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Philosophy and Logic

set Theory

John
Burgess

Downloaded from https://www.cambridge.org/core. IP address: 194.176.167.34, on 27 Jan 2022 at 13:37:29, subject to the Cambridge Core terms

of use, available at https://www.cambridge.org/core/terms https://doi.org/10.1017/9781108981828

ElementsinPhilosophyandLogic

FrederickKroon

TheUniversityofAuckland

SETTHEORY

JohnP.Burgess

PrincetonUniversity

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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}

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 <

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