Set theory with a universal set

122 Pages·1994·4.1 MB·English

by  Jamieson, Michael W

SETTHEORYWITHAUNIVERSALSET
By
MICHAELW.JAMIESON
ADISSERTATIONPRESENTEDTOTHEGRADUATESCHOOL
OFTHEUNIVERSITYOFFLORIDAINPARTIALFULFILLMENT
OFTHEREQUIREMENTSFORTHEDEGREEOF
DOCTOROFPHILOSOPHY
UNIVERSITYOFFLORIDA
1994
ACKNOWLEDGEMENTS
Ioweagreatdealofgratitudetomyadvisor,Prof. RickSmith,notonlyforsome
ofthekeyideaswhichhavemadethisprojectwork,andforhishelpinputtingthis
workinaproperperspective,butalsoforhissupport,andforhisopenmindednessin
becomingassociatedwithit. Thanksalsogototheothermembersofmygraduate
committee, Prof. WilliamMitchell, Prof. Douglas Cenzer, Prof. Gregory Ray,
fortheirhelp, andespeciallyto Prof. Jean Larsonforextraordinarydiligencein
conscientiouslyreadingthisworkinsomephaseswhichwerequiteunpleasant. Ialso
mustthankmyteachersthroughoutmyacademiccareerforhelpingshapeinmymind
amathematicaluniversewhichcontinuestoholdmyinterest.Iwanttothankallmy
academicandnonacademiccoworkersfortheirgood-naturedhelp. Finally,Iwantto
thankmyfamilyandwonderfulfriendsfortheirinvaluablesupportthroughoutthis
endeavor.
u
TABLEOFCONTENTS
ACKNOWLEDGEMENTS u
ABSTRACT iv
CHAPTERS
1. INTRODUCTIONANDSURVEY 1
2. BASICCONSTRUCTIONS 22
2.1. BasicConstructions 23
2.2. TheMainConstruction 31
2.3. GeneralResults 44
3. COMPREHENSIVEEXTENSIONALS-SUBALGEBRAS 53
4. AMODELOFSTRICTLYQUANTIFIER-FREECOMPREHENSION 68
4.1. PreliminaryDiscussion 68
4.2. TheDirectLimitConstruction 82
4.3. StrictlyQuantifier-freeComprehensionwithParameters .... 91
5. CONCLUSION 106
REFERENCES 113
BIOGRAPHICALSKETCH 115
in
AbstractofDissertationPresentedto
theGraduateSchooloftheUniversityofFlorida
inPartialFulfillmentoftheRequirementsfortheDegree
ofDoctorofPhilosophy
SETTHEORYWITHAUNIVERSALSET
By
MichaelW.Jamieson
August1994
Chairman: Dr. RickL.Smith
MajorDepartment: Mathematics
This dissertation presentsanew resolutionoftheset theoreticparadoxes of
RussellandCantor. Theworkcontainsamodelforatheoryofsetswhichimplies
that thereisauniversalset u. Inthelanguageofequalityandasetmembership
relation-E-,astrictlyquantifier-freeformulaisaquantifier-freeformulawhoseatomic
subformulashavenovariablewhichoccursmorethanonce.Theaxiomsofthetheory
areextensionalityandthecomprehensionschemeforstrictlyquantifier-freeformulas.
Anymodelofthistheorysatisfiesthesentences"theuniversalsetuexists"and"the
graphsoftheentiremembershipandidentityrelationsoveruexist". Therichest
modelpresentedisalsointernallycountable,i.e.itsatisfiesthesentences"Theordinal
—
ijoexists"and"Thereisafunction/:u w". TheauthordefinesalistofGodel-
likeoperations. Thislistcontainsavariantappropriatetothecontextofauniversal
setforeachoperationusedinconstructionofGodel'sLfromtheordinals,except
domainformationandunionofanarbitrarysetofsets. Godel-likeclosureimplies
strictlyquantifier-freecomprehension.
iv
ThismodelisconstructedinsideacountableatomicBooleanalgebraasadirect
—
limitofS-algebras,i.e. atomicsubalgebrasC^withone-to-onemapsSn'•Cn Cn
suchthatforallxGCntheelementSn(x)isanatomofCn. ThesemapsSn(-)induce
membershiprelations-En-byxEny<->Sn(x)<y. Thestructures(Cn,En)satisfy
thesentences"Sfj(x)isthesingletonofx"and"theuniverseisclosedunderbinary
union,intersectionandcomplement".Godel-likeclosureandinternalcountabilityare
preservedinanextensionalsubstructurewhichisconstructedbyiterativelyapplying
thefactthateverycountableBooleanalgebraisprojective.
CHAPTER1
INTRODUCTIONANDSURVEY
Thisdissertationpresentsamodelofthelanguageofamembershiprelation-E-
withthefollowingproperties.
(i) Themodelsatisfiesacomprehensionaxiomschemewhichappliesto
thosequantifier-freeformulaswitharbitraryparameterswhichcontain
no atomicsubformulavRv, i.e. no atomic subformulain whicha
variableisrepeated,
(ii) Equalityinthemodelisdeterminedbytheprincipleofextensionality
(iii) Themodelsatisfiesclosureunderpowersetformation,
(iv) Themodelsatisfiesthestatementau>exists",
(v) Themodelsatisfiesanaxiomofinternalcountability.
RussellobservedacontradictionintheoriginalFregesettheory. Ifristheclass
{x\x $l x}, thenr € r *-* r £ r. This argumentusesthecomprehensionaxiom
fortheformulaxljx,whichcontainsthesubformulaxExandisthereforeexcluded
fromthecomprehensionschemedescribedin(i). Theschemeherediffersfromthe
originalinconsistentschemeintwoways. First,thisschemeexcludesinstancesfor
quantifiedformulas,sincequantificationisbeyondthescopeofthiswork. Second,
thisschemealsoexcludesinstanceswithavariablerepeatedinanatomicsubformula.
Therationaleforthisexclusionisthatsuchformulasmayberegardedascontaining
animplicitquantification. Forexample,theformulaxljxislogicallyequivalentto
3y[xflyAx=y]. Thisrationaleisdiscussedlaterinthechapter.
DEFN 1.1. Strictlyquantifier-freeformulasinC(E,=)arethosequantifier-freefor-
mulaswithnosubformulavEvorv=vforanyvariablev. Strictlyquantifier-free
1
2
comprehensionwithparametersforC(E,=)istheaxiomschemewhichcontains,for
everystrictlyquantifier-freeformulaF(p,x), theaxiomVfByVx\xEy «- F(p,x)].
Notethatstrictlyquantifier-freecomprehensionallowsvariablerepetitionindistinct
atomicsubformulas,forexamplex6yAy€ x.
Chapter4containsanextensionalmodel(B,E)ofstrictlyquantifier-freecom-
prehension. Insuchmodelsonecan definetheBooleanalgebraoperations. The
axiomsofBooleanalgebrasarederivableinthetheorystrictlyquantifier-freecom-
prehensionplusextensionality.OperationssimilartomostofthosewhichGodelused
todefinetheclassLofconstructivesetsarealsodefinable,andthemodelinChapter
4isclosedundertheseoperations. Anystructureclosedundertheseoperationssat-
isfiestheschemestrictlyquantifier-freecomprehension.Theprooffollowsatheorem
byGodelaboutthestructureLofconstructiblesets[l].
ThemodelinChapter4satisfiestheadditionalaxiomthatthereisaone-to-one
function/intheuniversewhichembedstheuniverseintou.
DEFN 1.2. Astructure(C,E,=)isinternallycountableifitsatisfiesthefollowing:
(i) Thereisauniversalsetu,i.e.Vx[xEu]
(ii) Thereisanelementu>whoseelementswithrespecttoEformastruc-
tureisomorphictothenaturalnumbers,
—
(iii) Thestructure(C,E,=)satisfies3/[/:u u>].
Compared to internal countability, Russell's paradox is a somewhat simpler
issue,soitisconsideredfirst. However,thepossibilityoferrorintheZFtreatment
ofuncountabilityismuchmoreimportant.
Amodelofstrictlyquantifier-freecomprehensioncontainsauniversalsetu,the
membershiprelationeandtheidentityrelationtaselements. Thistheoryimplies
3
that~eflt={(x,y)\xljyAx=y}={(x,x)|xj^x}isanelementofthestructure.
TheformulaxEyAyEx,withvariablerepetitions,hasanextensionefle whichcan
beconstructedwithrelationinversesandintersection.Theideaofstrictlyquantifier-
freecomprehensionisthat toconstructtheextensionofxExrequiresadistinctly
differentcapability.Itrequirescollectionofthefixedpointsofarelation.Thisisan
intuitivelydistinctandmorecomplicatedkindoftask. Thedistinctionisanalogous
tothedifferencebetweeny=Ax,whichrepresentsthesimplematrixmultiplication
relation,andx=Ax,whichrepresentscollectionoftheeigenvectorswhoseeigenvalue
is1.
Aformaldistinctionbetweenthesetwokindsoftasks isevidentinthethe-
oryofcylindricalgebras[2],alsoknownasquantifiedBooleanalgebras. Acylindric
algebraisaBooleanalgebrawithoperationscQ,whicharecalledcylindrifications,
andelementsdag,whicharecalleddiagonalizationelements. Elementsoftheal-
gebracorrespondtoformulasinafirstorderlanguage. Acylindrificationoperation
ca correspondstoquantificationoveravariablexa. Thediagonalizationelements
correspondtoatomicformulasxa=xg.
ThecylindrificationoperationsbythemselvesdonotextendBooleanalgebras
sufficientlytorepresentthepredicatecalculus. Indeed,itisthediagonalarguments
whichrequiretheadditionaldiagonalizationelementsdag. Inacylindricalgebra,
an atomicformulap(xi,X2,X3) is associated with agenerator gp. The formula
p(xi,X2,xi), withvariable repetition, is not generallyassociated with adistinct
generator. Suchredundancycanbeallowed,butitproducesnon-extensionality. In-
stead,theatomicformulap(xi,X2,xi)canbegenerallyassociatedwiththenatural
representationofthelogicallyequivalentformula3x3[p(xi,X2,X3)Axi=X3]. That
representationisC3(<7pA^1,3). Nowtherepresentationoftheatomicformulawitha
repeatedvariabledoesinvolveaquantifier.
4
Thus,Russell'sconstructionisdonewithoutapparentquantification,becauseof
anincidentalfeatureofourparticularformulationoffirstorderpredicatecalculus.
Inthisformulation,relationconstructionandfixedpointcollectionappearsosimilar
thatthesettheory'snaivecomprehensionaxiomsdidnotdistinguishthem.Suppose
awell-formedfirstorderatomicformulaisdefinedasapredicatesymbolpfollowed
byavariablex\,andanother(i.e. adistinct)variablex%andanothervariable£3
andsoon,untilwehavethenumberofvariablesappropriateforp.Thenxyxsimply
isnotawell-formedformula.
Whilethismaynotbenaturalintheformalismoffirstorderpredicatecalculus,
thecorrespondingdistinctiondoes arisenaturallyincylindricalgebras. Thiscan
happenbecausetheequivalenceoffirstorderpredicatecalculuswiththetheoryof
cylindricalgebrasisnotrequiredtopreservenaturality. Twopointsfavorcylindric
algebrasinthisissue. OneistheinconsistencywhichresultswhenoneacceptsxEx
asanatomicformula.Theotheristheintuitionaboutthedistinctionwithconcrete
examples,likey = Ax and x = Ax. This workbegins developmentoftheview
that,asaconsequenceofsuppressingthisdistinction,thepresentationofidealset
theorysimplysaid somethingunintended. Infact,it happenedtosaysomething
false,somethingprovablyfalse(hence,anoutrightinconsistency).
Constructions similarto Russell's occur in arguments based on uncountable
infinities.TheseargumentscontaincertainstatementswhicharesimilartoRussell's
axiomofcomprehension. Itispossibletosalvagethesestatements,whichthiswork
takestobemisstatements,sincethesestatementsdonotcontradictthemselves.Still,
thereremainsthequestionwhethertheycontradictthetruth. Russell'sandCantor's
constructionsareverysimilar. Indeed,Russell'sparadoxwasintendedtodistillthe
essenceofCantor'sdiagonalarguments. ConsiderationofRussell'ssetisthenatural
5
initialpoint ofattack, becauseitisasimplerconstructionwithamuchstronger
result.
Itisdesirableforsettheorytotreattheseparadoxesmuchthesame. Chapters
2and4containmodels(C,E,=)ofinternalcountability. Somodelsexistforthe
quantifier-freepartoftheconstructionoftheparadoxicalelements. Thereforethis
work attributes thecontradictionswhich arisein thetraditionalanalysisofboth
paradoxestomistreatmentofquantification.
Toseehowthesuggested restrictionsrelatetothestrengthofthetheory,it
ishelpfultoconsiderananalogous,morewell-understoodsituationinSecondOrder
Arithmetic.There,allsetscontainonlynaturalnumbers.Church'sThesischaracter-
izesthosearithmeticrelationswhosedefinitionsgiveeffectivelycomputabledecision
proceduresforthem.Theeffectivelydecidablerelationsarethosewhicharedefinable
bothbyanexistentialformula(aformulawithasingleexistentialquantifier)andby
auniversalformula(aformulawithasingleuniversalquantifier). Inarithmetic,the
quantifier-freepartofauniversalorexistentialformulaisallowedtorefertoaddi-
tion,multiplication,equalityofnaturalnumbersandmembershipnGA. Thus,the
classofeffectiverelationsisindeedmuchmorelimitedthanthefullclassofdefinable
relations.
Aquantifier-freearithmeticdefinitioniseffectivelydecidable,relativetowhat-
eversetparametersoccurintheformula. Equalityofnaturalnumbersisdefinedby
m=n. Soequalityofnaturalnumbersiseffectivelydecidable. Equalityoftwosets
ofnaturalnumbersisdefinedbyx=y*-*Vz[z£x«-zGy],anditisnotgenerally
definablewithasingleexistentialquantifier. Soitisnoteffectivelydecidable,and
reasonablemodelsofsettheoryneednotcontainit.
Insecondorderarithmetic,equalityofsetsofnatural numbersisroughlyof
thecomplexityofthehaltingproblemrelativeto theelementsx and y. Second