Set Theory: With an Introduction to Real Point Sets PDF
Preview Set Theory: With an Introduction to Real Point Sets
Abhijit Dasgupta Set Theory With an Introduction to Real Point Sets Abhijit Dasgupta Set Theory With an Introduction to Real Point Sets AbhijitDasgupta DepartmentofMathematics UniversityofDetroitMercy Detroit,MI,USA [email protected] ISBN978-1-4614-8853-8 ISBN978-1-4614-8854-5(eBook) DOI10.1007/978-1-4614-8854-5 SpringerNewYorkHeidelbergDordrechtLondon LibraryofCongressControlNumber:2013949160 Mathematics Subject Classification (2000): 03E10, 03E20, 03E04, 03E25, 03E15, 28A05, 54H05, 03E30,03E75,03E50,03E55,03E60,03E02 ©SpringerScience+BusinessMediaNewYork2014 Thisworkissubjecttocopyright.AllrightsarereservedbythePublisher,whetherthewholeorpartof thematerialisconcerned,specificallytherightsoftranslation,reprinting,reuseofillustrations,recitation, broadcasting,reproductiononmicrofilmsorinanyotherphysicalway,andtransmissionorinformation storageandretrieval,electronicadaptation,computersoftware,orbysimilarordissimilarmethodology nowknownorhereafterdeveloped.Exemptedfromthislegalreservationarebriefexcerptsinconnection with reviews or scholarly analysis or material supplied specifically for the purpose of being entered and executed on a computer system, for exclusive use by the purchaser of the work. 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Printedonacid-freepaper SpringerispartofSpringerScience+BusinessMedia(www.birkhauser-science.com) To mymother and tothememoryofmyfather Preface Most modern set theory texts, even at the undergraduate level, introduce specific formal axiom systems such as ZFC relatively early, perhaps because of the (understandably real) fear of paradoxes. At the same time, most mathematicians and students of mathematics seem to care little about special formal systems, yet maystillbeinterestedinthepartofsettheorybelongingto“mathematicsproper,” i.e.,cardinals,order,ordinals,andthetheoryoftherealcontinuum.Thereappears to bea gulfbetweentextsofmainstreammathematicsandthose ofsettheoryand logic. This undergraduateset theory textbook regards the core material on cardinals, ordinals, and the continuumas a subject area of classical mathematics interesting in its own right. It separates and postpones all foundational issues (such as paradoxesandspecialaxioms)into an optionalpartat the end.The main material is thus developed informally—not within any particular axiom system—to avoid getting bogged down in the details of formal development and its associated metamathematicalbaggage.Ihopethiswillmakethistextsuitableforawiderange ofstudentsinterestedinanyfieldofmathematicsandnotjustforthosespecializing inlogicorfoundations.Atthesametime,studentswithmetamathematicalinterests will find an introduction to axiomatic ZF set theory in the last part, and some glimpsesinto keyfoundationaltopicsin the postscriptchaptersat the end of each part. Anotherfeatureofthisbookisthatitscoverageoftherealcontinuumisconfined exclusivelyto the reallineR. All abstractor generalconceptssuchas topological spaces,metricspaces,andeventheEuclideanspacesofdimension2orhigherare completely avoided. This may seem like a severe handicap, but even this highly restrictedframeworkallowstheintroductionofmanyinterestingtopicsinthetheory ofrealpointsets.Infact,notmuchsubstanceinthetheoryislostandafewdeeper intuitionsaregained.Asevidencedbytheteachingofundergraduaterealanalysis, the studentwho is first firmly groundedin the hard and concretedetailsof R will betterenjoyandhandletheabstractionfoundinlater,moreadvancedstudies. The book grew out of an undergraduate course in introductory set theory that I taught at the University of Detroit Mercy. The prerequisitefor the core material vii viii Preface ofthebookisapost-calculusundergraduateUScourseindiscretemathematicsor linearalgebra,althoughprecalculusandsomeexposuretoproofsshouldtechnically sufficeforPartsIandII. The bookstarts with a “prerequisites”chapteron sets, relations,and functions, includingequivalencerelationsandpartitions,andthedefinitionoflinearorder.The restisdividedintofourrelativelyindependentpartswithquitedistinctmathematical flavors. Certain basic techniques are emphasized across multiple parts, such as Cantor’s back-and-forth method, construction of perfect sets, Cantor–Bendixson analysis,andordinalranks. Part I is a problem-based short course which, starting from Peano arithmetic, constructs the real numbers as Dedekind cuts of rationals in a routine way with two possible uses. A student of mathematics not going into formal ZF set theory will work out, once and for all, a detailed existence proof for a complete ordered field. And for a student who might later get into axiomatic ZF set theory, the redevelopmentofPeanoarithmeticandthetheoryofrealnumbersformallywithin ZFwillbecomelargelysuperfluous.Onemayalso decidetoskipPartI altogether andgodirectlytoPartII. Part II contains the core material of the book:The Cantor–Dedekindtheory of thetransfinite,especiallyorder,thecontinuum,cardinals,ordinals,andtheAxiomof Choice.Thedevelopmentisinformalandnaive(non-axiomatic),butmathematically rigorous. While the core material is intended to be interesting in its own right, it alsoformsthefolkloreset-theoreticprerequisiteneededforgraduateleveltopology, analysis,algebra,andlogic.UsefulformsoftheAxiomofChoice,suchasZorn’s Lemma,arecovered. Part III of the book is about point sets of real numbers. It shows how the theory of sets and orders connects intimately to the continuum and its topology. In addition to the basic theory of R including measure and category, it presents more advanced topics such as Brouwer’s theorem, Cantor–Bendixson analysis, Sierpinski’s theorem, and an introduction to Borel and analytic sets—all in the contextoftherealline.Thusthereadergetsaccesstosignificanthigherresultsina concretemannerviapowerfultechniquessuchasCantor’sback-and-forthmethod. Asmentionedearlier,alldevelopmentislimited to thereals, buttheapparentloss ofgeneralityismostlyillusoryandthespecialcaseforrealnumberscapturesmuch oftheessentialideasandthecentralintuitionsbehindthesetheorems. Parts II and III of the book focus on gaining intuition rather than on formal development. I have tried to start with specific and concrete cases of examples and theorems before proceeding to their more general and abstract versions. As a result, some important topics (e.g., the Cantor set) appear multiple times in the book, generally with increasing levels of sophistication. Thus, I have sacrificed compactness and conciseness in favor of intuition building and maintaining some independencebetweenthefourparts. PartIVdealswithfoundationalissues. Theparadoxesarefirstintroducedhere, leadingtoformalsettheoryandtheZermelo–Fraenkelaxiomsystem.VonNeumann ordinalsarealsofirstpresentedinthispart. Preface ix Eachpartendswithapostscriptchapterdiscussingtopicsbeyondthescopeofthe main text, ranging from philosophicalremarksto glimpses into modernlandmark results of set theory such as the resolution of Lusin’s problems on projective sets usingdeterminacyofinfinitegamesandlargecardinals. Problems form an integral and essential part of the book. While some of them are routine, they are generally meant to form an extension of the text. A harder problemwillcontainhintsandsometimesanoutlineforasolution.Starredsections andproblemsmayberegardedasoptional. The book has enough material for a one-year course for advanced undergrad- uates. The relative independence of the four parts allows various possibilities for coveringtopics.Inatypicalone-semestercourse,IusuallybrieflycoverPartI,spend mostofthetimeinPartII,andfinishwithabriefoverviewofPartIV.Forstudents withmorefoundationalinterests,moretimecanbespentonthematerialofPartIV and the postscripts. On the other hand, for less foundationally inclined but more mathematicallyadvancedstudentswithpriorexposuretoadvancedcalculusorreal analysis,onlyPartsIIandIIImaybecoveredwithPartsIandIVskippedaltogether. Acknowledgments.IwanttothanktheUniversityofDetroitMercyforsupporting me with a sabbatical leave during 2011–2012 which made the writing of this book possible, and Professor László Kérchy, Editor of Acta Sci. Math. (Szeged), for kindly giving me permission to translate a section of von Neumann’s original Germanpaper[80].I alsowish tothankmystudents,myteacherPinakiMitrafor introducingmetosettheoryandlogic,andthelatesettheoristR.MichaelCanjar, who attended all my lectures in 2010 and provided daily feedback. Sadly, Mike leftusbeforethisbookcouldbefinished.ProfessorTarunMukherjee,towhommy lifelongdebtisbeyondmeasure,painstakinglyreadtheentiremanuscript,correcting many errors and giving his invaluable suggestions for improvement. Professor AndreasBlass was unbelievablykindand quickto give hisexpertcommentson a partofthebookeventhoughI(shamelessly)askedhimatthelastmomenttocheck itinanunreasonablyshorttime.Immenselyvaluableweretheextensiveanddeeply engagingfeedbackonthewholemanuscriptthatcamefromtheanonymousreferees, whichledmetodropandrewritesomesectionsandaddthelaterpostscripts.Iam also indebtedto Professor Prasanta Bandyopadhyayand Professor KallolPaul for their help, and to Professor Ioannis Souldatos for his thoughtful comments on a chapter.AtBirkhäuser,IgotpatienthelpfromTomGrassoininitialplanning,from KateGhezziandMitchMoultonduringwriting,andaboveallfromDr.AllenMann, a logicianhimself, in resolvingmanycrucialdifficulties.My friendandcolleague Dr.ShuvraDasprovidedguidance,advice,andwisdom.Veryspecialconcern,care, encouragement, and inspiration came from my friend Sreela Datta. My brother Anirbanandhisfamilyweretrulysupportive.MydaughterMalinifrequentlyhelped mewithmywriting.Finally,nothingwouldbepossiblewithoutmywifeSomawho isalwaysbymysidesupportingmeineverypossibleway. AnnArborandKolkata AbhijitDasgupta 2011–2013
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