Hilbert’s Tenth Problem
An Introduction to Logic, Number Theory, and Computability
M. Ram Murty
Brandon Fodden
EditorialBoard
SatyanL.Devadoss RosaOrellana
JohnStillwell(Chair) SergeTabachnikov
2010 MathematicsSubjectClassification.Primary11U05,12L05.
Foradditionalinformationandupdatesonthisbook,visit www.ams.org/bookpages/stml-88
LibraryofCongressCataloging-in-PublicationData
Names:Murty,MarutiRam,author. | Fodden,Brandon,1979–author.
Title:Hilbert’stenthproblem:anintroductiontologic,numbertheory,and computability/M.RamMurty,BrandonFodden.
Description:Providence,RhodeIsland:AmericanMathematicalSociety,[2019] | Series:Studentmathematicallibrary;volume88 | Includesbibliographical referencesandindex.
Identifiers:LCCN2018061472 | ISBN9781470443993(alk.paper)
Subjects:LCSH:Hilbert’stenthproblem. | Numbertheory–Problems,exercises, etc. | Mathematicalrecreations–Problems,exercises,etc. | Hilbert,David, 1862–1943. | AMS:Numbertheory–Connectionswithlogic–Decidability. msc | Fieldtheoryandpolynomials–Connectionswithlogic–Decidability. msc
Classification:LCCQA242.M89452019 | DDC512.7/4–dc23 LCrecordavailableathttps://lccn.loc.gov/2018061472
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∞ Thepaperusedinthisbookisacid-freeandfallswithintheguidelines establishedtoensurep ermanenceanddurability. VisittheAMShomepageat https://www.ams.org/ 10987654321242322212019
Analgebraofmind,aschemeofsense, Asymbollanguagewithoutdepthorwings, Apowertohandledeftlyoutwardthings Areourscantearningsofintelligence. TheTruthisgreaterandasksdeeperways.
-SriAurobindo,“DiscoveriesofScienceII”in CollectedPoems
Contents
Prefacexi Acknowledgmentsxiii Introduction1
Chapter1.CantorandInfinity5
§1.1.CountableSets5
§1.2.UncountableSets10
§1.3.TheSchr¨oder–BernsteinTheorem14 Exercises18
Chapter2.AxiomaticSetTheory23
§2.1.TheAxioms23
§2.2.OrdinalNumbersandWellOrderings28
§2.3.CardinalNumbersandCardinalArithmetic33 FurtherReading37 Exercises37
Chapter3.ElementaryNumberTheory41
§3.1.Divisibility41
§3.2.TheSumofTwoSquares50
viii Contents
§3.3.TheSumofFourSquares53
§3.4.TheBrahmagupta–PellEquation55 FurtherReading67 Exercises67
Chapter4.ComputabilityandProvability71
§4.1.TuringMachines71
§4.2.RecursiveFunctions82
§4.3.G¨odel’sCompletenessTheorems90
§4.4.G¨odel’sIncompletenessTheorems104
§4.5.Goodstein’sTheorem114 FurtherReading119 Exercises120
Chapter5.Hilbert’sTenthProblem123
§5.1.DiophantineSetsandFunctions123
§5.2.TheBrahmagupta–PellEquationRevisited131
§5.3.TheExponentialFunctionIsDiophantine137
§5.4.MoreDiophantineFunctions144
§5.5.TheBoundedUniversalQuantifier149
§5.6.RecursiveFunctionsRevisited155
§5.7.SolutionofHilbert’sTenthProblem159 FurtherReading164 Exercises165
Chapter6.ApplicationsofHilbert’sTenthProblem167
§6.1.RelatedProblems167
§6.2.APrimeRepresentingPolynomial171
§6.3.Goldbach’sConjectureandtheRiemannHypothesis180
§6.4.TheConsistencyofAxiomatizedTheories194 Exercises198
Chapter7.Hilbert’sTenthProblemoverNumberFields201
§7.1.BackgroundonAlgebraicNumberTheory201
§7.2.IntroductiontoZetaFunctionsand L-functions212
§7.3.ABriefOverviewofEllipticCurvesandTheir Lfunctions215
§7.4.Nonvanishingof L-functionsandHilbert’sProblem218 Exercises220
AppendixA.BackgroundMaterial223 Bibliography229 Index233
Preface
In1980,theseniorauthor(MRM)hadthegrandprivilegeofmeeting SarvadamanChowlaattheInstituteforAdvancedStudyinPrinceton,NewJersey.Chowlahadwrittenasmallbooktitled TheRiemannHypothesisandHilbert’sTenthProblem in1965andsothis wasanopportunitytoaskhimabouttheseeminglystrangetitleand howitcametobe.Wasthereaconnectionbetweenthetwo?Chowla replied,“Idon’tknow.Thesetwoproblemshavealwaysfascinated meandsoIchosethatasthetitle.”Hewentontosaythatthebook waslargelyaninspiredwork,writteninasinglenight,anditrepresentshisselectionofbeautifulpearlsfromnumbertheory.Itwasnot meanttobeatextbookbutmoreofaninvitationforfurtherstudy and“tostimulatethereader”.
Butthefactofthematteristhatthetwoproblems are related aswediscoveredonlymuchlaterintheworkofMartinDavis,Hilary Putnam,JuliaRobinsonandYuriMatiyasevi˘c.Infact,manyofthe famousHilbertproblemsareinterconnected.Thisinterconnectedness canbeusedasthefocusformathematicalinstruction.Anditcan bedonewithveryfewprerequisites.Thisisthe raisond’ˆetre ofthis book.
SomeoftheHilbertproblemssuchastheRiemannhypothesis (theeighthproblem)arestillopen.Theothersthathavebeensolved requiredformidablebackgroundandpreparation.Hilbert’stenth
problemisdifferentinthatabasicintroductiontoelementarynumbertheoryandmathematicallogicsufficestounderstandtheproof, andthiscanbedoneinarelativelyshorttime.Inadditiontothe grandarrangementofmathematicalideas,Hilbert’stenthproblem hasacolourfulcastofcharacters,manyofthemtragicheroes,who pondereddeeplyregardingtheenigmaofthehumanbeingandthe natureofmathematicaltruth.
Hilbert’stenthproblemanditssolutionrepresentinmicrocosm theriddleofhumanlifeitselfanditsmeaning.Thism´elangeofphilosophicalandmathematicalconundrumsarethemysteriesthatconfrontus.Inmanyways,thisbookisnotmeanttobeatextbook, butratheraninvitationtoexplorefurther.AsChowlawouldsay,we hope“tostimulatethereader”.
M.RamMurtyandBrandonFodden
KingstonandOttawa,Ontario
July2018
Introduction
In1900,atthesecondInternationalCongressofMathematicians (ICM),takingplaceinParis,DavidHilbert(1862–1943)presented alistoftwenty-threeproblemsthathefeltwerefundamentallyimportant,andwouldinfluencethedirectionofmathematicsinthe20th century.1 Inhisownwords(intranslation):2
Thesupplyofproblemsinmathematicsisinexhaustible,andassoonasoneproblemissolved numerousotherscomeforthinitsplace.Permit meinthefollowing,tentativelyasitwere,tomentionparticulardefiniteproblems,drawnfromvariousbranchesofmathematics,fromthediscussion ofwhichanadvancementofsciencemaybeexpected.
Inhistenthproblem,Hilbertaskedforanalgorithmthat,when givenanarbitraryDiophantineequation,willdeterminewhetherthe equationhasintegersolutionsornot.Thesolutiontothisproblem, thatthereisnosuchalgorithm,isoneoftheremarkableachievements of20th-centurymathematics.Whensuchdramaticadvancesina
1 TheaddresswasgiveninGerman,andhepresentedtenoftheproblems.Ina paperwritteninFrenchappearingintheproceedingsofthecongress,heincludedthe fulllistoftwenty-threeproblems.
2 Hilbert’saddresswastranslatedandpublishedinthe BulletinoftheAmerican MathematicalSociety ;see[Hil02].
DavidHilbert (PhotocourtesytheArchivesoftheMathematischesForschungsinstitutOberwolfach)
disciplinearemade,itisrarethatonecanexplainthesolutionin simpletermstothenonexpert.Fortunately,thisisnotthecasewith Hilbert’stenthproblem.Allthatisneededtounderstandhowthe solutionhasbeenputtogetherisanelementaryknowledgeofthe rudimentsoflogicandelementarynumbertheory,bothtopicsbeing atalevelaccessibletoanundergraduatestudentofmathematics.It isthepurposeofthismonographtoexplainthisworkinassimplea languageaspossible.Infact,wefeelitisasplendidwaytointroduce thestudenttobothlogicandnumbertheorythroughsuchamotivated introductionwithHilbert’stenthproblemasthefocus.
ThatHilbert’stenthproblemhasanegativesolution,inthatthe algorithmthatHilbertsoughtdoesnotexist,mighthavesurprised Hilbert.Still,heexpectedthatsuchthingsmayoccur.Inhis1900 address,hecommented:
Occasionallyithappensthatweseekthesolution underinsufficienthypothesesorinanincorrect sense,andforthisreasondonotsucceed.The problemthenarises:toshowtheimpossibilityof
thesolutionunderthegivenhypotheses,orinthe sensecontemplated.Suchproofsofimpossibility wereeffectedbytheancients,forinstancewhen theyshowedthattheratioofthehypotenuseto thesideofanisoscelesrighttriangleisirrational. Inlatermathematics,thequestionastotheimpossibilityofcertainsolutionsplaysapre-eminent part,andweperceiveinthiswaythatoldanddifficultproblems,suchastheproofoftheaxiomof parallels,thesquaringofthecircle,orthesolution ofequationsofthefifthdegreebyradicals,have finallyfoundfullysatisfactoryandrigoroussolutions,althoughinanothersensethanthatoriginallyintended.Itisprobablythisimportantfact alongwithotherphilosophicalreasonsthatgives risetotheconviction(whicheverymathematician shares,butwhichnoonehasasyetsupportedby aproof)thateverydefinitemathematicalproblem mustnecessarilybesusceptibleofanexactsettlement,eitherintheformofanactualanswertothe questionaskedorbytheproofoftheimpossibility ofitssolutionandtherewiththenecessaryfailure ofallattempts.
Inthisbook,wetouchonseveralofHilbert’sproblems.Hisfirst problem,thecontinuumhypothesis,isdiscussedinSections2.3and 4.3.Inhissecondproblem,Hilbertaskedforaproofoftheconsistencyoftheaxiomsofarithmetic.WediscussKurtG¨odel’smomentousresultonthisprobleminSection4.4.Webrieflymention Hilbert’sseventhproblem,onthetranscendenceof ab when a =0, 1 isalgebraicand b isirrationalandalgebraic,inSection1.2.Hilbert’s eighthproblemincludestheRiemannhypothesis,Goldbach’sconjecture,andthetwinprimeconjecture,whicharediscussedinSection 6.3.
Thescopeofthisbookisbroad.Aself-containedrigoroustreatmentofallthetopicscoveredbythisbookwouldmorethantriple itslength.Ourhopeistointroducethereadertonumeroustopicsin
logic,numbertheory,andcomputability.Theinterestedreadercan thenundertakefurtherstudyintheseareas.Tothatend,alistof referencesforfurtherreadingispresentedattheendofmostchapters.
Thefirstfourchaptersdeveloptherudimentarynotionsofset theory,elementarynumbertheory,andlogicneededforacomplete self-containedproofofHilbert’stenthprobleminChapter5.Some applicationsofthesolutiontoHilbert’stenthproblemarecoveredin Chapter6.Thismaterialisaccessibletotheundergraduatestudent andcanbecoveredinasemestercourse.3 Thefinalchapteraims tointroducetheaspiringstudenttocurrentresearchonthistopic, namelyHilbert’stenthproblemovernumberfields.Thischapterrequiresmoremathematicalmaturityandisintendedfortheadvanced student.Undoubtedly,thereismoreresearchtobedoneanditis ourhopethatthereaderisthustakentothefrontieroftheexisting knowledgeonthistopicsothatheorshemaysurveywhatisknown andwhatisunknown.
IfoneisjustlookingtounderstandthesolutiontoHilbert’stenth problem,whichisgiveninChapter5,thenSections3.4and4.2are necessary,providedoneiscomfortableusinganinformaldefinition ofanalgorithm.AmorecarefuldiscussionofalgorithmsandcomputabilityisgiveninSection4.1.InChapter6,someapplications ofthesolutiontoHilbert’stenthproblemaregiven.Theseusethe materialdevelopedinChapter2andSections4.3and4.4.However, wefeelthatbyreadingthroughtheentirebook,onewillgetabetter senseoftheinterplaybetweenthetopicscoveredbythisbookandan understandingofsomeofthemostimportantproblemsinlogicand numbertheoryofthepast150years.
3 Anappendixcontainingpreliminarymaterialisincluded.
CantorandInfinity
1.1.CountableSets
Therearesomephilosophersthatdenytheexistenceofinfinity.These arethe“finitists”.Theyarguethatsincewehaveneverseenan infinitecollectionofthings,infinitydoesnotexist.Whenpresented withnotionsofmathematicsthatleadonetotheideaofinfinity,they arguethatitmaybethattheuniverseisworkingmodulo p forsome largeprime p!
GeorgCantor(1845–1918)canbesaidtobethefounderofset theoryand,moregenerally,themathematicaltheoryofinfinitenumbers.HewasborninSt.Petersburgintoamerchantfamilythat settledinGermanyin1856.HestudiedinZ¨urichandthenBerlin whereheobtainedhisdegreein1867.In1869hebecamealecturer attheUniversityofHalleinGermanyandservedthereasaprofessor from1879to1913.Heputforththe continuumhypothesis (which willbedescribedattheendofthischapter)andattemptedtosolve it.Perhapsunderthestrainoftheseeffortsaswellasinitialoppositiontohisnewideasconcerninginfinity,hesufferedfromdepression whichmayhaveeventuallycontributedtohisdeath.Thecelebrated physicist,StephenHawking[Haw]wrote,“GeorgCantorscaledthe
1.CantorandInfinity
peaksofinfinityandthenplungedintothedeepestabyssesofthe mind:mentaldepression.”1
Cantor’sdoctoralthesiswasinnumbertheory.Later,heintroducedtheconceptsof ordinalnumbers and cardinalnumbers,which wediscussinChapter2.Usingthistheory,heprovedanumberof resultsthatcomparethesizesofinfinitesets,manyofwhicharegiven here.
Aset S issaidtobe countablyinfinite ifitcanbeputinoneto-onecorrespondencewiththenaturalnumbers(thatis,ifthere isabijectionbetween N = {0, 1, 2,...} and S ).A countable set iseitherfiniteorcountablyinfinite.Ifasetisnotcountable,it iscalled uncountable.Sincethefunctionthatsends n to2n isa
1 Hawkingeditedthebook GodCreatedtheIntegers inwhichhepennedshort essaysonabouttwodozenmathematicalgiants,withCantorbeingoneofthem.Unfortunately,theseessayswerenotcopyeditedproperlyandthereisaseriouserroron page1132.RespondingtoaquestionofDedekindastowhetheraninfinitesetcan bedefinedwithoutreferringtothenaturalnumbers,HawkingtriestogiveCantor’s reply.Thus,thefirstsentenceofthelastparagraphonpage1132shouldbe:“Cantor answeredhisfirstquestionbydefiningasetasbeinginfiniteifitcouldbeputintoa onetoonecorrespondencewithapropersubsetofitself.”
GeorgCantor (Photosource:Wikipedia)
bijectionbetween N andthesetofevennaturalnumbers,theset ofevennaturalnumbersiscountablyinfinite.Thesetofintegers, denotedby Z,isalsocountablyinfinitesincewemaydefineamap f : N → Z bysetting f (0)=0, f (1)=1, f (2)= 1, f (3)=2, f (4)= 2, f (5)=3, f (6)= 3,andsoon.
Moregenerally,weset
(n)=( 1)
, where x isthefloorfunction,whichreturnsthegreatestintegerless thanorequalto x.Thisfunctioniseasilyverifiedtobeinjectiveand surjective.
Whatabout Q,thesetofrationalnumbers?Coulditbethat Q iscountablyinfinite?Sinceanypositiverationalnumbercanbe writtenas a/b forsomenaturalnumbers a and b,weareledtoconsidertheproblemofdeterminingiftheset N × N oforderedpairsof naturalnumbersiscountablyinfinite.Thatis,isthereaone-to-one correspondencebetween N × N and N?Cantordiscoveredanexplicit function,givenby P (x,y )= (x + y )(x + y +1) 2 + x,
thatsetsupaone-to-onecorrespondencebetween N × N and N.This functioniscalledCantor’spairingfunction.Atableforthefirstfew valuesof P (x,y )isgivenbelow.
1.CantorandInfinity
Cantorfoundthepairingfunctionviaadiagonalmethodofenumeration.Thatis,hebeganhislistofpairsas (0, 0), (0, 1), (1, 0), (0, 2), (1, 1), (2, 0), (0, 3), (1, 2), (2, 1), (3, 0),....
Letusnotethatwecangroupthepairs(a,b)accordingtothesum a + b.Thereareonlyfinitelymanysuchpairsinanygroup.Correspondingtothesum k ,weseethatthereare k +1suchpairs.Now givenanorderedpair(x,y ),thegroupitliesinisdeterminedby k = x + y .Beforewereachthisgroup,thenumberoforderedpairs weencounteris
1+2+ + k = k (k +1) 2 .
Havingreachedthegroupwithsum k ,toreach(x,y )wehaveto proceedthrough (0,k ), (1,k 1),..., (x,y ), whichencompass x +1additionalpairs.Thusthepair(x,y )isinthe k (k +1) 2 + x +1= (x + y )(x + y +1) 2 + x +1
positioninthelisting.Sincewewantthefirstlistedpairtobemapped to0,thesecondtobemappedto1,andsoon,subtracting1yields thepairingfunction P (x,y ).2
Usingtheabovefunctions f and P ,itfollowsthat Z × Z isalso countablyinfinite,foronemayshowthat h : Z × Z → N given by h(x,y )= P (f 1 (x),f 1 (y ))isbijective.Byasimilarargument, A × B isalsocountablyinfiniteforcountablyinfinite A and B .By iteration,itfollowsthat Zn ,thesetofallordered n-tuplesofelements of Z,iscountablyinfiniteforanypositivenaturalnumber n,asis Nn .
SinceCantor’spairingfunction P (x,y )isbijective, F = P 1 is abijectivemapfrom N to N × N.Totheorderedpair(a,b)wemay associatethepositiverationalnumber(a +1)/(b +1),andthususe F tolistthepositiverationalnumbers,agreeingtoskipanypreviously
2 Inthiscontext,wementionaresultofRudolfFueter(1880–1950)andGeorge P´olya(1887–1985).Itisanopenquestiontodetermineallbijectivepolynomialmaps between N × N and N.FueterandP´olyashowedthatifwerestrictourattentionto quadraticpolynomials,thenessentiallyCantor’spairingfunction(uptopermutation) istheonlyone.Theirproofusesthetranscendenceof π ,asprovedbyFerdinandvon Lindemann(1852–1939),aswellasnontrivialanalyticnumbertheoryregardingerror termsinlatticepointenumerations.
listednumbers.Thislistingyieldsabijectionbetween N andthe positiverationalnumbers.Inparticular,since
F (0)=(0, 0),F (1)=(0, 1),F (2)=(1, 0),F (3)=(0, 2),
F (4)=(1, 1),F (5)=(2, 0),..., ourbijection g between N andthepositiverationalnumbersbegins as
g (0)=1/1=1,g (1)=1/2,g (2)=2/1=2,
g (3)=1/3,g (4)=3/1=3,... (2/2=1wasskippedsinceitwaspreviouslylisted).
Nowthatwehaveabijection g from N tothepositiverational numbers,wecandefineabijection q : N → Q asfollows.Wedefine q (0)=0, q (2n +1)= g (n),and q (2n +2)= g (n).Thus Q isa countablyinfiniteset.
Tosummarize,wehaveshownthefollowingtheorem.
Theorem1.1. Thefollowingsetsarecountablyinfinite: N, Z,and Q.
Notethatanyinfinitesubsetofacountablyinfinitesetisalso countablyinfinite.Theelementsofacountablyinfinitesetmaybe listedas a1 ,a2 ,a3 ,.... Thentheelementsofaninfinitesubsetmay belistedas an1 ,an2 ,an3 ,... for n1 ,n2 ,n3 ,...,aninfinitesubsequence of1, 2, 3,.... Thus,forexample,anyinfinitesetofrationalnumbers iscountablyinfinite.
If A and B arecountablyinfinite,then A ∪ B isalsocountably infinite.Thisisseenasfollows.Forsimplicity,weassumethesets aredisjoint.Let f : N → A and g : N → B bebijectivemaps.Define h : N → A ∪ B by h(2n)= f (n)and h(2n +1)= g (n).Itiseasyto showthat h isabijectivemap.
Therearenumbersthatarenotrationalnumbers.Theseare called irrationalnumbers.Forinstance, √2isirrational.Tosee this,supposewehavearationalnumber a/b,withthepropertythat (a/b)2 =2.Wemaysupposethat a/b isinlowestterms;thatis,there isnocommonfactorbetween a and b except1.Thenweget
a 2 =2b2 ,
1.CantorandInfinity
showingusthattheleft-handsideiseven.Thus a iseven,andwecan write a =2c forsomeinteger c.Wenowget4c2 =2b2 ,andcancelling thecommonfactorof2onbothsidesoftheequationyields 2c 2 = b2
Thisimpliesthattheright-handsideiseven,sothat b iseven.Thus wehaveboth a and b areeven,acontradiction.
√2isanexampleofan algebraicnumber.Anumber α issaidto be algebraic if α satisfiesanequationoftheform
n + an 1 αn 1 + + a1 α + a0 =0, with ai rationalnumbers.Thatis,thealgebraicnumbersarerootsof polynomialswithrationalcoefficients.Since √2isarootof x2 2, itisanalgebraicnumber.Onemayshowthatthesetofalgebraic numbersisacountablyinfiniteset,asisdoneintheexercisesatthe endofthechapter.
1.2.UncountableSets
Fromhismusingsoncountablesets,Cantorwentontoaskif R, thesetofrealnumbers,iscountable.Hisfirstproofthatthereals areuncountable,publishedin1874,usednestedintervals.Hismore famousproof,involvingthe diagonalargument,waspublishedin1891 andisgivenbelow.
Everyrealnumber x intheinterval(0, 1)= {x ∈ R :0 <x< 1} canbewrittenasaninfinitedecimal:
x =0.x1 x2 x3 .
Notethatadecimalexpansionendinginaninfinitesequenceof0’s 0.x1 x2 x3 ··· xm 1 xm 000 ··· with xm =0(calleda terminatingexpansion )alsohastheexpansion0.x1 x2 x3 xm 1 ym 999 ,where ym = xm 1.Ifweagreenevertoallowaninfinitesequenceof9’s asthe tail oftheexpansion,thenthedecimalexpansionisunique. Wecanestablishtheseassertionsasfollows.Takearealnumber 0 <x< 1.Then0 < 10x< 10,sowemaywrite
10x = x1 + y1 ,
where0 ≤ x1 ≤ 9,0 ≤ y1 < 1,and x1 isaninteger.Then
< 1
.
Iteratethisprocedurewith y1 .Thus 10y1 = x2 + y2 ,
where0 ≤ x2 ≤ 9,0 ≤ y2 < 1,and x2 isaninteger.Thus
Proceedinginthismanner,weget
where0 ≤ yn < 1.Weseeimmediatelythatthedecimalexpansion convergesto x.Toestablishuniqueness,letussupposethat
with0 ≤ xn ,yn ≤ 9.Let m bethesmallestnumberforwhich xm = ym .Withoutlossofgenerality,supposethat xm >ym .Then wehave
Thus
Hence0 <xm ym ≤ 1,whichimplies xm = ym +1.Thuswemust have yn =9,xn =0for n>m. Thusuniquenesscanfailonlyifone ofourdecimalexpansionseventuallyendsinaninfinitesequenceof 9’s.
WemaynowproveCantor’stheoremontheuncountabilityof R.
Theorem1.2. Theset R ofrealnumbersisuncountable.
Proof. Supposethattherealinterval(0, 1)werecountable.Wemay thenlistthem:
Nowconsiderthenumber
Inthisway,weavoidgettinga9or0asadigit,therebyavoiding repeating9’sandensuring r =0.Then r isin(0, 1)butcannot appearinourlistingabovesinceitdiffersfromeach rn inthe nth digit.Thisisacontradiction,andhencetherealinterval(0, 1)is uncountable.
Ifaset A isuncountableand A ⊆ B ,then B isalsouncountable.Toseethis,suppose B werecountable.Since A isinfinite, B tooisinfinite,andhencecountablyinfinite.Since A isaninfinite subsetofthecountablyinfiniteset B ,itmustbecountablyinfinite,a contradiction.Thus,since(0, 1)isanuncountablesubsetofthereal numbers,thesetofallrealnumbersisuncountable.
Supposethesetofirrationalnumberswerecountable.Since Q is countablyinfinite,wewouldthenhave R astheunionoftwocountable setsandhencecountable,acontradiction.Thusthereareuncountablymanyirrationalnumbers.
Arealnumberthatisnotalgebraiciscalleda transcendental number.Recallthatthesetofalgebraicnumbersiscountablyinfinite. Supposethesetoftranscendentalnumberswerecountable.Wewould thenhave R astheunionoftwocountablesetsandhencecountable, acontradiction.Thusthereareuncountablymanytranscendental numbers.
Inthissense,“most”realnumbersareirrational,andinfact transcendental.Cantorshowedtheuncountabilityofthetranscendentalnumbersin1874.Beforethis,theonlynumbersknowntobe transcendentalwerenumbersspecificallyconstructedtobeso(called Liouvillenumbers,namedafterJosephLiouville(1809–1882)),and e, whichwasshownbyCharlesHermite(1822–1901)tobetranscendentaljustoneyearearlier.ThusCantorprovedthatmostrealnumbers aretranscendentalatatimewhenonlyafewexampleswereknown! Thetranscendenceof π wasshownin1882byLindemann.InhisaddresstotheICMin1900,Hilbertgavehislistoftwenty-threeimportantunsolvedproblemsinmathematics.Inhisseventhproblem,he askedif a and b arealgebraicnumberswith a =0, 1and b irrational, doesitfollowthat ab istranscendental?Theanswerisyes,aswas provedindependentlyin1934byAlexanderGelfond(1906–1968)and TheodorSchneider(1911–1988).Therearestillmanyopenquestions regardingtranscendentalnumbers.Forexample,wedonotknowif thenumbers π + e or πe aretranscendental,althoughbothareexpectedtobe.Itcanbeprovedthatatleastoneofthemmustbe transcendental.ThisisanexerciseinChapter7.
Insteadofmerelyclassifyingsetsasfinite,countablyinfinite,and uncountable,wemayrefinethisbysayingthattwosets A and B have the samecardinality,written |A| = |B |,ifthereisabijectivemap betweenthem.Onemayshowthatthisisanequivalencerelation. Wesaythat A has cardinalitylessthanorequaltothatof B ,written |A|≤|B |,ifthereisaninjectivemapfrom A to B .Ifthereis aninjectivemapfrom A to B butnobijectivemapbetweenthe setsispossible,wesay A has smallercardinality than B andwrite |A| < |B |.WithTheorem1.1weshowedthatthat |N| = |Z| = |Q|. Sincetheinclusionmapfrom N to R isinjective,Theorem1.2shows that |N| < |R|.
Givenaset A,considerits powerset P (A)definedasthesetof allsubsetsof A.Itisclearthatthefunction f : A → P (A)defined by f (a)= {a} isinjective,andhence |A|≤|P (A)|.Cantorproved thefollowingtheorem.
Theorem1.3. Let A beaset.Thereisnobijectivemapbetween A and P (A),andhence |A| < |P (A)|.
Proof. Theproofisagainbycontradiction.Supposetherewerea bijectivemapfrom A to P (A).Toeach a ∈ A,wecanthenassigna uniqueset Ta ∈ P (A).Consider S = {a ∈ A : a/ ∈ Ta }
Clearly, S isasubsetof A.Thusitmustcorrespondtosome Tw with w ∈ A.Butthisleadstoacontradiction:
and
Inthisway,Cantorshowedthatthereisaninfiniteladderof infinitesets:
1.3.TheSchr¨oder–BernsteinTheorem
Insteadofseekingabijectivecorrespondencebetweentwosets A and B ,itissufficienttoestablishinjectivemaps f : A → B and g : B → A.Inotherwords,if |A|≤|B | and |B |≤|A|,then |A| = |B |.This isknownastheSchr¨oder–Bernstein3 theoremafterErnstSchr¨oder (1841–1902)andFelixBernstein(1878–1956).Beforeprovingthisin general,wefirstproveaspecialcase.If B ⊆ A,thentheinclusion mapfrom B to A isaninjection.Thus B ⊆ A implies |B |≤|A|.The followinglemmaistheSchr¨oder–Bernsteintheoreminthespecialcase whereonesetisasubsetoftheother.
Lemma1.4. Let A,B besetssuchthat B ⊆ A,andsupposethatwe haveaninjection f : A → B .Thenthereisabijection g : A → B .
3 Thereissomecontroversyonthenameofthistheorem.Itwasfirststated byCantorwithoutproofin1887.ItseemsRichardDedekindproveditin1887but didn’ttellanyoneaboutit.Itwasdiscoveredinhisnotesin1908.In1895Cantor publishedthefirstproof,buthisproofusestheaxiomofchoice,whichisdiscussed inthenextchapter.Dedekind’sunpublishedproofdidnotusetheaxiomofchoice. In1896Schr¨oderpublishedaproofsketchthatwasshowntobeincorrectafewyears later.In1897,Bernsteinprovedthetheorem—atage19!Afterwards,Bernsteinvisited Dedekind,whoapparentlythenindependentlyprovedthetheoremyetagain.
1.3.TheSchr¨oder–BernsteinTheorem 15
Proof. Toprovethis,wedefinesets D0 ,D1 ,... recursivelyasfollows.
D0 = A\B , D1 = f (D0 ), D2 = f (D1 ),andgenerally Dn+1 = f (Dn ). Nowdefinethemap g : A → B bysetting g (x)= f (x)if x isinsome
Dn and g (x)= x otherwise.If x isnotinany Dn ,theninparticular itisnotin D0 ,andso x isin B sothat g (x)= x ∈ B .Weclaimthat g isabijection.Toseethis,wehavetoshowthat g isinjectiveand surjective.Suppose g (x)= g (y ).Ifboth x and y areinsome Dn , thenweget f (x)= f (y ).Since f isinjective,wededuce x = y .If both x and y arenotinany Dn ,thenwehave x = g (x)= g (y )= y , soagain g isinjective.Nowconsiderthepossibilitythat x isinsome
Dn and y isnot.Then g (x)= g (y )implies f (x)= y .Since x is insome Dn ,itfollowsthat y isin Dn+1 ,acontradiction.Thus g is injective.Toseethat g issurjective,let b ∈ B .If b isnotinany Dn , then g (b)= b.If b isinsome Dn ,with n ≥ 1,then b ∈ f (Dn 1 )and so b isintherangeof g .If b ∈ D0 ,then b/ ∈ B .
Theorem1.5 (Schr¨oder–Bernsteintheorem). If f : A → B and g : B → A areinjective,thenthereisabijectionbetween A and B .
Proof. Thecomposition g ◦ f : A → g (B )isalsoinjectivesince
g (f (x))= g (f (y ))=⇒ f (x)= f (y )=⇒ x = y.
g (B )isasubsetof A andso,bythepreviouslemma,thereisa bijection h : A → g (B ).Since g isinjective, g 1 exists,andwehave amap g 1 : g (B ) → B .Define F : A → B by F (z )= g 1 (h(z )). Weshowthat F isbothinjectiveandsurjective.If F (z1 )= F (z2 ), then g 1 (h(z1 ))= g 1 (h(z2 )),andapplying g tobothsidesgives h(z1 )= h(z2 ).Since h isinjective,wededuce z1 = z2 .Let b ∈ B . Thereisan a ∈ A suchthat h(a)= g (b).Then F (a)= g 1 (h(a))= g 1 (g (b))= b.
Wehaveseenthat |N| < |R| and |N| < |P (N)|.Howdothe cardinalitiesof R and P (N)compare?WewillusetheSchr¨oder–Bernsteintheoremtoprovethattheyareinfactthesame.
Theorem1.6. |R| = |P (N)|
Proof. Considerthefunction f : P (N) → R definedby f (S )= 0.x1 x2 x3 ,where xi =0if i 1 ∈ S and xi =1if i 1 / ∈ S .
