Keywords:settheory,knowledgerepresentation,informationmanagement,commonsensereasoning,nonwellfoundedsets(hypersets)

Transcription

1 NonstandardSetTheoriesand DepartmentofComputerEngineeringandInformationScience,BilkentUniversity,Bilkent, VAROLAKMAN InformationManagement Ankara06533,Turkey ofthesealternativesettheoriestoinformationorknowledgemanagementaresurveyed. studyofvariousnonstandardtreatmentsofsettheoryfromthisperspectiveisoered.applications variousareasofarticialintelligence,inparticularintelligentinformationsystems.inthispaper,a Abstract.Themeritsofsettheoryasafoundationaltoolinmathematicsstimulateitsusein DepartmentofComputerEngineering,BosphorusUniversity,Bebek,Istanbul80815,Turkey MUJDATPAKKAN Keywords:settheory,knowledgerepresentation,informationmanagement,commonsensereasoning,nonwellfoundedsets(hypersets) 1.Introduction Settheoryisabranchofmodernmathematicswithauniqueplacebecausevariousother branchescanbeformallydenedwithinit.forexample,book1oftheinuentialworksof volumes.bourbakisaidin1949(goldblatt,1984)1: N.Bourbakiisdevotedtothetheoryofsetsandprovidestheframeworkfortheremaining Carthy(1983,1984)hasemphasizedtheneedforfundamentalresearchinAIandclaimed thataineedsmathematicalandlogicaltheoryinvolvingconceptualinnovations.inan openingaddress(mccarthy,1985),hestressedthefeasibilityofusingsettheoryinaibecausethereisconsiderablebeauty,economy,andnaturalnessinusingsetsforinformation ThisbringsupthepossibilityofusingsettheoryinfoundationalstudiesinAI.Mc- wholeofthemathematicsofthepresentday." theoryofsets:::[o]nthesefoundationsicanstatethaticanbuildupthe \[A]llmathematicaltheoriesmayberegardedasextensionsofthegeneral Wethenconsiderthealternativesettheorieswhichhavebeenproposedtoovercomethe applicationsofthisformaltheorytotheproblemsofintelligentinformationmanagement. modelingandknowledgerepresentation. ofsettheory,eachinnovatingdierentaspectssuchasurelements,cumulativehierarchy, limitationsofthestandardtheory.finally,wesurveyvarious\nonstandard"treatments details whichthereadercanndintextslike(halmos,1974),(fraenkeletal.,1973),and (Suppes,1972) andinsteadfocusontheunderlyingconcepts.whileweassumelittleor notechnicalbackgroundinsettheoryperse,wehopethatthereaderisinterestedinthe Inthispaper,werstgiveabriefreviewof\classical"settheory.Weavoidthetechnical self-reference,cardinality,well-orderings,andsoon.itisshownthatsuchtreatments whichareallveryrecentandsometimesesoteric arequiteusefultotheiiscommunity, 1

2 forthereareassortedtechnicalproblemsininformationmanagement(e.g.,commonsense G.Cantor'sworkonthetheoryofinniteseriesshouldbeconsideredasthefoundation 2.Earlydevelopmentsinsettheory reasoning,terminologicallogics,etc.)thatmayprotfromsuchnonstandardapproaches. oftheresearchinsettheory.incantor'sconception,asetisacollectionintoawholeof denite,distinctobjectsofourperceptionorourthought(theelementsoftheset).this histheoremscanbederivedfromthreeaxioms:extensionalitywhichstatesthattwosets itsmembers. iftheobjectisamemberofthatset.inotherwords,asetiscompletelydeterminedby propertyofdenitenessimpliesthatgivenasetandanobject,itispossibletodetermine areidenticaliftheyhavethesamemembers,abstractionwhichstatesthatforanygiven Choicewhichstatesthatifbisaset,allofwhoseelementsarenonemptysetsnotwoof propertythereisasetwhosemembersarejustthoseentitieshavingthatproperty,and whichhaveanyelementsincommon,thenthereisasetcwhichhaspreciselyoneelement incommonwitheachelementofb. Intheinitialstagesofhisresearch,Cantordidnotworkfromaxioms.However,allof selves."thispropertycanbedenotedas:(x2x)(orsimplyx62x)inthelanguageof rstorderlogic.theaxiomofabstractionitselfcanbeformulatedas dictioncouldbederivedfromtheaxiomofabstraction AxiomVinFrege'ssystem by considering\thesetofallthingswhichhavethepropertyofnotbeingmembersofthem- whichwasdevelopedoncantor'snaiveconception(vanheijenhoort,1967).thiscontra- itsevolution.in1902,b.russellfoundacontradictioning.frege'sfoundationalsystem Thetheorywassoonthreatenedbytheintroductionofsomeparadoxeswhichledto where'(x)isaformulainwhichyisfree.inthecaseofrussell'sparadox'(x)=x62x itself.thisisparadoxicalsincevbydenitionisthemostinclusiveset.thisisthesocalledcantor'sparadoxandledtodiscussionsonthesizeofcomprehensiblesets.strictly speaking,itwasfrege'sfoundationalsystemthatwasoverthrownbyrussell'sparadox, encompassedasawholeandtreatedasamathematicalobject." (Goldblatt,1984):\Somepredicateshaveextensionsthataretoolargetobesuccessfully theoryandinalternateapproaches.however,itisbelievedthataxiomaticsettheory wouldstillhaveevolvedintheabsenceofparadoxesbecauseofthecontinuoussearchfor foundationalprinciples. 2.1.Alternateaxiomatizations Thenewaxiomatizationstookacommonstepforovercomingthedecienciesofthenaive Paradoxesoftheprecedingsortwereinstrumentalinnewaxiomatizationsoftheset andwehave:8x9y[x2y$x62x].substitutingyforx,wereachthecontradiction y2y$y62y. knowncantor'stheoremstatesthatthepowersetofvhasagreatercardinalitythanv Anotherantinomyoccurredwiththe\setofallsets,"V=fx:x=xg.Thewell- 8x9y[x2y$'(x)]; \limitationofsize"constraint.later,j.vonneumannwouldclarifythisproblemasfollows notcantor'snaivesettheory.thelattercametogriefpreciselybecauseofthepreceding approachbyintroducingclasses.nbg,whichwasproposedbyvonneumann(1925)and 2

3 laterrevisedandsimpliedbyp.bernays(1937)andgodel(1940),wasthemostpopular ParadoxisavoidedbyshowingthattheclassY=fx:x62xgisaproperclass,notaset. ofthese.innbg,therearethreeprimitivenotions:set,class,andmembership.classes areconsideredastotalitiescorrespondingtosome,butnotnecessarilyall,properties.the classicalparadoxesareavoidedbyrecognizingtwotypesofclasses:setsandproperclasses. Aclassisasetifitisamemberofsomeclass.Otherwise,itisaproperclass.Russell's Visalsoconsideredasaproperclass.TheaxiomsofNBGaresimplychosenwithrespect tothelimitationofsizeconstraint. on.themembershiprelationisdenedbetweensetsofdierenttypes,e.g.,xn2yn+1. aresaidtobeoftype1,setswhoseelementsareoftype1aresaidtobeoftype2,andso Mofindividuals.TheelementsofMareoftype0.Setswhosemembersareoftype0 withitslanguageandaregenerallydubbedtype-theoretical.russellandwhitehead's Inthistheory,ahierarchyoftypesisestablishedtoforbidcircularityandhenceavoid paradoxes.forthispurpose,theuniverseisdividedintotypes,startingwithacollection TheoryofTypesistheearliestandmostpopularofthese(Whitehead&Russell,1910). Otherapproachesagainstthedecienciesofthenaiveapproachalternativelyplayed thetheoryoftypes,sincetheproblematicwisnotstratied. overcomesomeunpleasantaspectsoftheformer(quine,1937).nfusesonlyonekindof variableandabinarypredicateletter2formembership.anotioncalledstraticationis avoided. introducedtomaintainthehierarchyoftypes2.innf,russell'sparadoxisavoidedasin Therefore,x62xisnotevenavalidformulainthistheoryandRussell'sParadoxistrivially 2.2.ZFsettheory SimilartotheTheoryofTypesisQuine'sNewFoundations(NF)whichheinventedto elements. tensionalitywhichstatesthattwosetsareequalifandonlyif(i)theyhavethesame asobjectsand2formembership.equalityisdenedexternallybytheaxiomofex- (1922)byintroducinganewaxiom.ZFiscarriedoutinalanguagewhichincludessets zationwasbye.zermelo(1908).a.a.fraenkel(1922)observedaweaknessofzermelo's Zermelo-Fraenkel(ZF)istheearliestaxiomaticsysteminsettheory.Therstaxiomati- systemandproposedawaytoovercomeit.hisproposalwasreformulatedbyt.skolem onlysetwhoseexistenceisexplicitlystated.thepairsetaxiomstatestheexistenceofa setwhichhasamemberwhentheonlyexistingsetis;.sothesetf;gcannowbeformed nowandwehavetwoobjects;andf;g.theapplicationoftheaxiomrepetitivelyyields intheuniverseofthistheory.thecumulativehierarchyworksasfollows(tiles,1989). stepwisemannerbyvariousdenedoperators.hencetherearenoindividuals(urelements) intentionistobuildupmathematicsbystartingwith;andthenconstructfurthersetsina TheNullSetAxiomguaranteesthatthereisasetwithnoelements,i.e.,;.Thisisthe ZF'sessentialfeatureisthecumulativehierarchyitproposes(Parsons,1977).The whichstatestheexistenceofatleastoneinniteset,fromwhichotherinnitesetscanbe formed.thesetwhichtheaxiomassertstoexistisf;;f;g;f;;f;gg;f;;f;g;f;;f;ggg;:::g. unionofalreadyexistingsets.thussff;;f;gg;ff;;f;gggg=f;;f;g;f;;f;ggg.however itshouldbenotedthatallthesesetswillbenitebecauseonlynitelymanysetscanbe formedbyapplyingpairsetandsumsetnitelymanytimes.itistheaxiomofinnity whichstatestheexistenceofsetscontaininganynitenumberofelementsbydeningthe anynitenumberofsets,eachwithonlyoneortwoelements.itisthesumsetaxiom 3

4 withthe;andextendstoinnity.itcanbenoticedthatcumulativehierarchyproduces ThecumulativehierarchyisdepictedinFigure1.Thus,theZFuniversesimplystarts allnitesetsandmanyinniteones,butitdoesnotproduceallinnitesets(e.g.,v). WhiletherstveaxiomsofZFarequiteobvious,theAxiomofFoundationcannot Figure1:Thecumulativehierarchy beconsideredso.theaxiomstatesthateverysethaselementswhichareminimalwith respecttomembership,i.e.,noinnitesetcancontainaninnitesequenceofmembers beenguaranteedbythepreviousaxiomsandwhichsatisfyaproperty': whichrequireaninnityofiterationsofanoperationtoformsets.italsoforbidssets :::2x32x22x12x0.Innitesetscanonlycontainsetswhichareformedbyanite numberofiterationsofsetformation.hencethisaxiomforbidstheformationofsets theproblematicsetx=fxgcannotbeshowntoexist3.theaxiomofseparationmakes itpossibletocollecttogetherallthesetsbelongingtoasetwhoseexistencehasalready whicharemembersofthemselves,i.e.,circularsets.russell'sparadoxisavoidedsince offunctionsfortheformationofsetsbutstillhastherestrictionoftheoriginalaxiomof descriptiontogetherintoaset,asassumedbycantor.itonlygivespermissiontoform Theaxiomdoesnotallowtosimplycollectallthethingssatisfyingagivenmeaningful subsetsofasetwhoseexistenceisalreadyguaranteed.italsoforbidstheuniverseofsets tobeconsideredasaset,henceavoidingthecantor'sparadoxofthesetofallsets.the AxiomofReplacementisastrongerversionoftheAxiomofSeparation.Itallowstheuse 8x9u[x2u$x2v&'(x)]: cardinality. Axiomisanimportantaxiom,becauseCantor'snotionofaninnitenumberwasinspired axiomschemes.therefore,zfisnotnitelyaxiomatizable4. byshowingthatforanyset,thecardinalityofitspowersetmustbegreaterthanits Separation.Itshouldbenotedthatthesetwoaxiomsareinfactnotsingleaxiomsbut usedinaproof.zfwiththeaxiomofchoiceisknownaszfc. set.theformaldenitionofthepoweroperation,p,isp(x)=fy:yxg.thepower TheAxiomofChoiceisnotconsideredasabasicaxiomandisexplicitlystatedwhen ThePowerAxiomstatestheexistenceofthesetofallsubsetsofapreviouslydened startingwith;anditeratingthepowersetoperationpwherearankfunctionr()is hasaformaltreatment.theclasswfofwellfoundedsetsisdenedrecursivelyinzf denedfor2ord,theclassofallordinals5: Itshouldbenotedthattheinformalnotionofcumulativehierarchysummarizedabove R(0)=;, 4

5 Figure2:TheWFuniverseintermsofordinals α. ω R(α). justiedbythecommonacceptanceofthestatementthattheuniverseofzfisequivalent R(+1)=P(R()), 2 totheuniverseofwf. TheWFuniverseisdepictedinFigure2whichbearsaresemblancetoFigure1.Thisis R()=S<R()whenisalimitordinal, WF=SfR():2Ordg. 1 ZFandNBGproduceessentiallyequivalentsettheories,sinceitcanbeshownthat 0 NBGisaconservativeextensionofZF,i.e.,foranysentence',ifZFj=',thenNBG j='.themaindierencebetweenthetwoisthatnbgisnitelyaxiomatizable,whereas ZFisnot.Still,mostofthecurrentresearchinsettheoryisbeingcarriedoutinZF. Nevertheless,ZFhasitsowndrawbacks(Barwise,1975).Whilethecumulativehierarchy providesapreciseformulationofmanymathematicalconcepts,itmaybeaskedwhether simplefactsshouldhavesimpleproofs,isquiteoftenviolatedinzf(barwise,1975).for tohavearound,e.g.,circularsets.clearly,thetheoryisweakinapplicationsinvolving self-referencebecausecircularsetsareprohibitedbytheaxiomoffoundation. itislimiting,inthesensethatitmightbeomittingsomeinterestingsetsonewouldlike pairshx;yisuchthatx2aandy2b,reliesonthepowersetaxiom6. example,vericationofatrivialfactliketheexistenceinzfofab,thesetofallordered thisdistinctiondisappearsinzf.besides,theprincipleofparsimony,whichstatesthat tureofthesetsdenedinitareoccasionallylost.forexample,beingaprimenumber between6and12isadierentpropertythanbeingasolutiontox2 18x+77=0,but Strangelyenough,ZFistoostronginsomeways.Importantdierencesonthena- AdmissiblesetsareformalizedinarstordersettheorycalledKripke-Platek(KP).BarwiseweakenedKPbyreadmittingtheurelementsandcalledtheresultingsystemKPU (Barwise,1975).Urelementsaretheobjects(orindividuals)withnoelements,i.e.,they 3.1.Admissiblesets 3.Alternateapproaches canoccurontheleftof2,butnotontheright.theyarenotconsideredinzfbecausezf 5

6 Figure3:Theuniverseofadmissiblesets V (α) M constructionprocess. decidedtoincludethem. axiomspair,union,and0-separation7treattheprinciplesofsetconstruction.these TheaxiomsofExtensionalityandFoundationareaboutthebasicnatureofsets.The isstrongenoughtolivewithoutthem.butsincekpuisaweakversionofkp,barwise importantaxiomof0-collectionassuresthatthereareenoughstagesinthe(hierarchical) veaxiomscanbetakenascorrespondingtozfaxiomsofthesameinterpretation.the TheuniverseofadmissiblesetsoveranarbitrarycollectionMofurelementsisdened KPUisformulatedinarstorderlanguageLwithequalityand2.Ithassixaxioms. M recursively: VM(+1)=P(M[VM()), VM(0)=;, thatthekpuuniverseislikethezfuniverse(excludingtheexistenceofurelements),since itsupportsthesameideaofcumulativehierarchy. ThisuniverseisdepictedinFigure3,adaptedfromBarwise(1977).Itshouldbenoticed IfMisastructure8forL,thenanadmissiblesetoverMisamodelUMofKPUof VM=SVM(). VM()=S<VM(),ifisalimitordinal, spectstheprincipleofparsimony.(thelatterclaimwillbeprovedinthesequel.)butit theformum=(m;a;2),whereaisanonemptysetofnonurelementsand2isdened stillcannotdealwithself-referencebecauseofitshierarchicalnature. inma.apureadmissiblesetisanadmissiblesetwithnourelements,i.e.,itisamodel ofkp. 3.2.Hypersets ItwasD.Mirimano(1917)whorststatedthefundamentaldierencebetweenwellfoundedandnonwellfoundedsets.Hecalledsetswithnoinnitedescendingmembership KPUisaneleganttheorywhichsupportstheconceptofcumulativehierarchyandre- sequencewellfoundedandothersnonwellfounded.nonwellfoundedsetshavebeenextensivelystudiedthroughdecades,butdidnotshowupinnotableapplicationsuntilaczel (1988).Thisisprobablyduetothefactthattheclassicalwellfoundeduniversewasa rathersatisfyingdomainforthepracticingmathematician \themathematicianinthe street"(barwise,1985).aczel'sworkonnonwellfoundedsetsevolvedfromhisinterestin 6

7 Figure4:a=fb;fc;dgginhypersetnotation a {c,d} b c d relationbetweenxandy(i.e.,x2y). modelingconcurrentprocesses.headoptedthegraphrepresentationforsetstouseinhis sentation(aczel,1988),whereanarrowfromanodextoanodeydenotesthemembership theory.aseta=fb;fc;dggcanbeunambiguouslydepictedasinfigure4inthisrepre- Figure5:TheAczelpictureof Ω fromthezfcandaddingtheafaresultsinthehypersettheoryorzfc /AFA.(ZFC andnonwellfoundedotherwise.aczel'santi-foundationaxiom(afa)statesthatevery graph,wellfoundedornot,picturesauniqueset.removingtheaxiomoffoundation(fa) isthepictureoftheuniqueset=fg. sincegraphsofarbitraryformareallowed,includingtheonescontainingpropercycles, onecanrepresentself-referringsets(barwise,1992).forexample,thegraphinfigure5 withoutthefaisdenotedaszfc.)whatisadvantageouswiththenewtheoryisthat Aset(picturedbyagraph)iscalledwellfoundedifithasnoinnitepathsorcycles, nodesarethenitepathsoftheapg9whichstartfromthepointoftheapg,whoseedges arepairsofpathshn0!!n;n0!!n!n0i,andwhoserootisthe pathn0oflengthoneiscalledtheunfoldingofthatapg.theunfoldingofanapgalways picturesanysetpicturedbythatapg.unfoldingtheapginfigure5resultsinaninnite structuraldierencebetweenthem.therefore,allofthethreegraphsinfigure6depict tree,analogousto=fffggg. Thepictureofasetcanbeunfoldedintoatreepictureofthesameset.Thetreewhose theuniquenonwellfoundedset. AccordingtoAczel'sconception,fortwosetstobedierent,thereshouldbeagenuine AczeldevelopshisownExtensionalityconceptbyintroducingthenotionofbisimula- 7

8 Figure6:Otherpicturesof Ω Ω Ω Ω Ω Ω Ω relationrg1g2satisfyingthefollowingconditions: tion.abisimulationbetweentwoapg's,g1withpointp1andg2withpointp2,isa Figure7:TheAFAuniverse A F A universe 2.ifnRmthen 1.p1Rp2 foreveryedgen!n0ofg1,thereexistsanedgem!m0ofg2suchthat well - founded universe universe,becauseitincludesthenonwellfoundedsetswhicharenotcoveredbythelatter. thismeansthattheypicturethesamesets.itcanbeconcludedthatasetiscompletely determinedbyanygraphwhichpicturesit10. TheAFAuniversecanbedepictedasinFigure7,extendingaroundthewellfounded Twoapg'sG1andG2aresaidtobebisimilar,ifabisimulationexistsbetweenthem; foreveryedgem!m0ofg2,thereexistsanedgen!n0ofg1suchthat n0rm EquationsintheAFAuniverse.Aczel'stheoryincludesanotherimportant universeofhypersetswithatomsfromanothergivenseta0suchthataa0andxis usefulfeature:equationsintheuniverseofhypersets. denedasa0 A.TheelementsofXcanbeconsideredasindeterminatesrangingover X-sets.Asystemofequationsisasetofequations theuniverseva.thesetswhichcancontainatomsfromxintheirconstructionarecalled LetVAbetheuniverseofhypersetswithatomsfromagivensetAandletVA0bethe fx=ax:x2x^axisanx-setg 8

9 foreachx2x.forexample,choosingx=fx;y;zganda=fc;mg(thusa0= fx;y;z;c;mg),considerthesystemofequations setsbutnoatomsaselements),oneforeachx2x,suchthatforeachx2x,bx=ax. Here,isasubstitutionoperation(denedbelow)andaisthepuresetobtainedfroma bysubstitutingbxforeachoccurrenceofanatomxintheconstructionofa. Asolutiontoasystemofequationsisafamilyofpuresetsbx(setswhichcanhaveonly z=fm;xg: x=fc;yg; TheSubstitutionLemmastatesthatforeachfamilyofpuresetsbx,thereexistsaunique y=fc;zg; operationwhichassignsapuresetatoeachx-seta,viz. TheSolutionLemmacannowbestated(Barwise&Moss,1991)11.IfaxisanX-set, setsbxsuchthatforeachx2x,bx=ax. thenthesystemofequationsx=axhasauniquesolution,i.e.,auniquefamilyofpure thereexistsauniquefunctionfforallx2xsuchthat thesetofindeterminates,gafunctionfromxtop(x),andhafunctionfromxtoa, Thislemmacanbestatedsomewhatdierently(Pakkan,1993).LettingXagainbe a=fb:bisanx-setsuchthatb2ag[fx:x2a\xg: Obviously,g(x)isthesetofindeterminatesandh(x)isthesetofatomsineachX-setax ofanequationx=ax.intheaboveexample,g(x)=fyg,g(y)=fzg,g(z)=fxg,and h(x)=fcg,h(y)=fcg,h(z)=fmg,andonecancomputethesolution f(x)=ff(y):y2g(x)g[h(x): systemofequations Asanotherexampledueto(Barwise&Etchemendy,1987),itmaybeveriedthatthe f(x)=fc;fc;fm;xggg; f(y)=fc;fm;fc;yggg; f(z)=fm;fc;fc;zggg: x=fc;m;yg; y=fm;xg; existenceofsomegraphs(thesolutionsoftheequations)withouthavingtodepictthem situationtheory(barwise&perry,1983),databases,etc.sinceitallowsustoassertthe inginformationwhichcanbecastintheformofequations(akman&pakkan,1993),e.g., hasauniquesolutionintheuniverseofhypersetsdepictedinfigure8withx=a,y=b, andz=c. Thistechniqueofsolvingequationsintheuniverseofhypersetscanbeusefulinmodel- z=fx;yg: withgraphs.wenowgiveanexamplefromdatabases AFAandrelationaldatabases.Relationaldatabasesembodydataintabular formsandshowhowcertainobjectsstandincertainrelationstootherobjects.asan 9

10 Figure8:Thesolutiontoasystemofequations c a b C M Figure9:Arelationaldatabase FatherOf MotherOf John Bill Sally Tim John Kitty Kathy Bill Tom Tim Kathy Kitty 10 BrotherOf Bill Kitty

11 Figure10:TheSizeOfrelationfortheprecedingdatabase fsizeofg.adatabasemodelmforrel0iscorrectiftherelationsizeofmcontainsall 3 pairshr;niwherer2relandn=jrj,thecardinalityofr.sucharelationcanbeseen RMisanitebinaryrelationthatholdsinmodelM. 3 infigure10.itmayappearthateverydatabaseforrelcanbeextendedinauniqueway domainsomesetrelofbinaryrelationsymbolssuchthatforeachrelationsymbolr2rel, relations:fatherof,motherof,andbrotherof.adatabasemodelisafunctionmwith exampleadaptedfrom(barwise,1990),thedatabaseinfigure9includesthreebinary IfonewantstoaddanewrelationsymbolSizeOftothisdatabase,thenRel0=Rel[ 1 toacorrectdatabaseforrel0.unfortunately,thisisnotso. 4 databases,thenthesolutionoftheequation SizeOfSizeOfn.ButthisisnottrueinZFCbecauseotherwisehSizeOf;ni2SizeOf. causeifmiscorrect,thentherelationsizeofstandsinrelationsizeofton,denotedby IfHypersetTheoryisusedasthemeta-theoryinsteadofZFCinmodelingsuch AssumingtheFA,itcanbeshownthattherearenocorrectdatabasemodels.Be- goodknowledgeofthatworldandbeabletomakeinferencesoutoftheirknowledge.the 4.Commonsensesettheory Ifwewanttodesignarticialsystemswhichwillworkintherealworld,theymusthavea (whichcanbefoundbyapplyingthesolutionlemma)isthedesiredsizeofrelation. x=fhrm;jrmji:r2relg[fhx;jrelj+1ig addsthatthisisaverydicultthing.hestatesthat\aprogramhascommonsenseif programistodeneanaivecommonsenseviewoftheworldpreciselyenough,butalso fromthisknowledgeareknownascommonsense.anyintelligenttaskrequiresittosome degreeanddesigningprogramswithcommonsenseisoneofthemostimportantproblems inai.mccarthy(1959)claimsthatthersttaskintheconstructionofageneralintelligent itautomaticallydeducesforitselfasucientlywideclassofimmediateconsequencesof commonknowledgewhichispossessedbyanychildandthemethodsofmakinginferences mathematics,canalsoberepresentedwithsets.forexample,thenotionof\society"can anythingitistoldandwhatitalreadyknows." ibleunit,orascomposedofotherparts,asinmathematics.relationships,againasin beconsideredtobearelationshipbetweenasetofpeople,rules,customs,traditions,etc. Whatisproblematichereisthatcommonsenseideasdonothaveveryprecisedenitions (adaptedfromwebster'sninthnewcollegiatedictionary): sincetherealworldistooimprecise.forexample,considerthedenitionof\society" Itappearsthatincommonsensereasoningaconceptcanbeconsideredasanindivis- 11

12 belefttotheexperienceofthereaderwithallthesecomplexentities. appeartobeascomplexasthedenitionitself.sothisdenitionshouldprobablybetter Nevertheless,whetherornotasettheoreticaldenitionisgiven,setsareusefulfor Itshouldbenotedthatthenotions\tradition,"\institution,"and\existence"also activitiesandinterestsachoicetocometogethertogivesupporttoandbe supportedbyeachotherandcontinuetheirexistence." \Societygivespeoplehavingcommontraditions,institutions,andcollective \traditions"disjointfromthesetof\laws"(onecanquicklyimaginetwoseparatecircles conceptualizingcommonsenseterms.forexample,wemaywanttoconsiderthesetof reasoningpointofviewandexaminetheset-theoreticprincipleswhichcannotbeexcluded ofavenndiagram).wemaynothaveawellformedformula(w)whichdeneseither fromsuchreasoning. set-theoretictechnicalitieshaveaplaceinourtheory,werstlookfromthecommonsense concentrateonthepropertiesofsuchatheory.insteadofdirectlycheckingifcertain importantandsimplycorrespondstothesetformationprocessofformalsettheories,i.e., thecomprehensionprinciple. ofthesesets.suchaformationprocessofcollectingentitiesforfurtherthoughtisstill 4.1.Desirableproperties Havingdecidedtoinvestigatetheuseofsetsincommonsensereasoning,wehaveto setisacollectionofindividualssatisfyingaproperty.thisiswhatexactlycorresponds totheunrestrictedcomprehensionaxiomofcantor.however,wehaveseenthatthis toallowurelements.thisseemsliketherightthingtodobecauseinanaivesense,a Werstbeginwiththegeneralprinciplesofsetformation.Therstchoiceiswhether formedoutofpreviouslyformedentities,thenotionofcumulativehierarchyimmediately beconsideredanindividualitself.sincewearetalkingabouttheindividualsasentities comestomind. areledtothequestionwhenthecollectionofallindividualssatisfyinganexpressioncan leadstorussell'sparadoxinzf.theproblemariseswhenweuseasetwhosecompletion isnotoveryetintheformationofanotherset,oreveninitsownformation.thenwe Theory). stage(shoeneld,1977)(butnotnecessarilyattheprecedingstage,asinrussell'stype lements(ifincludedinthetheory)andthesetswhichhavebeenformedatsomeprevious Inthecumulativehierarchy,anysetformedatsomestagemustbeconsistingoftheuretratedbythehierarchicalconstructinFigure11,wherewehavebricksasindividuals,and intuitionandissupportedbymanyexistingtheories,viz.zfandkpu.itcanbeillus- maketowersoutofbricks,andthenmakewallsoutoftowers,andsoon(perlis,1988). Thecumulativehierarchyisoneofthemostcommonconstructionmechanismsofour beamemberofitself,sinceitalsoisanonprotorganization?thisisnotanunexpected suchsetsareusedintheirownformation.circularityisobviouslyacommonmeansof individualsandthesetofallnonprotorganizationsisalsoaset;alltheseareexpressible inthecumulativehierarchy.butwhatifthesetofallnonprotorganizationswantsto commonsenseknowledgerepresentation.forexample,nonprotorganizationsaresetsof event(figure12)becausethisumbrellaorganizationmaybenetfromhavingthestatus ofanonprotorganization(e.g.,taxexemption,etc.)(perlis,1988). Atthispoint,theproblemofsetswhichcanbemembersofthemselvesarises,since

13 bricks towers made of bricks Figure11:Aconstructionalacumulativehierarchy walls made of towers rooms made of walls buildings made of rooms... Figure12:Nonprotorganizations Red Cross 13? UNICEF Amnesty NPO NPO International

14 sets.thisisanimportantissueinrepresentationofmeta-knowledgeandisaddressedin warematchedwithasetofnamesofw,therebyallowingself-referencebytheuseof frameworksforsemanticsisespeciallyemphasized.however,suchformalizationswhich (Feferman,1984)and(Perlis,1985).Inthesereferences,amethodwhichreies creates thenamehasstrongrelationshipwiththeformula,ispresented.inthisway,anysetof asyntactictermfromapredicateexpression awintoanameforthewassertingthat names.in(feferman,1984),theneedfortype-free(admittinginstancesofself-application) Thusweconcludethatapossiblecommonsensesettheoryshouldalsoallowcircular alsocapturethecumulativehierarchyprinciplearenotverycommon.amongtheories revisedsofar,aczel'stheoryistheonlyonewhichallowscircularity.byproposinghis preservedthehierarchicalnatureoftheoriginalaxiomatization. thatcanenterintorelationswithotherparts(barwise&perry,1983).theirinternal Situationsforknowledgerepresentation.Situationsarepartsofthereality 4.2.Applications structuresaresetsoffactsandhencetheycanbemodeledbysets.therehasbeena Anti-FoundationAxiom,AczeloverrodetheFAofZFwhichprohibitscircularsets,but considerabledealofworkonthisespeciallybybarwisehimself(barwise,1989a).he usedhisadmissiblesettheory(barwise,1975)astheprincipalmathematicaltoolin thenthisonesurelycontainsitself. announcementwillnotberepeated."ifannouncementsareassumedtobesituations, situation,says,bysaying\thissituation,"ourutteranceisalsoapartofthatsituation. Asanotherexample,onesometimeshearsradioannouncementsconcludingwith\This whichweutterthestatement\thisisaboringsituation."whilewearereferringtoa Circularsituationsarecommoninourdailylife.Forexampleconsiderthesituationin problemsandthendiscoveredthataczel'stheorycouldbeasolution(barwise,1989c). thebeginning.however,inthehandlingofcircularsituations,hewasconfrontedwith persetsandsatisfying12: BarwisedenedtheoperationM(tomodelsituationswithsets)takingvaluesinhy- ifbisnotasituationorstateofaairs,thenm(b)=b, if=hr;a;ii,thenm()=hr;b;ii(whichiscalledastatemodel),wherebisa authorsconcentrateontheconceptoftruth.inthisstudy,twoconceptionsoftruthare totheabsenceofauniversalsetinzf). Usingthisoperation,Barwiseprovesthatthereisnolargestsituation(corresponding Wealsoseeatreatmentofself-referencein(Barwise&Etchemendy,1987),wherethe ifsisasituation,thenm(s)=fm():sj=g. functiononthedomainofasatisfyingb(x)=m(a(x)), examined,primarilyonthebasisofthenotoriousliarparadox13.theauthorsmakeuse from(barwise&etchemendy,1987)),wherehe;p;iidenotesthatthepropositionphas Englishintenwords"wouldberepresentedinAczel'stheoryasinFigure13(adapted wetakeetobethepropertyof\beingexpressibleinenglishintenwords"). ofaczel'stheoryforthispurpose.astatementlike\thissentenceisnotexpressiblein thepropertyeifi=1,anditdoesnothaveitifi=0(whichisthecaseinfigure13if 14

15 p = <E,p,0> = <E,<p,0>> Figure13:TheAczelpictureof\ThissentenceisnotexpressibleinEnglishintenwords" {E} {E,<p,0>} <p,0> Commonknowledge(mutualinformation).TwocardplayersP1andP2 E {p} {p,0} 0 wouldknowthatp1hasanace.because,ifp1doesnotknowp2hasanace,havingheard answer\no."iftheyaretoldthatatleastoneofthemhasanaceandaskedtheabove questionagain,rsttheybothwouldanswer\no."butuponhearingp1answer\no,"p2 aregivensomecardssuchthateachgetsanace.thus,bothp1andp2knowthatthe thatatleastoneofthemdoes,itcanonlybebecausep1hasanace.obviously,p1would followingisafact: Whenaskedwhethertheyknewiftheotheronehadanaceornot,theybothwould :EitherP1orP2hasanace. p initiallywasknownbyeachofthem,butitwasnotcommonknowledge.onlyafterit information?thisisknownasconway'sparadox.thesolutionreliesonthefactthat beingtoldthatatleastoneofthemhasanacemusthaveaddedsomeinformationto reasonthesameway,too.so,theywouldconcludethateachhasanace.therefore, on.thisiterationcanberepresentedbyaninnitesequenceoffacts(wherekisthe relation\knows"andsisthesituationinwhichtheabovegametakesplace,hence2s): becamecommonknowledge,itgavemoreinformation(barwise,1989d). form:p1knows,p2knows,p1knowsp2knows,p2knowsp1knows,andso thesituation.howcanbeingtoldafactthateachofthemalreadyknewincreasetheir hk;p1;si,hk;p2;si,hk;p1;hk;p2;sii,hk;p2;hk;p1;sii,:::however,consideringthe systemofequations Hence,commonknowledgecanbeviewedasiteratedknowledgeofofthefollowing thesolutionlemmaassertstheexistenceoftheuniquesetss0ands[s0satisfyingthese y=s[fhk;p1;yi;hk;p2;yig; x=fhk;p1;yi;hk;p2;yig; 15

16 equations,respectively,where Figure14:Finitecardinalitywithoutwell-ordering whichcontainsjusttwoinfonsandiscircular Possiblemembership.Onefurtheraspecttobeconsideredis\possible"membershipwhichmighthavemanyapplications,mainlyinlanguageorientedproblems.This Then,thefactthatsiscommonknowledgecanmoreeectivelyberepresentedbys0 s0=fhk;p1;s[s0i;hk;p2;s[s0ig: conceptcanbehandledbyintroducingpartialfunctions functionswhichmightnothave processingofsyntacticinformation,andsituationsemantics(misloveetal.,1990). helpfulinprovidingrepresentationsfordynamicaspectsoflanguagebymakinguseof whichdealswithmodalsentences,i.e.,sentencesofnecessityandpossibility),dynamic correspondingvaluesforsomeoftheirarguments.acommonsensesettheorymaybe partiality.forexample,partialityhasapplicationsinmodality(thepartoflinguistics disjunction).asolutiontothisproblemistorepresentthissituationasapartialset,one neitherassertionaboutthenameoftheboycanbeassuredonthebasisofs(becauseofthe sinwhichyouhavetoguessthenameofaboy,viz. Thissituationcanbemodeledbyasetoftwostatesofaairs.Theproblemhereisthat Wehadmentionedabovethatsituationscanbemodeledbysets.Considerasituation nothavetosupporteitherspecicassertion.anothernotioncalledclarication,whichis withtwo\possible"members.inthiscasesstillsupportsthedisjunctionabovebutdoes akindofgeneralinformation-theoreticordering,helpsdeterminetherealmembersamong sj=theboy'snameisjonortheboy'snameisjohn. possibleones.ifthereexistsanothersituations0,wheres0j=theboy'snameisjon,then Cardinalityandwell-ordering.Imagineaboxof16blackand10whiteballs s0iscalledaclaricationofs. (Figure14).Weknowthatthereare26ballsinthebox,orformally,thecardinalityof thesetofballsintheboxis26.aftershakingthebox,wewouldsaythatthattheballs intheboxarenotorderedanymore,oragainformally,thesetoftheballsdoesnothave awell-ordering.this,however,isnottrueinclassicalsettheory,becauseasetwithnite cardinalitymusthaveawell-ordering(zadrozny,1989). purposesindailyspeech.forexample,ifaskedaboutthenumberofballsintheboxin Figure14,onemighthavesimplyanswered\Manyballs!"So,atleastinprinciple,dierent observethatpeoplealsouseotherquantierslike\many"or\morethanhalf"forcounting Whiletheformalprinciplesofcountingarepreciseenoughformathematics,wecan 16

17 (Zadrozny,1989). shouting"shouldprobablydeclinetoanswerquestionssuchas\whoistherstone?" countingmethodscanbedevelopedforcommonsensereasoning.itisnaturaltoexpect, forexample,thatasystemwhichcanrepresentastatementlike\agroupofkidsare Axiomshouldnotbenecessary.WeobservethisinKPUsettheorywheretheproofis obtainedviadenitionsandsimpleaxioms14(barwise,1975). istenceofasimplefactlikethecartesianproductoftwosetsab,requiredtheuseof ofoneelementofthesetaandoneelementofthesetb.toprovethis,thestrongpower thepoweraxiominzf6.thesetobtainedinthismannerjustconsistsofpairsformed expectationfromacommonsensesettheory.wehaveobservedthattheproofoftheex- Wealsoexpectourtheorytoobeytheparsimonyprinciple.Thisisaverynatural port,cf.hisremark:\theadhoccharacteroftheaxiomsofzermelo-fraenkelsettheory 5.ApplicationsofnonstandardtheoriesinIIS Cowen(1993)showed,inalandmarkpaper,thatresearchersinbothsettheoryandcomputersciencearestudyingsimilarobjects:graphs,conjunctivenormalforms,setsystems, NaDSettheory,thattheabsolutecharacterofZForNBGsettheoriesisnoteasytosup- oftenbeofuseintheother.gilmore(1993)providesevidence,withintheframeworkofhis etc.andtheirinterrelations.themoralis,hebelieves,advancesinoneareacanthen Barwise,andYasukawaandYokota. andclaimedthatautomateddevelopmentofsettheorycouldbeimproved.wewillnow inautomatedtheoremproving.brown(1978)gaveadeductivesystemforelementaryset reviewsomeessentialresearcheorts,byperlis,zadrozny,misloveetal.,dionneetal., anewclausalversionofnbg,comparingitwiththeonegivenin(boyeretal.,1986), theorywhichisbasedontruth-valuepreservingtransformations.quaife(1992)presented andtheequivalentclass-settheoryofgodel-bernaysnaturallyleadstoskepticismabout theirfundamentalroleinmathematics."settheoryhasalsobeenthesubjectofresearch theorywhichhenamedcst0:9y8x[x2y$(x)&ind(x)]: theory(perlis,1988).herstproposedanaxiomschemeofsetformationforanaiveset 5.1.Perlis'swork viduals."however,indcansometimesbecriticallyrich,i.e.,ifisthesamewithind HereisanyformulaandIndisapredicatesymbolwiththeintendedextension\indi- Perlis'sapproachwastodevelopaseriesoftheoriestowardsacompletecommonsenseset itself,thenymaybetoolargetobeanindividual.(thisisthecaseofcantor'saxiom obtainedentities."cst1isconsistentwithrespecttozf.unfortunately,itcannotdeal usingtheso-calledackermann'sscheme: ofabstraction.)therefore,atheoryforahierarchicalextensionforindisrequired.to withself-referringsets. HereHC(x)canbeinterpretedas\xcanbebuiltupasacollectionfrompreviously supportthecumulativehierarchy,perlisextendedthistheorytoanewonecalledcst1 HC(y1)&:::&HC(yn)&8x[(x)!HC(x)]!9z[HC(z)&8x[x2z$(x)]] 17

18 Gilmore-Kripke(Gilmore,1974),whichformsentitiesregardlessoftheiroriginsandselfreferentialaspects.GKsettheoryhasthefollowingaxiomschemewhereeachw(x) hasareication[(x)]withvariablesfreeasinanddistinguishedvariablex PerlisnallyproposedCST2whichisasynthesisoftheuniversalreectiontheoryof axiomstoaugmentgk:(ext1)xy$extx=exty whereydoesnotappearin15.thereisalsoadenitionalequivalence(denotedby) axiom: GKisconsistentwithrespecttoZF(Perlis,1985).Perlisthenproposedthefollowing wz$8x(x2w$x2z): y2[(x)]$(y) theirmembers.thus,whilegkprovidestherepresentationofcircularity,theseaxioms supportthecumulativeconstructionmechanism.thistheorycandealwithproblemslike Theseaxiomsprovideextensionalconstructions,i.e.,collectionsdeterminedonlyby (Aext)y1;:::;yn2HC&8x(x!x2HC)!ext[]2HC&8x(x2[]$(x)) (Ext3)x2HC!9y(x=exty) (Ext2)xextx nonprotorganizationmembershipdescribedearlier(perlis,1988). mentionedearlier(zadrozny,1989). schemebasedonbarwise'skpuforcardinalityfunctions,hencedistinguishingreasoning aboutwell-orderingsfromreasoningaboutcardinalitiesandavoidingtheboxproblem beseparatelymodeledinanexistingsettheory.inparticular,heproposedarepresentation ratherincommonsensetheoriesinvolvingdierentaspectsofsets.hethinksthatthesecan 5.2.Zadrozny'swork Zadroznydoesnotbelieveina\supertheory"ofcommonsensereasoningaboutsets,but EisafunctionfromasubsetofSEintoVV.Itisassumedthatx2yithereexists anedgebetweenxandy.hedenestheedgescorrespondingtothemembersofasetas hisconceptionisatriplehv;se;eiwherevisasetofvertices,seisasetofedges,and ZadroznyinterpretssetsasdirectedgraphsanddoesnotassumetheFA.Agraphin whichcantakevalueslike1;2;3;4,or1;2;3;about-ve,or1;2;3;many).thelastelement ofnumeralswhichislinkedwithsetsbyexistenceofacountingroutinedenotedby#,and preservingmappingfromtheedgesem(s)ofasetsintothenumeralsnums(anentity anaturalnumbernontoaset,i.e.,afunctionfromanumberontothenodesofthe graphoftheset.however,zadroznydenesthecardinalityfunctionasaone-to-oneorder Inclassicalsettheory,thecardinalityofanitesetsisaone-to-onefunctionfrom EM(s)=fe2ES:9v[E(e)=(v;s)]g: k=fa;b;fx;yg;dgwiththreeatomsandonetwo-atomsetisshowninfigure15,adapted oftherangeofthefunctionisthecardinality.therepresentationofthefourelementset from(zadrozny,1989).thecardinalityofthesetisabout{ve,i.e.,thelastelement ofnumswhichistherangeofthemappingfunctionfromtheedgesoftheset.(the cardinalitymightwellbe4ifnumswasdenedas1;2;3;4.)zadroznythenprovestwo 18

19 cfigure15:one-to-oneorderpreservation k importanttheoremsinwhichheshowsthatthereexistsasetxwithnelementswhichdoes theelementsofwhichdonotformaset. nothaveawell-orderingandthereexistsawell-orderingoftypen,i.e.,withnelements, about-five... vis-a-vissettheorycanbefoundin(zadrozny&kim,1993).inthiswork,itisshown a b d multimediaobjectsusingsettheoryandmereology.mereologyisunderstoodasatheory ofpartsandwholes,viz.ittriestoaxiomatizethepropertiesoftherelationpartof.the thatitispossibletowriteformalspecicationsfortheprocessofindexingorretrieving paperarguesthatbothsettheoryandmereologyhavespecicrolestoplayinthetheory MorerecentworkofZadroznytreatingdierentaspectsofcomputationalmereology x y ofmultimedia,fortheybothprovideuswithalanguagetoaddressmultimediaobjects, albeitwithcomplementaryroles:theformerisusedtoindex,thelattertodescribeknowledgeaboutindexes.thisobservationleadstoasimpleformalismregardingthestorage hidesomeofitselements.thereexistsaprotoset?whichisemptyexceptforpackaging. protosets,whichisageneralizationofhf thesetofwellfoundedhereditarilynitesets andretrievalofmultimediamaterial. 5.3.Misloveetal.'swork Mislove,Moss,andOles(1990,1991)developedapartialsettheory,ZFAP,basedon 16.Aprotosetislikeawellfoundedsetexceptthatithassomekindofpackagingwhichcan Fromanitecollectionx1;:::;xn,onecanconstructtheclearprotosetfx1;:::;xngwhich containotherelements,too.wesaythatxisclariedbyy,xvy,ifonecanobtainy hasnopackaging,andthemurkyprotoset[x1;:::;xn]whichhassomeelements,butalso packaging.forexample,amurkysetlike[2;3]contains2and3aselements,butitmight graphofaczel)andsuchthatthereisadecorationdofgwiththerootdecoratedasx, hasapicture,asetgwhichisapartialsetgraph(correspondingtotheaccessiblepointed AFA)totherelationset.Thetwoaxiomsare(i)Pict,whichstatesthateverypartialset consistsoftwoaxiomsandzfaset,therelativizationofallaxiomsofzfa(zf+aczel's fromxbytakingsomepackaginginsidexandreplacingthisbyotherprotosets. tualmembership),2(forpossiblemembership),andset(forsetexistence).thetheory PartialsettheoryhasarstorderlanguageLwiththreerelationsymbols,2(forac- 19

20 and(ii)psa,whichstatesthateverysuchghasauniquedecoration.partialsettheory mostoftheworkinsemanticsoftermsubsumptionlanguagesadoptsanextensionalview. suggestedthatamoreintensionalviewofconceptdescriptionsshouldbetaken.ingeneral, 5.4.Dionneetal.'swork Dionne,Mays,andOles(1992)proposeanewapproachtointensionalsemanticsofterm subsumptionlanguages.theirworkisinspiredbytheresearchofwoods(1991)who ZFAPisthesetofalltheseaxioms.ZFAPisaconservativeextensionofZFA. Thus,conceptsareinterpretedassetsofobjectsfromsomeuniverse.Rolesofconcepts areinterpretedasbinaryrelationsovertheuniverse.conceptdescriptionsarecomplex predicatesandoneinquireswhetheragiveninstancesatisesacomplexpredicate. isdevelopedasanewapproachtosemanticsoftermsubsumptionlanguages.itisshown man&schmolze,1985).theyoerageneraldiscussionofcycles,viz.howtheyarise,and ofnonwellfoundedsets,especiallythebisimulationrelation.aso-calledconceptalgebra howtheyarehandledink-rep.therestrictedlanguagetheyuseallowssimpleconcepts, tions,involvingtermsofthisconceptlanguage.essentialuseismadeofaczel'stheory conceptsformedbyconjunctions,androlesofconceptswhosevaluerestrictionsareother concepts.intheirvision,aknowledgebaseisasetofpossiblymutuallyrecursiveequa- Dionneetal.considerasmallsubsetofK-REP,aKL-ONEstyleoflanguage(Brachcept. 5.5.Barwise'swork Barwise(1989b)attemptedtoproposeasettheory,SituatedSetTheory,notjustforuse denitions,i.e.,conceptswithdierentnamesthatsemanticallystandforthesameconsumptiontesting eveninthepresenceofcycles.conceptalgebrasalsoallowformultiple thatfortheabovesubsetofk-rep,thesealgebrasaccuratelymodeltheprocessofsub- inai,butforgeneraluse.hementionedtheproblemscausedbythecommonviewofset theorywithauniversalsetv,butatthesametimetryingtotreatthisuniverseasan extensionalwhole,lookingfromoutside(whichhenames\unsituatedsettheory").his proposalisahierarchyofuniversesv0v1v2:::whichallowsforauniverseof universesatisfyingthatdescription. thatforanygivendescriptionofthesetsofallsetsv,therewillalwaysbeapartial oneiscurrentlyworkingin.thisproposalsupportsthereectionprinciplewhichstates alowerleveltobeconsideredasanobjectofauniverseofahigherlevel.heleavesthe canstepbackandworkin.therefore,thenotionsof\set,"\properclass,"andthesettheoreticnotions\ordinal,"\cardinal"areallcontextsensitive,dependingontheuniverse orcircular.therearenoparadoxesinthisviewsincethereisalwaysalargeruniverseone axiomswhichtheseuniverseshavetosatisfytoone'sconceptionofset,beitcumulative i.e.,objectswithpartialinformation.(theuniverseofhypersetsona[xisdenotedas VAofhypersetsoverasetAofatomstomodelnonparametricobjects,i.e.,objectswith HypersetTheoryforthispurpose.Forthispurpose,heusedtheobjectsoftheuniverse VA[X],analogoustotheadjunctionofindeterminatesinalgebra.) completeinformationandthesetxofindeterminatestorepresentparametricobjects, Barwise(1989c)alsostudiedthemodelingofpartialinformationandagainexploited 20

21 object[ref=`akmanetal.(1993)']/ journal=`proc.natl.acad.sci.utopia', title=`lecturesincommonsensesettheory', volume=78, pages= , [date=1993/4/24, year=1993 ].Figure16:AnobjectinQUIXOTE kind=paper, authors={`v.akman',`m.pakkan',`m.surav'}, thenpar(a)=;sinceadoesnothaveanyparameters.barwisethendenesananchor wheretc(a)denotesthetransitiveclosureofa,thesetofparametersofa.ifa2va, asafunctionfwithdomain(f)xandrange(f)va Awhichassignssetsto indeterminates.foranya2va[x]andanchorf,a(f)istheobjectobtainedbyreplacing Foranyobjecta2VA[X],Barwisecallstheset eachindeterminatex2par(a)\domain(f)bythesetf(x)ina.thisisaccomplishedby solvingtheresultingequationsbythesolutionlemma. par(a)=fx2x:x2tc(a)g; assignparametricobjects,notjustsets,toindeterminates.forexample,ifa(x)isaparametricobjectrepresentingpartialinformationaboutsomenonparametricobjecta2va ParametricanchorscanalsobedenedasfunctionsfromasubsetofXintoVA[X]to aboutitsstructure. theformb(y)(anotherparametricobject),thenanchoringxtob(y)resultsintheobject a(b(y))whichdoesnotgivetheultimateobjectperhaps,butisatleastmoreinformative andifonedoesnotknowthevaluetowhichxistobeanchored,butknowsthatitisof hlabel;operator;valuei. jectconsistsofanidentier(oid)andproperties,eachattributeofwhichisatriple QUIXOTEfordeductiveandobject-orienteddatabases.InQUIXOTE,anob- YokotaandYasukawa(1992)proposedaknowledgerepresentationlanguage,called 5.6.YasukawaandYokota'swork handsideistherelatedproperties.anobjectconsistsofanoidanditsproperties,and canbewrittenasasetofattributetermswiththesameoidasfollows: summarized.considerfigure16wherethelefthandsideof\/"isanoidandtheright QUIXOTEisgiven.Thisworkisalongthelinesof(Dionneetal.,1992)andwillnowbe In(Yasukawa&Yokota,1991),apartialsemanticsforthesemanticsofobjectsin Anoidcanbedenedintensionallybyasetofrulesasfollows(notetheProlog-likestyle): path[from=x;to=y](arc[from=x;to=y] o=[l1=a;l2=b]()o=[l1=a];o=[l2=b] 21

22 Here,path[from=X;to=Y]istransitivelydenedfromasetoffactssuchas evenifitisgeneratedindierentenvironments.furthermore,circularpathscanbede- arc[from=a;to=b],etc.,andtheoidisgeneratedbyinstantiatingxandyasa resultoftheexecutionoftheprogram.thisguaranteesthatanobjecthasauniqueoid theconstraintx=o[[l=x].thesemanticsofoidsisdenedonasetoflabeledgraphs path[from=x;to=y](arc[from=x;to=z];path[from=z;to=y] asasubclassofhypersets.thus,anoidismappedtoalabeledgraphandanattribute isrelatedtoafunctiononasetoflabeledgraphs.sincethemetatheoryiszfc /AFA, allofthefamiliarset-theoretictechniquestodealwithcircularphenomenaarebroughtto bear. 6.Conclusion Settheorycanbeusefulinintelligentinformationmanagement.Themethodologymay change,ofcourse.auniversalsettheory,answeringmanytechnicalquestionsininformationmanagement,canbedevelopedbymeansofproposingnewaxiomsormodifying beexaminedandmodiedbasedonexistingsettheories.nomatterwhatproposalis existingones.alternativelyandmoreconservatively,dierentset-theoreticconceptsmay IIScommunity. Acknowledgments followed,webelievethatfurtherresearchinthiseldshouldbetotheadvantageofthe CouncilofTurkeyundergrantno.TBAG{992. Asusual,wearesolelyresponsibleforthisnalversion. Zadrozny(IBMT.J.WatsonResearchCenter)formoralsupportandtechnicaladvice. WewouldliketothankananonymousrefereeoftheJournalofIntelligentInformation Systemsforcriticalcommentswhichwerecrucialinrevisingtheoriginalmanuscript.Larry KerschbergandMariaZemankova,twooftheEditors-in-ChiefofJIIS,wereextremelykind andaccommodatingwhenwerstsubmittedthepaper.wearealsoindebtedtowlodek Therstauthor'sresearchissupportedinpartbytheScienticandTechnicalResearch 22

23 Notes 2.Awwissaidtobestratiedifintegersareassignedtothevariablesofwsuchthatall 1.Ontheotherhand,anothergreatmathematicianofthiscentury,R.Thom,saidin1971:\The 1984). oldhopeofbourbaki,toseemathematicalstructuresarisenaturallyfromahierarchyofsets, 3.LetusrecallRussell'sParadox.Weletrbethesetwhosemembersareallsetsxsuchthat occurrencesofthesamefreevariableareassignedthesameinteger,allboundoccurrences fromtheirsubsets,andfromtheircombination,isdoubtless,onlyanillusion"(goldblatt, xisnotamemberofx.thenforeverysetx,x2rix62x.substitutingrforx,we donotyethavetheobjectz,andhencecannotuseitasamemberofz.thesamereasoning ofavariablethatareboundbythesamequantierareassignedthesameinteger,andfor showsthatcertainothersetscannotbemembersofz.forexample,supposethatz2y. Theexplanationisnotdicult.Whenweareformingasetzbychoosingitsmembers,we obtainedthecontradiction. Forexample,(x2y)&(z2x)isstratiedas(x12y2)&(z02x1). everysubformulax2y,theintegerassignedtoyisequaltotheintegerassignedtox+1. ofthemembersof?wewouldliketohaveanarmativeanswertothisquestion.still, setsandisacollectionofstagessuchthateachmemberofxisformedatastagewhichis stage.nowthequestionbecomes:givenacollectionofstages,isthereastageafterall Stagesareimportantbecausetheyenableustoformsets.Supposethatxisacollectionof amemberof.ifthereisastageafterallofthemembersof,thenwecanformxatthis Thenwecannotformyuntilwehaveformedz.Henceyisnotavailableandthereforecannot theanswercannotalwaysbe\yes";ifisthecollectionofallstages,thenthereisnostage formedin\stages."foreachstages,therearecertainstageswhicharebefores. beamemberofz.carryingthisanalysisabitfurther,wearriveatthefollowing.setsare 5.Alinearordering<ofasetaisawell-orderingifeverynonemptysubsetofahasaleast 4.AtheoryT(anysetofformulasclosedunderimplication,i.e.,forany',ifTj=',then set.anordinalnumberstandsforanordertypewhichisrepresentedbywell-orderedsets. element.informally,anorderedsetissaidtobewell-orderedifthesetitself,andallits First,callasetatransitiveif8x(x2a!xa).Then,asetisanordinalnumber(or nonemptysubsetshavearstelementundertheorderprescribedforitselementsbythat aftereverystagein. '2T)isnitelyaxiomatizableithereisaniteT0Tsuchthatforevery ordinal)ifitistransitiveandwell-orderedby2.itshouldbenotedthatincaseofnitesets, thenotionsofcardinalnumberandordinalnumberarethesame.theclassofallordinal int,t0j=. 6.Wehavetoprovethetheorem denedi2.if=+1,theniscalledasuccessorordinal;elseitisalimitordinal. setofnaturalnumbers,isthenextlimitordinal(figure2). TheAxiomofInnityguaranteestheexistenceoflimitordinalsotherthan0.Infact,!,the numbersisdenotedbyord.therelationship<betweentwoordinalsand,<,is toshowthatthecartesianproductsetexists.themainpointoftheproofisthatifx=hy;zi, andify2aandz2b,thenx2p(p(a[b)).then,bytheaxiomofseparation, Thetheoremtobeprovedisequivalentto(1)withoutthestatementx2P(P(a[b)).Then wemustshowthattheequivalencein(1)stillholdswhenthatstatementiseliminated. Given(1),itfollowsthat 9c8x[x2c$x2P(P(a[b))&9y9z(y2a&z2b&x=hy;zi)]: 9c8x[x2c$9y9z(y2a&z2b&x=hy;zi)] 23x2c (2)

24 implies Thenbythefollowingtheorem(whichcanbeprovedbythePowerAxiom): ffyg;fy;zgg,andbythehypothesesy2aandz2b,wehave: itisobviousby(1)that(3)implies(2).by(3)andthedenitionoforderedpairsx= Toprovetheconverseimplication,wemustshowthat(3)impliesx2P(P(a[b)),since fygfa[bgandfy;zgfa[bg: 9y9z[y2a&z2b&x=hy;zi]: 7.Thecollectionof0formulasofalanguageListhesmallestcollection containingthe weconclude: atomicformulasoflandinductivelydenedas: Thus,ffyg;fy;zggP(a[b),i.e.,xP(a[b),andagainby(4),wehavex2P(P(a[b)). fyg2p(a[b)andfy;zg2p(a[b): b2p(a)$ba; (4) (b)if'; (a)if'inin,then:'isalsoin. 9.Aczelusestaggedgraphstorepresentsets,i.e.,eachchildlessnodeinthegraphistagged 8.AstructureforarstorderlanguageLisapairhM;Ii,whereMisanonemptysetcalled thedomainofthestructureandiisaninterpretationfunctionassigningfunctionsand predicatesovermtothesymbolsinl. byanatomor;.apointedgraphisataggedgraphwithaspecicnoden0calledits (c)if'isin,then8u2v'and9u2v'arealsoin forallvariablesuandv. arein,then('^ )and('_)arealsoin. as: n0!n1!!n.adecorationofagraphisafunctiond(n)foreachnoden,dened point.apointedgraphisaccessible(denotedasapg)ifforeverynodenthereisapath 10.TheuniquenesspropertyofAFAleadstoanintriguingconceptofextensionalityforhypersets.Theclassicalextensionalityparadigm,thatsetsareequalitheyhavethesame D(n)=tag(n); fd(m):misachildofng;otherwise. ifnhasnochildren, ityofsay,a=f1;agandb=f1;bgbecauseitjustassertsthata=bia=b,atriviality Apictureofasetisanapgwhichhasadecorationinwhichthesetisassignedtothepoint. members,worksnewithwellfoundedsets.however,thisisnotofuseindecidingtheequal- 11.TheSolutionLemmaisanelegantresult,butnoteverysystemofequationshasasolution. (Barwise&Etchemendy,1987).However,intheuniverseofhypersets,aisindeedequaltob example,thepair sincetheyaredepictedbythesamegraph.toseethis,consideragraphgandadecoration Firstofall,theequationshavetobeintheformsuitablefortheSolutionLemma.For thesameasdexceptthatd0(x)=b.d0mustalsobeadecorationforg.butbythe uniquenesspropertyofafa,d=d0,sod(x)=d0(x),andthereforea=b. DassigningatoanodexofG,i.e.,D(x)=a.NowconsiderthedecorationD0exactly y=f1;xg; x=fy;zg; 24

25 12.Astateofaair(a.k.a.infon)hR;a;iiisatriplewhereRisann-aryrelation,aisan beinismeant.wheni=1,thatstateofaairiscalledafactandthepolarityisusually appropriateassignmentofobjects,andiisthepolarity,1ifthereisatleastoneinstanceof addedtozfc. Rholdingofa,and0otherwise.Byastateofaair,astatethataairsmayormaynot Asanotherexample,theequationx=P(x)cannotbesolvedbecauseCantorhasproved(in cannotbesolvedsinceitrequiresthesolutiontobestatedintermsoftheindeterminatez. omitted.forexample,thestateofaairhsleeping;tom;gardeniisafactiftomissleeping ZFC )thatthereisnosetwhichcontainsitsownpowerset nomatterwhataxiomsare 13.AccordingtotheLiarParadox(alsoknownastheEpimenidesParadox),Epimenides,the 14.Thepredicateofa;b;u,whichisdenedas\uisanorderedpairhy;ziwithy2aandz2b" inthegarden. Cretan,said\AllCretansareliars."Nowthisstatementcannotbetruesincethiswould hy;zi2wyforallz2b.applying0-collectionagain,wehave: cwithhy;zi2cforally2aandz2b.thisfollowsfrom0-collectionasfollows.given shouldbetrue,leadingtoacontradiction. falseeither,sincethiswouldimplythatcretansarenotliars,hencewhatepimenidessays y2b,thereexistsasetd=hy;zi.so,by0-collection,thereexistsasetwysuchthat makeepimenidesaliar,leadingtothefalsityofhisstatement.thestatementcannotbe inkpu,is0.hence,0-separationcanbeusedonceitisknownthatthereexistsaset 15.Theisthenotationforrstwritingall!symbolsintermsof&and:,thenpassingnegationsinthroughtopredicateletters,andnallyreplacingeachoccurrenceofasubformula 8y2a9w8z2b9d2w(d=hy;zi) 16.Thetransitiveclosureofaset,denotedbyTC(a),isdenedbyrecursionasfollows: sothereisac1suchthatforally2a,z2b,hy;zi2wforsomew2c1.thus,ifc=sc1, thenhy;zi2cforally2aandz2b. :x2[y]intheresultbyx2[:y]. Hence,TC(a)=a[Sa[S2a:::Foranyinnitecardinalm,thesetH(m)isdenedas lessthanm.fh(n):n=1;2;:::gisthesetofhereditarilynitesets.hence,everyelement H(m)=fx:jTC(x)j<mg.TheelementsofH(m)aresaidtobehereditarilyofcardinality ofahereditarysetisahereditaryset. S0a=a, Sn+1a=S(Sna), TC(a)=SfSna:n=1;2;:::g. 25

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28 Authors'biographies VarolAkmanisassociateprofessorofComputerEngineeringandInformationScience hewasafulbrightscholarandaresearchassistantatrensselaerpolytechnicinstitute, puterengineering(1980)frommiddleeasttechnicaluniversity,ankara.in1980{1985, atbilkentuniversity,ankara,turkey.hisresearchisconcentratedinmathematicallogic andcommonsensereasoning,andadditionally,philosophyandnaturallanguagesemantics. B.S.degreeinElectricalEngineering(1979,highhonors)andanM.S.degreeinCom- eninformatica,amsterdam(1986{1988)andapost-doctoralpositionwiththevakgroep PriortoBilkent,heheldaseniorresearcherpositionwiththeCentrumvoorWiskunde ofturkey.healsoreceivedtheprof.mustafan.parlarfoundation'sscienticresearch wasconferredthe1989researchprizeofthescienticandtechnicalresearchcouncil awardin1990.akman'sdoctoraldissertationwaspublishedin1987byspringer-verlag. NewYork,andreceivedaPh.D.degreeinComputerandSystemsEngineering.Akman Informatica,RijksuniversiteitUtrecht,theNetherlands(1985{1986).Akmanreceiveda boardmemberofthejournalcomputers&graphicsandthebilkentuniversitylecture Heisalsothesenioreditorofa1989Springer-Verlagvolumeondesign.Heisaneditorial SeriesinEngineering(researchmonographsmarketedbySpringer-Verlag).Heisinthe M.S.degreeinComputerEngineeringandInformationScience(1993)fromBilkentUniversity,Ankara.Pakkan'smaster'sthesisstudiesnonwellfoundedsetsfromacomputational ArticialIntelligence,andEuropeanFoundationforLogic,Language,andInformation. sentationandnaturallanguagesemantics.pakkanreceivedab.s.degreeincomputer Engineering(1990,honors)fromAegeanUniversity,Izmir,Turkey.Hethenreceivedan BosphorusUniversity,Istanbul,Turkey.Hisareasofinterestincludeknowledgerepre- MujdatPakkanisadoctoralstudentintheDepartmentofComputerEngineering, programcommitteesofvariousconferencesandisamemberofamericanassociationfor perspective. 28