismissed
had
his contemporaries
had
found such a correspondence.This
that
he
explained
(1831-1916), Cantor
as
follows:
can
be
expressed
"correspondence"
Let(xl,x2)be a pointintheunitsquare,and let O-X1 lX 2X1,3 ... andOX2 1X2 22X2,3* be decimal
Map the point(x1, x2) to the pointon the real line whose
expansionsof xl and x2 respectively.
is O.X1 1X2,1X1,2X2,2 ... (See e.g.[19,p. 187].)
decimalexpansion
Dedekind pointedout that therewas a problemwith this approach.The decimal
expansionsof rationalsare not unique, so to avoid duplicationwe must not allow
expansionsof some type, say expansionsthat contain infinitestringsof zeros.
However, by disallowingexpansionswith infinitestringsof zeros, the irrational
number0.11010201010201010102... could neverbe obtainedunderCantor'scorrespondence.
This reasoningdoes howevergive us an injectionof [0,1] X [0,1] into[0,1]. It is
trivialto findan injectionof[0,1] into[0,1] X [0,1]. These twofacts,togetherwiththe
Theorem(if thereare injectionsofthe set A intothe set B and
Schroeder-Bernstein
betweenA and B; see
thenthereis a bijectivecorrespondence
B intoA respectively,
between[0,1]
e.g. [20]), allow us to concludethatthereis a bijectivecorrespondence
duringtheperiodin question
and [0,1] X [0,1]. However,set theorywas in itsinfancy
and it would be 20 years beforeE. Schroederand Felix Bernsteinindependently
provedthe theoremthatbears theirnames[16, p. 172-173] and occasionallyCantor's
name as well (e.g. [21, 22]). So thiswas notan optionforCantor.
exhibita bijection.To do thishe modifiedhis
Instead,Cantorneeded to explicitly
[23]. Denote the continuedfraction
previousapproachto use continuedfractions
1
al +
a2
where a1, a2,a3,... > 0 are integers.
by [a1, a2,a3,...]
1
+a
number,
if,and onlyif,it representsan irrational
Since a continuedfraction
is infinite
is unique [see e.g. 24], Cantorcould set up the
in which case the representation
correspondence
([a,,,
al2,...],
, [a 1,
a2,2, ...]
[a2,1,
a2 1, .. an1, al,2, a2,2,.
[an,1an,2
...
'
, an,2)
I
in (0, 1)n = (0, 1) X (0, 1) X ... x (0, 1) and irrationals
betweenn-tuplesofirrationals
of the previousapproachand gives a bijective
in (0, 1). This avoids the difficulties
_
between([0, 1] Q)n and [0,1] - Q. Cantorthentookgreatlengths
correspondence
between[0,1] and [0,1] - Q. Repeated
to provetherewas a bijectivecorrespondence
gives a bijective
applicationof thisfactcombinedwiththe previouscorrespondence
and
[0,
1].
between[0, 1]n
correspondence
Secondly,it is knownthatCantorstudiedbinaryexpansions.In fact:
denotedby o, could be representedas
Cantorrecognisedthatthe powerof the linearcontinuum,
well by [thepowerof]the set of all representations:
x=
wheref(v)
=
2
+
+
2v
+
0 or 1 [foreach integerv] [19, p. 209].
This content downloaded from 132.178.2.64 on Mon, 14 Oct 2013 11:42:52 AM
All use subject to JSTOR Terms and Conditions
140
MATHEMATICS MAGAZINE
There is, so it seems,no substantiveevidence about how Cantorcame upon the
Cantor set and Cantor function.However, given Cantor's route into point set
topology,his arithmetic
introduction
of the Cantorset and Cantorfunction,
and his
facilitywitharithmetic
methods,as we have just illustrated,
it is feasiblethatit is
withinthe arithmetic
framework
of binaryand ternaryexpansionsthatCantorcame
upon the Cantorset and Cantorfunction.
REFERENCES
1. R. L. Wheeden and A. Zygmund,Measure and Integral:An Introduction
to Real Analysis,Marcel
Decker,Inc., New York,1977,p. 35.
2. T. Hawkins,Lebesgue's Theoryof Integration:
Its Originsand Development,
Chelsea PublishingCo.,
Madison,WI, 1975.
3. H. L. Royden,Real Analysis,MacmillanPublishingCo., New York,1988,p. 64.
4. H. J. S. Smith,On the integration
of discontinuous
functions,
Proc. LondonMath. Soc. (1) 6 (1875),
140-153.
5. G. Cantor,Beweis, daB eine fir jeden reellen Werth von x durch eine trigonometrische
Reihe
gegebeneFunctionf(x) sich nuraufeine einzigeWeise in dieserFormdarstellenliBt,Part1, Crelle
Jl. Math. 72 (1870), 139-142. Reprinted[7, pp. 80-83].
6. G. Cantor,Beweis, daB eine fir jeden reellen Werth von x durch eine trigonometrische
Reihe
gegebeneFunctionf(x) sich nuraufeine einzigeWeise in dieserFormdarstellenliBt,Part2, Crelle
Jl. Math. 73 (1871), 294-296. Reprinted[7, pp. 84-86].
7. G. Cantor,GesammelteAbhandlungen
mathematischen
und philosophischen
Inhalts,E. Zermelo(ed.),
New York,1980.
Springer-Verlag,
8. G. Cantor,Uber die Ausdehnungeines Satzes aus der Theorieder trigonometrischen
Reihen,Math.
Ann.5 (1872), 123-132. Reprinted[7, pp. 92-102].
9. W. Purkert,Cantor'sPhilosophicalViews,in The Historyof ModernMathematics,
Vol. 1: Ideas and
TheirReception,D. E. Rowe and J. McCleary(eds.), AcademicPress,Boston,1989.
10. G. Cantor,Uber unendliche,lineare Punktmannichfaltigkeiten,
Part 1, Math. Ann. 15 (1879), 1-7.
Reprinted[7, pp. 139-145].
11. G. Cantor,Uber unendliche,linearePunktmannichfaltigkeiten,
Part2, Math.Ann.17 (1880), 355-358.
Reprinted[7, pp. 145-148].
12. G. Cantor,Uber unendliche,linearePunktmannichfaltigkeiten,
Part3, Math.Ann.20 (1882), 113-121.
Reprinted[7, pp. 149-157].
13. G. Cantor,Uber unendliche,linearePunktmannichfaltigkeiten,
Part4, Math. Ann.21 (1883), 51-58.
Reprinted[7, pp. 157-164].
14. G. Cantor,Uberunendliche,linearePunktmannichfaltigkeiten,
Part5, Math.Ann.21 (1883), 545-591.
Reprinted[7, pp. 165-209].
15. G. Cantor,Uberunendliche,linearePunktmannichfaltigkeiten,
Part6, Math.Ann.23 (1884), 453-488.
Reprinted[7, pp. 210-246].
16. J.W. Dauben, Georg Cantor:His Mathematicsand Philosophyof the Infinite,PrincetonUniversity
Press,Princeton,NJ,1990.
17. W. Purkertand H. J. Ilgauds,GeorgCantor:1845-1918, BirkhauserVerlag,Basel, 1987.
18. G. Cantor,De la puissancedes ensemblesparfaitsde points,Acta Math.4 (1884), 381-392. Reprinted
[7, pp. 252-260].
19. J. W. Dauben, The developmentof Cantorianset theory,in From the Calculus to Set Theory:
1630-1910, I. Grattan-Guinness
(ed.), Gerald Duckworth& Co., London,1980.
20. G. F. Simmons,Introduction
to Topologyand ModernAnalysis,R. E. KriegerPublishingCo., Malabar,
FL, 1983,pp. 28-30.
21. K. Hrbacekand T. Jech,Introduction
to Set Theory,Marcel Decker,Inc., New York,1984,p. 72.
New York,1982,
22. G. Takeutiand W. M. Zaring,Introduction
to AxiomaticSet Theory,Springer-Verlag,
p. 86.
23. G. Cantor,Ein Beitragzur Mannigfaltigkeitslehre,
CrelleJl. Math.84, (1878), 242-258. Reprinted[7,
pp. 119-133].
24. G. H. Hardyand E. M. Wright,An Introduction
to the TheoryofNumbers,ClarendonPress,Oxford,
1989,pp. 129-140.
This content downloaded from 132.178.2.64 on Mon, 14 Oct 2013 11:42:52 AM
All use subject to JSTOR Terms and Conditions