Three topics in set theory : finiteness and choice, cardinality of compact spaces, and singular Jónsson cardinals

96 Pages·2000·4 MB·English

by  De La Cruz, Omar, 1970-

THREETOPICSINSETTHEORY:
EINITENESSANDCHOICE,CARDINALITYOECOMPACTSPACES,
ANDSINGULARJONSSONCARDINALS
By
OMARDELACRUZ
ADISSERTATIONPRESENTEDTOTHEGRADUATESGHOOL
OFTHEUNIVERSITYOFFLORIDAINPARTIALFULFILLMENT
OFTHEREQUIREMENTSFORTHEDEGREEOF
DOGTOROFPHILOSOPHY
UNIVERSITYOFFLORIDA
2000
Idedicatethisworktomyfatherandmother,Hugo,Oana,andmygoodfriends.
—Amiparecer, estenegocioendospaletas
lededarareyo,yesasi: eltalhombrejura
quevaamorirenlahorca,ysimuereenella,
juroverdad, yporlaleypuestamereceserlitreyque
paselapuente;ysinoleahorcan,juro
mentira,yporlamismaleymerecequeleahorquen.
Cervantes,DonQuijotedelaMancha.
ACKNOWLEDGMENTS
Firstandforemost,IwouldliketothankDr.W.J.Mitchell,whohasgivenme
invaluablehelpfromthebeginningofmystudiesattheUniversityofFlorida. Iam
ingreatdebttohimforhisgenerosity,hiswisdom,hispatience,and,especially,for
hisinfiniteknowledgeofsettheory.
Ialsowanttoexpressmygratitudetothefollowingpersonsandinstitutions:
ThemembersofmySupervisoryCommittee,Dr.J.A.Larson,Dr.D.A.Cenzer,
Dr.R.L.Smith,andDr.G.B.Ray,whoprovidedvaluablecommentsandcorrections
atseveralstagesofmywork; theDepartmentofMathematicsattheUniversityof
Florida, its staff, anditsfaculty, especially Dr. J. Martinez, Dr. J. Keesling, and
Dr.B.Brechner. HugoDelaCruz,whohasgivenmehisunconditionalsupportallmy
life;Dr.C.A.DiFrisco,whohasgivenmehisadvice,help,andfriendshipformany
years;CONICITofVenezuela,foradditionalfinancialaidinmyfirstsemesters;and
mycolleagues,fortheirfriendshipandhelp,especiallyDr.JeffLeaning,Dr.Chawne
Kimber,Dr.WarrenMcGovern,MarfaRinehart,FarzanRiazati(whopaperedawall
oftheofficewesharedatthetimewithArnieMiller’slistofquestions; oneofthe
questionsthereledtowhatisnowGhapter2),ZiaUddin,andDr.StaceyChastain.
Specialthanksgotomyparents,whogavemethecuriosityandconfidencethat
haveledmehere,andtoCanaMocioalcaforhergreathelpandpatience.
IV
TABLEOFCONTENTS
ACKNOWLEDGMENTS
iv
LISTOFFIGURES vii
ABSTRACT
viii
INTRODUCTION
1
CHAPTERS
1 FINITENESSANDTHEAXIOMOFCHOICE 5
1.1 Introduction 5
1.2 NotionsofFiniteness 6
1.3 AxiomsofChoiceforFiniteFamilies 11
1.4 IndependenceProofs: PermutationModels 17
1.5 AZFModel 25
1.6 TheEmbeddingTheorem 31
2 COUNTABLECOMPACTSPACES
INTHEABSENCEOFAC 37
2.1 Introduction 37
2.2 TheOriginalResultandSomeVariations 38
2.3 ASimpleModel 42
2.4 ACountableConnectedCompactHausdorffSpace 45
3 JONSSONCARDINALSANDSINGULARITY 63
3.1 Introduction 63
3.2 BasicConcepts 63
3.3 TheTheorem 65
3.4 ConstructionoftheIterations 67
3.5 TheIndiscerniblesGeneratedby(A4^) 69
GONGLUSIONS 74
APPENDIXA GARDINALNUMBERSINSETTHEORY
WITHOUTCHOIGE 76
A.l DefinitionofGardinalNumbers 76
V
A.2 OrderingofCardinalNumbers 77
A.3 OperationsonCardinalNumbers 77
APPENDIXB ELEMENTARYCONCEPTSOFPLTOPOLOGY 79
B.l BasicDefinitions 79
B.2 Subdivisions,Joins,StarsandLinks 80
B.3 SimplicialMapsandPiecewiseLinearMaps 81
REFERENCES 83
BIOGRAPHICALSKETCH 85
VI
LISTOFFIGURES
Figure page
1.1 RelationsBetweenNotionsofFiniteness 11
1.2 Someimplicationsbetweenprinciples 18
2.1 Themap0i,inthetwo-dimensionalcase 53
W
2.2 Theset (three-dimensionalview) 60
vii
AbstractofDissertationPresentedtotheGraduateSchool
oftheUniversityofFloridainPartialFulfillmentofthe
RequirementsfortheDegreeofDoctorofPhilosophy
THREETOPICSINSETTHEORY:
EINITENESSANDCHOICE,CARDINALITYOECOMPACTSPACES,
ANDSINGULARJONSSONCARDINALS
By
OmarDelaCruz
August2000
Chairman: Dr.WilliamJ.Mitchell
MajorDepartment: Mathematics
Inthisdissertationweworkonthreedifferenttopicsinsettheory.
Thefirsttopicdealswithweakchoiceprinciplesoftheform: Every “finite"family
ofnon-emptysets has a choicefunction, where “finite” stands for one ofseveral
differentdefinitionsoffinitenessthatarenotequivalentunlessweassumetheaxiom
ofchoice(AC).Severalrelationsofimplicationandindependenceareestablished.
In the second chapter we construct a model ofset theory without AC where
thereexistsacountablecompactHausdorffspacewithnoisolatedpoints,andamore
elaboratemodelwhere thereexistsacountablecompact Hausdorffspace which is
connected.
ThelasttopicisthestudyofsingularJonssoncardinalsusingthetechniquesof
coremodeltheory WeextendaresultofMitchelltoincludesingularcardinals.
viii
INTRODUCTION
Thisworkaddressesseveralquestionsregardingthreeseparatetopicsinsettheory.
Thefirsttwotopics,whichcorrespondtoChapters 1 and2,dealwithsomeofthe
unexpectedphenomenathatcanarisewhentheaxiomofchoice(AC)isnotassumed.
Thethirdtopicisthestudyofsomelargecardinalsof“mediumsize,” inparticular
theso-calledJonssoncardinals,usingthetechniquesofcoremodeltheory.
In the first chapter we study the regulating influence of AC and some ofits
weakerconsequencesonthenotionoffiniteness. Wheninthehistoricaldevelopment
ofmathematicstheuseofactualinfinitiesbecameprogressivelyacceptable,theneed
tofindprecisedefinitionsfortheintuitivenotionsoffinitenessandinfinitybecame
pressing. Amongthe earlyinquiresintothesubjects werethoseofBernoulliand
Dedekind, but until the development ofset theory by Cantor there was no solid
frameworkforthestudyofinfinitequantities. Thecurrentlyaccepteddefinitionof
finiteness (whichautomaticallyprovidesadefinitionofinfinity,bynegation) isthe
following: asetisfiniteifithasnelements,forsomepositiveintegernumbern.
OneoftheearliestsystematicstudiesoffinitenesswasdonebyTarski [Tar24].
Oneofhismotivationswastofindapurelysettheoreticaldefinitionofafiniteset,
inordertoavoidtheappealtothenotionofnumber. Afterthedevelopmentofthe
VonNeumannordinals,whichallowedmathematicianstoconsidernumbersassets
(andthereforenolongeranexternalnotion),thismotivationbecamelessimportant.
However,themanydefinitionsthatheincluded,alongwiththerelationsofimplication
betweenthemthathefound,plusthefactthatallthesedefinitionsbecomeequivalent
underAC,turnedouttobeveryinfluentialintherealizationoftheimportanceof
ACinresolvingmanyfundamentalmattersinmathematics.
1
2
The maincontributionofthischapteristhestudyofseveralweakversions of
ACoftheform “everyfinitefamilyofnon-emptysetshasachoicefunction.” Ifby
“finite” weunderstandthecommonlyaccepteddefinition,thestatementabovecan
beprovedwiththeusualaxiomsofsettheory(ZF),withoutanyuseofAC. Butif
weallowtheuseoflessrestrictivedefinitionsoffiniteness,theresultingstatements
cannotbeprovedinZFunlesssomeextraprincipleisassumed(usuallyaconsequence
ofAC).Severaldifferentresultsareobtained,establishingrelationsofimplicationand
independencebetweenthesestatementsandseveralwellknownchoiceprinciples.
InChapter2wepresentsomeexamplesoftheunusualbehaviorthatcanarise
whenACfails,eveninwellstudiedareasofmathematicsliketopology.
Someofthebest understood objectsin topologyarecompact Hausdorfftopo-
logicalspaces. However,therichstructureofthesespacesandthemanyproperties
thatmathematicianshavebecomeusedtoexpectingfromthemdependonAC.The
particularpropertythatwestudyhereisthefollowing:
EverycompactHausdorffspacewithnoisolatedpoints
hascardinalitygreaterthanorequaltothatofthecontinuum. (1)
ThiscanbeprovedeasilyusingAC;however,weshowherethatthisresultcannot
beprovedfromZFalone,byprovidingamodelofZFwherethereexistsatopology
onujwhichiscompactHausdorffandhasnoisolatedpoints.
The result above answers a question posed by Marianne Morillon (see Miller
[Mil93]). However,itcanbeprovedthattheexampleprovidedisanextremallydis-
connectedspace. InordertoansweranotherquestionofMorillon(personalcommu-
nication,1998),themaintheoremofthechapterproducesanexampleofacountable
compactHausdorffspacethatisconnected.
Bothoftheexamplesobtainedinthischapterarecompactspacesbutlackachoice
functionthat,givenanopencover,choosesaparticularfinitesubcover. Thisleadsus
toconsiderthenotionofeffectivecompactness,whichrequirestheexistenceofsuch