SetTheory
ElementsinPhilosophyandLogic
DOI:10.1017/9781108981828 Firstpublishedonline:January2022
JohnP.Burgess PrincetonUniversity
Authorforcorrespondence: JohnP.Burgess, jburgess@princeton.edu
Abstract: Settheoryisabranchofmathematicswithaspecialsubject matter,theinfinite,butalsoageneralframeworkforallmodern mathematics,whosenotions figureineverybranch,pureandapplied. ThisElementwillofferaconciseintroduction,treatingtheoriginsofthe subject,thebasicnotionofset,theaxiomsofsettheoryandimmediate consequences,theset-theoreticreconstructionofmathematics,and thetheoryoftheinfinite,touchingalsoonselectedtopicsfromhigher settheory,controversialaxiomsandundecidedquestions,and philosophicalissuesraisedbytechnicaldevelopments.
Keywords: sets,infinity,continuum,cardinals,ordinals
©JohnP.Burgess2022
ISBNs:9781108986915(PB),9781108981828(OC)
ISSNs:2516-418X(online),2516-4171(print)
1HistoricalRoots
Althoughinretrospectothers(BernardBolzano,RichardDedekind)canbe viewedasprecursors,settheorywaslargelythecreationofasingleindividual, GeorgCantor,beginninginthe1870s,andhiskeywork(Cantor,1915)remains highlyreadabletothisday.Helaunchedthe fieldwithtworesultsonquestions withancientroots.
1.1StringstoOrdinals
Pythagoreansnotedthatifthelengthsofotherwisesimilarstringsareintheratio 2:1,theshortersoundsanoctavehigher.Why?Becauseitvibratestwiceas quickly.Inmodernmathematicallanguage,ifthegraphofthedisplacementofthe centerofthestringwithtimeapproximates y ¼ cos x forthelonger,itwill approximate y ¼ cos2x fortheshorter.Norealstringvibratessosimply,anda betterapproximationforthelongstringwouldbe y ¼ a1 cos þ a2 cos2x; with theamplitude a1 ofthe “fundamental” muchlargerthantheamplitude a2 ofthe “overtone.” Bytheeighteenthcentury,workersinanalysis,thebranchofmathematicsbeginningwithcalculus,weredealingwithinfinitetrigonometricseries:
The “vibratingstringcontroversy” engagingLeonhardEulerandothersconcernedhowwideaclassoffunctionscanberepresentedinthisform.Thedispute exposed,beyondendemicdeficienciesofrigorinthetreatmentofinfiniteseries, lackofacommonunderstandingaboutwhatismeantbya function.Theensuing nineteenth-centuryrigorizationofanalysis,besidesbanninganyliteralinfinities orinfinitesimals,explainingcontextscontainingthesymbol ∞ withoutassuming ittodenoteanythinginisolation, fixedonthemaximallygeneralnotionof function,underwhich any correlationbetweeninputsandoutputscounts,as longasthereisoneandonlyoneoutputperinput.Improvedrigoreventuallyled toconsensusabouttheexistenceoftrigonometricseriesrepresentations.
Butwithexistencetherecomeuniquenessquestions.Couldafunctionhave twodifferent representations?Doestheconstantfunctionzerohaveanyother thanthetrivialonewith an ¼ bn ¼ 0forall n?BernhardRiemannshowedit doesnotifthesequenceconvergesforall x.Butwhatifoneallowsan exceptionalpointforwhichconvergenceisnotassumed?EnterCantor.It turnsoutthateventhentrivialityholds(and,asaconclusion,wegetwhatwe didnotassumeasapremise,convergenceevenattheexceptionalpoint). Indeed,onecanallowtwoorany finitenumberofexceptionalpoints.One canevenallowinfinitelymanyaslongastheyareall isolated fromoneanother,
meaningthatforeachexceptional x thereisapositive ε withno other exceptionalpointsbetween x ε and x þ ε.Onecanevenallowa doubly exceptional point,notisolatedfromotherexceptionalpoints.Indeed,onecanallowtwoor any finitenumber.Onecanevenallowinfinitelymanyaslongastheyare isolatedfromoneanother.Onecanevenallowa triply exceptionalpoint.Andso on.Andasonegoeson,itbecomesnaturaltoswitchfromspeakingintheplural oftheexceptionalpointstospeakinginthesingularofthe setE ofwhichthey are elements.Whatitmeanstotreat E asasingleitemistothinkofoperations beingapplicabletoit.TherelevantoperationonsetsCantorcalled derivation, discardingisolatedpoints.Let E0 be E itself,andlet Enþ1 bethederivedsetof En .Reimann’sresultwasthatuniquenessholdsif E0 ¼ ∅ ,theemptyset,with noelements.Cantor ’sresultswerethatuniquenessholdsifanyof E1 ; E2 ; E3 ; isempty.Moreover,ifwelet Eω betheintersectionofthe En , thesetof x belongingtoallofthem,uniquenessstillholdsif Eω ¼ ∅ . Moreover,theresultscontinue,withsetsindexedby:
andmore.HereareCantor ’s transfiniteordinalnumbers,and,asthenotation suggests,heintroducedanarithmeticforthem,withaddition,multiplication, andexponentiation.
1.2QuadraturetoCardinals
Euclidshowsmanygeometrical figurescanbeconstructedwithstraightedge andcompass,indicatingthestepsinvolvedandprovingtheyleadtothedesired result.Thusonecan duplicatethesquare,orconstruct,giventhesideofa square,thesideofasquareoftwicethearea,justbytakingthediagonalofthe originalsquare.Toshowaconstruction not possibleismoredifficult,and requiresananalysisavailableonlywiththemoderncoordinatemethods, whichtransformgeometricintoalgebraicproblems.Thus duplicatingthe cube,constructing,giventhesideofacube,thesideofacubeoftwicethe volume,turnsoutequivalenttoobtainingakeynumber, 2 3 p ,fromrational numbersbyaddition,subtraction,multiplication,division,andextractionof squareroots.Andthiswasprovedimpossibleinthe1830s,disposingofan ancientproblem.For quadratureofthecircle,constructingforagivencirclea squareofequalarea,thekeynumberis π.Now,although 2 3 p isnotobtainablein thewayindicated,itisatleastan algebraic numberinthesenseofasolutiontoa polynomialequation:
withrationalcoefficients ai ,namely, x3 –2 ¼ 0.Itwasconjectured,however, that π isnotevenalgebraicinthissense.JosephLiouvilleshowednonalgebraic or transcendental numbersexist.Then e,thebasisofthenaturallogarithms,was showntobeonebyCharlesHermite,and, finally, π byFerdinandvon Lindemann.Betweentheselasttwo,Cantorshowedthatthevastmajorityof realnumbersaretranscendental.
Sincethesetsofalgebraicsandtranscendentalsareinfinite,tosayonehas moreelementsthantheotherrequiresadefinitionofwhenthe transfinite cardinal,ornumberofelementsofoneinfiniteset, A,isequalorunequalto thatofanother, B.Cantortookashisstandardofequalitytheexistenceofa bijection between A and B,arelationunderwhicheachelementof A is associatedwithexactlyoneelementof B,andviceversa.Inthecaseoftheset N ofnaturalnumbers,theexistenceofabijectionwithaset B meansthatthe elementsof B canbe enumerated orlistedinasequenceindexedby0,1,2, ..., asin Table1.Aninfinitesetwhoseelementscanbesoenumeratediscalled denumerable,whileasetthatis either denumerable or finiteiscalled countable.
ThenumberofelementsofadenumerablesetCantorcalled ℵ0 (pronounced “alephnought”).Whatthetableshowsisthatsignedintegersandpositive rationalsbothhavecardinalorsize ℵ0 ;sodothesignedrationals.Nowadays, a finitesequenceofkeystrokesistransmittedelectronicallyasasequenceof zerosandones,thebinarynumeralforsomenaturalnumberthatmaybe consideredacodeforthesequence.Thismakesthesetofsuchsequences denumerable,inorderofincreasingcodenumber.Then,sinceapolynomial equationofdegree n hasatmost n solutions,eachalgebraicnumbercanbe denotedbyanexpressionsuchas “thesecondsmallestsolutionto 2x3 9x2 6x þ 3 ¼ 0” andgivenacodenumberaccordingly.Buttheir denumerabilitywasestablishedincorrespondencebetweenDedekindand Cantorlongbeforethedigitalagebegan.
Bycontrast,Cantorshowedthatthewholeset R ofrealnumbers(andhence thesetoftranscendentals,leftoverwhenweremovethealgebraics)is not denumerable.Nocountablesetcancontainevenjustthosewhosedecimal
Table1 Denumerablesets
Set Enumeration
Naturalnumbers012345678 ... Integers01 12 23 34 4
Positiverationals1/11/22/11/32/33/23/11/43/4
Table2 Thediagonalargument
expansioninvolvesonly0sand1s;orwhatisthesame,allinfinitezero-one sequences;orwhatisthesame,allsetsofnaturalnumbers,eachsuchbeing representablebythezero-onesequencewithoneinthe nthplaceifandonlyif n isintheset.Thisheestablishedbyhisfamous diagonalargument.Supposewe haveanenumerationofsomeset S ofinfinitezero-onesequences,asin Table2. Godownthediagonal,markedwithasterisks.Takeinorderforeach n thedigit appearinginthe nthplaceinthe nthrowofthetable.Thisgives0100 ... .Now swapthezerosandtheones.Thisgives1011 ,asequencethatdoesnot belongtothedenumerableset S,sinceitdiffersinthe nthplacefromthe nth sequence.Cantorcalledthecardinaloftherealnumbersorpointsoftheline c Analogouslytotheresultsin Table1 inthisdiscussion,heshowedthatthe positiverealnumbers,orevenjustthoseina finiteinterval,alsohavecardinal c, asdopairsofrealnumbers,orequivalentlycomplexnumbers.Healsointroducedanarithmetic,withaddition,multiplication,andexponentiation,forhis cardinals.
Cantor ’saudaciousintroductionof ω and ℵ whenmathematicianshadjust finishedexplainingaway ∞ provokedareaction.ButCantor ’stheorywon acceptanceamongleadersintherisinggenerationfairlyquickly(asexamples theyputforth,suchastheone-,two-,andthree-dimensional Cantorset, Sierpinskicarpet,and Mengersponge,whoseimagesappearalloverthe Internettoday,capturedtheimaginationofamateurs).TheleadingmathematicianDavidHilbertinsisted: “NooneshallexpelusfromtheparadiseCantor createdforus.”
2TheNotionofSet
Manyobjectionsturnedoncertain paradoxes.Cantor,unlikehiscontemporary GottlobFrege,nevermadetheassumptionsthatledtotheseparadoxes,buthe didnotmakeclearenoughwhatassumptionshe was making.Hissuccessors hadtobemoreclearandexplicit.Explicitaxiomatizationbeganinthe first decadeofthetwentiethcenturywithErnst Zermelo(1908/1967).Hissystem,
withadditionsandamendments,mainlybyAbraham Fraenkel(1922/1967), remainsthatacceptedtoday,whenitisrecognizedthattheparadoxesresult mostlyfromconfusingthenotionofsetbehindtheaxiomsof Zermelo–Fraenkel settheorywithChoice (ZFC)withotherideas.
2.1Collections
Theexpression “amultiplicityofobjects” beginssingularbutendsplural,and maybeunderstoodasreferringeithertoa plurality,amany,ortoa universal,a oneasopposedtoamany.Universalsinclude properties,whichare intensional, meaningthattwomaybedifferentevenwhilehavingexactlythesame instances,aswiththestockexample beingacoininmypocket and beinga pennyinmypocket,whicharedistinctpropertiesevenifIhavenocoinsinmy pocketbutpennies.Theyalsoinclude aggregates completelydeterminedby theircomponents.Onekind,topicofatheorycalled mereology,isa fusion ofa pluralityofcomponentpartsintoasinglewhole,inawaythatpermitsdifferent pluralitiestohavethesamefusion,asdotheeightranksandtheeight filesofa chessboard,thefusionbeingtheselfsamechessboardineithercase.Bycontrast wehave collections,inwhichmanyaregatheredintoaonewithoutlosingtrack ofwhichmanytheywere.
Thenotionofcollectionin Frege(1893) wasthatofan extension.Herewe startwithallobjects,andtakewhathecalleda concept (associatedwitha predicate),anddivideobjectsintothosethatfallundertheconcept(satisfythe predicate)andthosethatdonot.Thecollectionofthosethatdoistheextension oftheconcept,sothattheextensionsoftwoconceptsarethesameifandonlyif theconceptsare coextensive,havingexactlythesamethingsfallingunderthem. Graphically,wemayrepresenttheunboundedrangeofallobjectswithwhich westartasanunboundedblankpage,andrepresenttheextensionasgivenbya dividinglineorcurveseparatingobjectsinsidefromobjectsoutside,asin Figure1.ButforFrege,theextensionisitselfanobject:Ifrepresentedbya dot,thatdotmustfallonthepageononesideortheotherofthedivision – but which?Thatisthequestionindicatedbythequestionmarksinthe figure.
BertrandRussellraisedanembarrassingissueabouttheextension R ofthe concept:it isanextensionthatasanobjectisoutside,notinside,itself.Inthe caseoftheuniversalextension, V,theextensionof isself-identical, V isinside itselfsince everything isinside V.Inthecaseoftheemptyextension ∅ ,the extensionof nonself-identical, ∅ isoutsideitselfsince nothing isinside ∅ . Hence ∅ isinside,and V isoutside,theRussellextension R.Butjustasthe statement thisverystatementisfalse seemstobetrueifitisfalseandfalseifitis
true,so R seemstobeinsideitselfifoutsideitself,andoutsideifinside.Thisis the Russellparadox as Russell(1902) putittoFrege.
Contrastingwiththisinconsistent “topdown” notionofextensionisthe “bottomup” notionofan ensemble.Herewestartwithagiven “universeof discourse,” whichmightberepresentedbyabox,andapredicatewill,likeacurve inaVenndiagram,markofftheensembleofthingsintheuniversethatdosatisfy itfromthingsintheuniversethatdonot.Theensembledoes not,however,itself belongtotheuniverse.Adotrepresentingitwouldlieoutsidethebox,asin Figure2.Implicithereisthepossibilityof iteration.Wecanaddanewboxatop theoriginal,toaccommodateallthedotsrepresentingensemblesofthingsinthe lowerbox,andthenmore.Buttherearetwowaystoimplementthisidea.
Onthe layered approachofthe theoryoftypes,derivingfrom Russell(1908) bywayofFrank Ramsey(1925),wehaveahierarchywith individuals atthe bottomtypezero,collectionscalled classes oftypezeroitemsattypeone, classesoftypeoneitemsattypetwo,andsoon.Evenifweassume no itemsat typezero,therewillbeoneitemattypeone,theemptyclass ∅ 1 oftypezero items,andthentwoitemsattypetwo,theemptyclass ∅ 2 oftypeoneitems,and thesingletonclass ∅ 1 g2 oftheoneitemattypeone.Attypethree,therewill befouritems,asin Table3.Withoneitemattypezero,therewillbetwoattype
Figure2 Anensemble
Figure1 Anextension
Table3 Thelayeredhierarchy
4SixteenItems
0NoItems
one,thenfour,thensixteen.Butwithonly finitelymanyindividuals,therewill onlyeverbeonly finitelymanyitemsofanyonetype.Formathematical purposes,Russellassumedinfinitelymanyindividuals.
2.2Sets
Bycontrast,wehavethe cumulative approach,wheresuccessiveboxesare nested,likeChineseboxesorRussiandolls,eachhigheroneaddinganewlevel ofcollectionscalled sets.Inboxzeroareindividualsor Urelemente;atlevel one,setswhoseelementsareindividuals;inboxone,individualsandlevel-one sets;atleveltwo,anynewsetswhoseelementscomefromboxone;inboxtwo, box-oneandlevel-twoitems;andsoon.
InZFC,weconsideronly pure sets,withoutindividuals.Therethenwillbe noitemsatlevelzero,oneitem,theemptyset ∅ ,atlevelone,inboxone.Asfor leveltwo,fromtheoneiteminboxonecanbeformedtwosets:theemptyset ∅ anditssingleton ∅ gf ,buttheformerwealreadyhave,soonlythelatterisnew. Inboxthreewillbefouritems,twonewatlevelthree.Inboxfourwillbesixteen items,twelvenewatlevelfour.Andsoon,asin Table4.
Afterall finitelevels,wemayrecognizeabox ω containingeverythingof finitelevelbutnothingnew,andthenformalevel ω þ 1forsetswhoseelements comefromlevel ω,meaningfromany finitelevel,butdonotthemselvesappear atanysuchlevel,containingastheydosetsofarbitrarilyhigh finitelevel.We canthencontinuethroughthetransfiniteordinals.Zermeloat firstclaimedfor hisaxiomsonlythattheypermittednoneoftheknowndeductionsofcontradictions,andseemedadequatetodevelopCantor ’ssettheory(astheyarewith Fraenkel’sfriendlyamendments).Onlylater(asin Zermelo,1930)didsomethinglikethepictureinthetableemerge.
Theidealofrigoristhatoneshouldlistinadvanceall primitives,notions assumedmeaningfulwithoutdefinition,and postulates oraxioms,results assumedtruewithoutdemonstration,andgiventheseprinciplesallfurther
Table4 Thecumulativehierarchy
+1{Ø,{Ø},{{Ø}},{{{Ø}}},…}andManyOtherNewItems NoNewItems
4TwelveNewItems
3{{Ø}},{Ø,{Ø}}
2{Ø}
1Ø
0NoItems
Table5 Primitivelogicalnotions
SymbolOperation Reading
: Negation “not”
˄ Conjunction “and”
˅ Disjunction “or”
8 Universalquantification “forall”
∃ Existentialquantification “forsome” or “thereexists”
notionsorresultsshouldbelogicallyderived,bydefinitionordeduction.Inset theory,thereisjustoneprimitive,writtenwithastylizedepsilonsymbol, x 2 y, read “ x isanelementof y ” or “ x isin y ” or “ y contains x. ” Allothernotionsmust bedefinedintermsofthisandthelogicalnotionofidentityusingthelogical operatorsin Table5.A formula Φ isbuiltupfrom atomic formulas x 2 y and x ¼ y usingthe fiveoperationsinthetable.
Someminimalfamiliaritywithlogicalnotionsandnotationsmustbe assumedhere(foraquickreview,see Boolos,Burgess,andJeffrey,2002, chapters9and10),includinganabilitytorecognizesimplelogicallaws.In particular,familiarityisassumedwiththedistinctionbetween “free” and “bound” occurrencesofvariablesinaformula,thosethatarenotandthose thatarecaughtbyaquantifier.Forexample,intheformulaassertingthenonemptinessof x,namely ∃yðy 2 xÞ,the x isfreebutthe y isbound.Thelattercould bechangedto z withoutchangingthemeaning.Otherlogicalandset-theoretic notionsmaybedefinedintermsofwhatwehavesofar,asin Tables6 and 7,but officiallythesearemereabbreviations.
Table6 Definedlogicalnotions
AbbreviationDefinition
Φ Ψ :ΦVΨ
OperationReading
Conditional “if Φ then Ψ” Φ ≡ ΨΦ Ψ ðÞ ˄ Ψ Φ ðÞ
x 6¼ y :x ¼ y
Biconditional “Φ ifandonlyif Ψ” or “ Φ iff Ψ”
Nonidentity “ x isdistinctfrom y ”
∃!xΦðxÞ ∃x 8 yðΦðyÞ ≡ x ¼ yÞ Unique existence “thereexistsaunique”
Table7 Definedset-theoreticnotions
AbbreviationDefinition
Reading
x = 2 y : x 2 y “ x isnotanelementof [ornotin] y ”
x ⊆ y 8z ðz 2 x z 2 yÞ “ x isasubsetof[or includedin] y ”
8x2y ΦðxÞ8xðx 2 y ΦðxÞÞ “forall x in y ... ”
∃x2y ΦðxÞ ∃xðx 2 yΛΦðxÞÞ “forsome x in y ”
3TheZermelo–FraenkelAxioms
TheaxiomsofthesystemZFCwillbepresentednext,inbothwordsand symbols,tobeassumedwithoutproof,butnotwithoutsomethingintheway ofinformal,intuitivejustification.
3.1Statement
The firstaxiomsayssetswiththesameelementsarethesame.Ithastwo equivalentformulations:
Extensionality ð1Þ8zðz 2 x ≡ z 2 yÞ x ¼ y; ð2Þ x ⊆ y ∧ y ⊆ x x ¼ y
Byconvention,indisplayingformulasinitialuniversalquantifiersareomitted, sowhatismeantisreally 8x8y()wherewhatisexplicitlywrittenis.As(2) suggests,proofsofidentitiesmostoftencomeintwoparts,provinginclusionin twodirections.Extensionalityimpliesthatifthereisaset y whoseelementsare allandonlythesets x satisfyingacondition Φ,itisunique.Thatuniqueset,ifit exists,isdenoted xj ΦðxÞg f ,andwehave z 2 xj ΦðxÞg f if Φ(z).Frege’sinconsistentassumptionwouldbeanaxiomof comprehension,accordingtowhich
xj ΦðxÞg f always existsfor any condition Φ.Appliedtothecondition x = 2 x this wouldgivetheRussellparadox,anditisnotassumedinZFC.
Thesecondaxiomsaysthatifwe alreadyhave someset u,wecanatleast separateoutfrom u thoseofitselementsthatsatisfyacondition Φ toform x 2 ujΦðxÞg : f
Separation ∃y8xðx 2 y ≡ðx 2 u ∧ Φ x ðÞÞÞ
Thisisnotasingleformula,butratheraruletotheeffectthatanythingofa certain form countsasanaxiom.Thecasesfordifferent Φ arecalled instances of the scheme ofseparation.(Zermelo’soriginalformulationwasvaguer.)Note thatseparationimpliesthereisno universalset ofallsetsV ¼ xj x ¼ xg f .If therewere,wecould,byseparation,obtaincomprehension.
Furtheraxiomsstatetheexistenceofcertainspecificsets:
Pairing ∃y ðu 2 y ∧ v 2 yÞ
Union ∃y 8z 2 X 8x 2 zðx 2 yÞ:
Withwhatwehavesofar,somebasicexistenceresultsthenbecomededucible, thosein Table8.(Theexpression “family” usedinthetablemaybeusedforany setofsets.)
Separationgivesustheemptyset,sincegivenanyset u atall – andeven purelogicassumesthereisatleastoneiteminthedomainourquantifi ers rangeover,whichinthepresentcaseconsistsofsets – separationgives x 2 ujx = 2 ug f ,whichisempty.Italsogivestwofoldintersections,andbythe alternativedefinition,familyintersections,ifthefamily X hasatleastone member u;alsodifferences.Nowgiven y containing u and v,wecanseparate outtheelementsof y identicaltooneofthosetwo,sopairingwithseparation givestheunorderedpair.Unionwithseparationgivesusfamilyunion.The unorderedtripleandtwofoldunionwethengetusingthealternativedefinitions.
Thedifference u v isalsocalledthe relative complementof v in u.An absolute complement v ¼ xj x = 2 vg f cannotexist,because v ∪ v wouldbe thenonexistentV.
Thenexttwoaxiomsarethese:
Power ∃y8xðx ⊆ u x 2 yÞ:
Infinitity ∃yðØ 2 y ∧ 8x 2 yð xg2 yÞÞ: f
Powerwithseparationgivesthe powerset PxðÞ¼ yjy ⊆ x g f andalso y ⊆ xj Φ y ðÞg¼ y 2 PxðÞjΦ y ðÞg: f f
Unorderedpair
Unorderedtriple
Table8 Moredefinednotions
Infinityguaranteestheexistenceofasetthatcontainsallof ∅ and ∅ g f and ∅ g ffg andsoon,andhenceisinfinite;alternativeformulationsarepossible; moredetaileddiscussionispostponed.
Alsopostponedisdetaileddiscussion,beyonditsmerestatement,ofthe widelyknownaxiomofchoice(AC),picturedin Figure3
Setswhoseintersectionisnonemptyaresaidto meet or overlap;thosewhose intersectionisemptyarecalled disjoint,andafamilyanytwomembersofwhich aredisjointiscalled pairwisedisjoint,afamilyofnonempty,pairwisedisjointsets iscalleda partition (ofitsunion),andthemembersofthefamilythe cells thereof. Axiomofchoiceassertsthatforany partition thereisa selector,asetcontaining exactlyoneelementfromeachcell(representedinthe figurebythescattered dots).Aloneoftheaxioms,ACassertstheexistenceofasetsatisfyingacertain condition,withoutgivenadefinitionofsuchasetas xjΦ x ðÞg f forany Φ:
Choice 8X ð8x 2 Xx 6¼ ∅ ðÞ
Fraenkel’sdistinctiveadditiontoZermelo’saxioms,replacement,isascheme sayingthatiftoeachelement x ofaset u thereisassociatedaunique y satisfying acondition Φ x; y ðÞ – callit φðxÞ – wemayreplaceeach x in u by φðxÞ andform theset φ x ðÞjx 2 ug f .Actually,itisenoughtoassumethereisasetcontainingall φðxÞ for x 2 u andthenapplyseparationtogetthesetofall andonly the φðxÞ for x 2 u.So,thenewassumptionweneedisthis:
3.2Motivation
While “ intuition ” maynotbeappealedtoinproofsoftheorems,stillwhere axiomsareconnectedwithanintuitivepicture,itmayatleastsuggest
Figure3 Partitionandselector
conjectures,besidesbeingasourceofconfidenceintheconsistencyofan axiomsystem,beyondthemereinductiveconsiderationthatnocontradiction hasbeenfoundsofar.Forsuchreasons,in terestattachestotherelationship betweentheaxiomsofZFC(beyondextensionality)andthecumulative hierarchypicture.
Forseparation,thenonexistenceofauniversalset V isclear,sincethe elementsofasetthatatagivenlevelcomefromlowerlevels.Bycontrast,if aset x appearsatagivenlevel,thenitselementsallappearatlowerlevels, includingsuchofthemassatisfysomecondition Φ,andhencethesetofallsuch willappearatalevelnohigherthanthatof x itself.
Forpairing,if u appearsatsomeleveland v atsomelevel,oneoftheselevels willhavetobenoearlierthantheother,andboth u and v willbepresentatthat level,andso u; vg f shouldappearattheverynext.Forunion,if X appearsat somelevel,everyelementappearsatsomeearlierlevelandeveryelementof suchanelementatsomestillearlier,soallelementsofelementswillbepresent atlevelsbelowthatof X,andtheset ∪ X presentbythesamelevelas X.For power,if u appearsatsomelevel,thenwehaveseenallitssubsetsarepresentby thatlevel,andso P ðu Þ shouldappearattheverynext.Forinfinity,itassertsno morethantheexistenceofsuchasetasweseeatlevel ω þ 1in Table4 in Section2.2.
Forchoice,ifapartition X occursatsomelevel,itiseasilyseenanyselector foritwillappearbythatsamelevel.But is thereanyselector?Theassumption thatthehierarchyismaximally “wide,” admittingatagivenlevel all setsthat couldconceivablybeformedfromelementsatlowerlevels,meansthatwe shouldnotbeimposinganyrequirementof definability asapreconditionforset existence.Historically,objectionstoAChavegenerallyrestedonimplicit impositionofsomesuchprecondition,sothecumulativehierarchypicture excludesthemajor antichoiceargument.Butthatisnotquitetosaythatit providesasubstantive prochoiceargument,andtheaxiomremains,toadegree, controversial.Althoughitisnolongercommonforworkingmathematiciansto startheoremswhoseproofdependsonAC,settheoristskeeptrack.
Forreplacement,manyfeeltheunderstandingthatthecumulativehierarchy issupposedtobemaximally “high,” admitting all levelsthatcouldconceivably beadmitted,supportstheaxiom.Butheretheinfluencemaybefeltofwhat somewouldclaimisa further thought,adoctrineof limitationofsize,according towhichallthatcanpreventapluralityofsetsfrombeingcollectedtogetherinto asetwouldbetherebeing toomany ofthem.(Cantordistinguishedthe inconsistent multiplicitiesthatcannotbecollectedintoawholefromthe consistent onesthatcanbytheformers’ being absolutelyinfinite where thelatterareonly transfinite.)Theideawouldbethatin φ x ðÞjx 2 ug f there
wouldnotbetoomanyelementstoformaset,sincetherewouldbenomore thantherearein u,whichalready is aset.See Boolos(1971) forcritical discussion.
ThereremainsanaxiomnotalwayscountedaspartofZFC – and,in particular,notsocountedinatleastonewidelyusedintroductorytextbook –althoughsocountedhere.Ithastwoequivalentformulations.
Inwords,if x hasanyelementsatall,thenithasanelement y thatis epsilon minimal,meaningthatthereisnootherelement z with z 2 y.Theaxiom,which alsogoesbythealias regularity,isdirectlysuggestedbythecumulative hierarchypicture:If x hasanyelements,itmusthaveanelement y oflowest possiblelevelforanelementof x,andsucha y willbeepsilonminimal.Some immediateconsequences:
Thereisnoset x with x 2 x.
Therearenosets x, y with x 2 y 2 x.
Therearenosets x, y, z with x 2 y 2 z 2 x
Whynot?Because xg f or x; yg f or x; y; zg f ,asthecasemaybe,wouldhave noepsilon-minimalelement.Theaxiomalsoexcludestheexistenceofany infinitedescendingchainwith x1 2 x0 and x2 2 x1 and x3 2 x2 andsoon. AlongsideorthodoxsettheoryZFC,thereexistheterodox “alternative” set theories. Incurvati(2020) surveysseveral,includingtwothatpermitinfinite descendingsequences:a “graph” conceptionduetoPeterAczelanda “stratified” conceptionduetoW.V.Quine.(Healsoconsidersa “paraconsistent” conception thatacceptscomprehensionandtheRussellparadox,butadoptsadeviantlogicin hopesofquarantiningthecontradiction.)SeealsoHolmes(2017).
4ImmediateConsequences
Someconsequencesoftheaxiomswereestablishedwellbeforesettheory becameaseparatesubject.
4.1TheAlgebraofSets
AnimportantsteptowardmodernlogicwastakenbyGeorgeBoole,whose LawsofThought (1854)containsformulasinalgebraicsymbolismeachadmittingtworeadings:asaprincipleoflogicandaswhatwerecognizeretrospectivelyasaoneofsettheory.Thustheformula a b ¼ b a expressesboth thelogicallawofthecommutativityofconjunction, Φ ∧ Ψ iff Ψ ∧ Φ,and
Foundation ð1Þ8xðx 6¼
2 yÞÞ
2Þ8xðx 6¼ ∅ ∃y 2 x ðy ∩ x ¼ ∅ ÞÞ:
theset-theoreticlawofthecommutativityofintersection, x ∩ y ¼ y ∩ x. Supposeweareworkingforatimeonlywithsubsetsofsomegivenset I,and allowourselvestowrite –x for I x.Thenthe firstbatchoftheoremsofZFC consistsofequationsofso-called Booleanalgebra for ∩ and ∪ and –.The proofofsuchanequationconsistsinapplyingextensionalityaftershowingthat anyitemwillbelongtotherightsideifitbelongstotheleftside;andtheproofof that consistsinunpackingthedefinitionsandapplyingalawoflogic,thevery lawthatinBoole’snotationwouldbeexpressedbythesamealgebraicalformula astheset-theoreticresultwearetryingtoprove,thus:
Anynumberoffurtherlawsofthealgebraofsetsarefoundin Table9,inpairs of “dual” lawsonthesamerow.Another,notinthetable,isthelaw x ¼ x; correspondingtothelawof doublenegation: :: Φ iff Φ.
Furtherlawsinvolvinginclusionappearin Table10.Manyoftheselawsmaybe familiarfromschool,wheretheymighthavebeenillustratedbyVenndiagrams. Evenanintroductorytextbookofsettheory,althoughitmightruntohundredsof pages,wouldleavetheverificationofmostas “exercisesforthereader”– withthe goodexcusethat,inanycase,onecanonlyreallylearnamathematicalsubjectby doingexercises – andinthismuchshorterElement,wheretheaimmustbelessto trainthereader in thantoinformthereader about atechnicalsubject,theywill all be soleft.(Theproofsdonot all havetobeproceed “elementwise,” asinthe commutativelawexample.Onceonehasaccumulatedafewlaws,otherscanbe derivedfromthem “algebraically,” withoutgoingbacktothedefinitions.)
4.2TheAlgebraofRelations
Boole ’slogiccoversabitmorethanAristoteliansyllogistic,beingaversionof themodernlogicof one-place predicates.Itisstillnotenoughtoanalyze seriousmathematicalarguments,whichgenerallyinvolve two-place predicates(suchas 2).Thelogicofmany-placepredicatesinpresent-daytextbooks derivesfrom Frege(1879) conceptually,andGiuseppePeanoandothers notationally,butevenbeforethem,therewereattemptstodeveloppredicate logicinBoole ’salgebraicstyle.
Toincorporaterelationtheoryintosettheorywemustidentifyrelationswith setsofsomekind.The firststepistoignorethedistinctionbetweenarelation R suchas parentof andwhatissometimescalledthe “graph” oftherelation,the setoforderedpairs(a, b)with a aparentof b.Wewrite Rab or aRb or(a, b) 2 R indifferently.Thesecondstepistoidentifyanorderedpair(a, b)withasetof somekind,mostcommonlyusingthe Wiener–Kuratowski definition:
Table9
Booleanlaws
NameSymbolicstatement
Commutative
Associative
Distributive
Complement
Table10 MoreBooleanlaws
Name Symbolicstatement
Reflexivity x ⊆ x
Antisymmetryif x ⊆ y ∧ y ⊆ x x ¼ y
Transitivityif x ⊆ y ∧ y ⊆ z x ⊆ z
Extrema
∅⊆ x ∧ x ⊆ y
Complementarity x ⊆ y ≡
Latticelaws
a; b ðÞ¼ ag; a; bg fg:
Itwouldbeidletopretendthisrevealswhatorderedpairshavebeenallalong. Itisanattempttodefinesomethingwithallthefeaturesoforderedpairs needed formathematics,withoutgoingbeyondsettheory.Itsacceptabilitydependson prioranalysisofjustwhat is neededformathematics.Theconsensusisthatthe existenceforevery a and b ofauniqueorderedpair(a, b),togetherwiththe followingfundamentallawofpairs,willdo:
FundamentalLawofPairs a; b ðÞ¼ c; d ðÞ iff a ¼ b ∧ c ¼ d :
GiventheWiener–Kuratowskidefinition,theexistenceoftheorderedpair followsbythreeapplicationsofpairingtoget{a}and{a, b}andthen(a, b). Theproofofthefundamentallawwillbeleftasanexercise.Onealsoneeds,for any A and B,theexistenceoftheCartesianproduct:
⊗
Therearetwointerestinglydifferentproofs.The firstbeginsbynoting thatwehavealreadytheexistenceoftheunionof A and B,ofthepower setofthatunion,andofthepowersetofthepowerset;whilealsoeachof ag f and a; bg f isasubsetoftheunionandhence(a, b )de finedthe Wiener– Kuratowskiwayisasubsetofitspowerset.Separationthen giveswhatwewant:
Thesecondbeginsbyapplyingreplacementtwicetoconclude:
Table11 Definedrelation-theoreticnotions
Name Symbol Definition
Domain dom Raj ∃b aRbg f
Range ran Rbj ∃aaRbg f
Restriction
Image
RjCx; b ðÞ2 Rjx 2 C g f
R½C
Inverse R 1
Composition
bj ∃x 2 C xRbg f
b; a ðÞj aRbg f
R ∘ S a; c ðÞj ∃b 2 BaRb Λ bRc ðÞg f
ag ⊗ B ¼ a; b ðÞj b 2 Bg existsforeach a 2 A: f f ag ⊗ B fj a 2 Ag exists: f
Then A ⊗ B ¼ ∪ ag ⊗ B fj a 2 Ag f exists.
Thereisahostofdefinitionsthatcannowbemade,someassembledin Table 11.(Itistraditionaltoillustratesomeofthembykinshiprelations.Thusthe inverseof parentof is childof andthecompositionof sisterof and parentof is auntof.)Notethatinthetablethe a andthe b indom R andran R will automaticallybelongto ∪∪ R undertheWiener–Kuratowskidefinition,so domainandrangeexistbyseparation.Weleavetheexistencequestioninthe othercasestothereader.Thereareothertermsinuse:theinverseof R is alternativelycalledthe converse,while R 1 ½D iscalledthe preimage of D,and dom R ∪ ran R the field of R.
Studentsofmathematicsencounterthesedefinitionsgraduallyinthecourse ofstudyingthisorthatbranchofmathematics,ratherthaninablocinaseparate courseonsettheory.Readersencounteringthelotallatoncemaythinkof learningthemaslikelearningvocabularyinaforeignlanguage,andtryto absorbafeweachday.
Thesenotionsareconnectedbyanendlesslistoflittlelaws,suchas R½C ¼ ran RjC ; thatfollowatoncefromthedefinitions.Suchlawsoccupy pageafterwearypageinthe firstvolumeofWhiteheadandRussell’smonumental PrincipiaMathematica (1910).Afewareusuallysingledoutforspecial mention:
Thereisalsospecialvocabularyforspecialfeaturesarelationmayormaynot possess,shownin Table12 (whereindefiningconditionsaresupposedtohold
Table12 Propertiesofrelations
Name Definition
Reflexive
aRa
Irreflexive :aRa
Symmetric
Antisymmetric
Transitive
Connected(reflexivecase)
Connected(irreflexivecase)
aRb bRa
aRb ˄ bRa a ¼ b
aRb ˄ bRc aRc
aRb ˅ bRa
aRb ˅ a ¼ b ˅ bRa
for alla, b, c inthe fieldoftherelation).Officially,arelationisa set ofordered pairs,sosincethereisno set of all pairs(x, y)suchthat x 2 y theassumption thatthereiscanwithoutmuchdifficultybeshowntoimplytheexistenceofa universalset – elementhoodisnotarelationintheofficialsense;neitheris inclusion ⊆ .Wecanstillcallthemrelationships,andapplytheterminologyin thetable.(ThereisavariantNBGofZFCinwhichtheyaretreatedmore formallyas “classes,” collectionsassumedoverandabovesets.)Thusinclusion isreflexive,elementhoodirreflexive.
4.3Functions,Orders,Equivalences
Forfuturereference,definitionswillbecollectednowpertainingtothreekinds ofrelationubiquitousinmathematics.Thismaterial,admittedlyabitdryuntil wearereadytotakeupsubstantiveexamples,maybeskimmedandreferred backtoasneededlater.A function isarelation R suchthatforany a indom R thereisa uniqueb inran R with aRb.Oftenoneuseslowercaseletters f, g for functions.Theunique b with afb iscalledthe value oroutputfor argument or input a,anddenoted f(a).If,additionally,forany b intherangethereisexactly one a inthedomainwith f ðaÞ¼ b; thefunction f iscalled injective .Thenotation f : A → B indicatesthat f isafunctionwithdom f ¼ A andran f ⊆ B.Ifran f ¼ B,thenthefunctioniscalled surjective withrespectto B,while bijective meansbothinjectiveandsurjective,andafunctionthatisin-orsur-orbijective iscalledan in- or sur- or bijection.(Olderterminologywas “one-to-one” and “onto” and “correspondence.”)Incertaincontexts,itprovesconvenienttowrite thevaluesofafunction X withdom X ¼ I notas X ðiÞ butas Xi .Withthis notation,wewritetherangeas Xi j i 2 I g f andcallitan indexedfamily with indexsetI.TheDeMorgananddistributivelawsof Table9 in Section4.2, amongothers,generalizetoindexedfamilies:
A two-place functionissimplyafunctionwhoseargumentsareorderedpairs, butwewrite fa; b ðÞ ratherthan f ð a; b ðÞÞ forsimplicity.Notethat,officially,a functionisa set oforderedpairs,sowecannotcallintersectionanduniontwoplacefunctions.Wecallthem operations,andapplythesameterminologyof “associative” and “commutative” andsoontothemastotwo-placefunctions.
Itiseasilyseenthat identity onaset A; i ¼ a; a ðÞja 2 Ag f isafunction;also thattheinverse f 1 ofafunction f isafunctionif f isaninjection.Also,the composition f ∘ g forfunctions f and g isafunctionifran f ⊆ dom g.Agood exerciseistoverifythattheidentityfunctionisabijection,thattheinverseofa bijectionisabijection,andthatthecompositionoftwobijectionsisabijection. Thenifwedefinetwosetstobe equipollent or equinumerous ifthereisa bijectionbetweenthem,asCantordid,itfollowsthatequipollenceorequinumerosityisareflexive,symmetric,andtransitiverelationship.
Noticethatunderthedefinitionsusedsofar,beginningfromwhatisthemost naturaldefinitionofcompositionwhenworkingonthegeneraltheoryofrelations,wegetforfunctionsthat f ∘ g ðÞ a ðÞ¼ g ð faðÞÞ,whereinthenotationthe orderof f and g getswitched.Themoreusualapproachinmainstreammathematics,whichratherseldomconsiderscompositionofrelationsotherthanfunctions,modifiesthedefinitionsoastogettheresult g ∘ f ðÞ a ðÞ¼ g ð faðÞÞ. Apartialorderisarelationthatisreflexive,antisymmetric,andtransitive. Oftenwewrite ≤ orasimilarsymbolforapartialorder,andthenuserelated notationsinmoreorlessobvioussenses:
A minimal element x of aset X isonesuchthatforno y in X is y < x.A minimum or least isonesuchthatforall y in X wehave x ≤ y.Theterms maximal and maximum or greatest areusedanalogously.Theminimal versus minimum distinctioncollapses,allowingbothtobeabbreviated min,forconnectedpartial orders,called total or linear ordersorsimply orders.A chain inapartialorderis asubset C ofits fieldconnectedby ≤ .A wellorder isoneinwhichevery nonemptysubsetofthe fieldhasaleastelement.Asetis wellorderable ifthere existssomewellorderonit(inwhichcase,therewillalsoexistothers). Sometimes,itismoreconvenienttostartwiththenotionofa strict order <,a relationthatisirreflexive,transitive,andconnected,andthinkof ≤ asdefinedin termsof <