The Real Numbers: An Introduction to Set Theory and Analysis PDF
Preview The Real Numbers: An Introduction to Set Theory and Analysis
Undergraduate Texts in Mathematics John Stillwell The Real Numbers An Introduction to Set Theory and Analysis Undergraduate Texts in Mathematics Undergraduate Texts in Mathematics SeriesEditors: SheldonAxler SanFranciscoStateUniversity,SanFrancisco,CA,USA KennethRibet UniversityofCalifornia,Berkeley,CA,USA AdvisoryBoard: ColinAdams,WilliamsCollege,Williamstown,MA,USA AlejandroAdem,UniversityofBritishColumbia,Vancouver,BC,Canada RuthCharney,BrandeisUniversity,Waltham,MA,USA IreneM.Gamba,TheUniversityofTexasatAustin,Austin,TX,USA RogerE.Howe,YaleUniversity,NewHaven,CT,USA DavidJerison,MassachusettsInstituteofTechnology,Cambridge,MA,USA JeffreyC.Lagarias,UniversityofMichigan,AnnArbor,MI,USA JillPipher,BrownUniversity,Providence,RI,USA FadilSantosa,UniversityofMinnesota,Minneapolis,MN,USA AmieWilkinson,UniversityofChicago,Chicago,IL,USA Undergraduate Texts in Mathematics are generally aimed at third- and fourth- year undergraduate mathematics students at North American universities. These texts strive to provide students and teachers with new perspectives and novel approaches.Thebooksincludemotivationthatguidesthereadertoanappreciation ofinterrelationsamongdifferentaspectsofthesubject.Theyfeatureexamplesthat illustratekeyconceptsaswellasexercisesthatstrengthenunderstanding. Forfurthervolumes: http://www.springer.com/series/666 John Stillwell The Real Numbers An Introduction to Set Theory and Analysis 123 JohnStillwell DepartmentofMathematics UniversityofSanFrancisco SanFrancisco,CA,USA ISSN0172-6056 ISBN978-3-319-01576-7 ISBN978-3-319-01577-4(eBook) DOI10.1007/978-3-319-01577-4 SpringerChamHeidelbergNewYorkDordrechtLondon LibraryofCongressControlNumber:2013946655 MathematicsSubjectClassification:03-01,26-01 ©SpringerInternationalPublishingSwitzerland2013 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.springer.com) To Elaine Preface Every mathematician uses the real number system, but mathematics students are seldom told what it is. The typical undergraduate real analysis course, which is supposed to explain the foundations of calculus, usually assumes a definition of R, or else relegatesit to anappendix.By failingto reachthe realfoundation(pun intended),real analysis runs the risk of looking like a mere rerun of calculus, but with more tedious proofs. A serious look at the real numbers, on the other hand, openstheeyesofstudentstoanewworld—aworldofinfinitesets,wheretheneed for new ideas and new methodsof proofis obvious. Not only are theoremsabout therealnumbersinterestinginthemselves,theyfitintothefundamentalconceptsof realanalysis—limits,continuity,andmeasure—likeahandinaglove. However,anybookthatrevisitsthefoundationsofanalysishastoreckonwiththe formidableprecedentofEdmundLandau’sGrundlagenderAnalysis(Foundations ofAnalysis)of1930.Indeed,theinfluenceofLandau’sbookisprobablythereason thatsofewbookssince1930haveevenattemptedtoincludetheconstructionofthe realnumbersin anintroductiontoanalysis. On theone hand,Landau’saccountis virtuallythelastwordinrigor.Theonlywaytobemorerigorouswouldbetorewrite Landau’sproofsincomputer-checkableform—whichhasinfactbeendonerecently. Ontheotherhand,Landau’sbookisalmostpathologicallyreader-unfriendly.Inhis PrefacefortheStudenthesays“Pleaseforgeteverythingyouhavelearnedinschool; foryouhaven’tlearnedit,”andinhisPrefacefortheTeacher“Mybookiswritten, asbefitssucheasymaterial,inmercilesstelegramstyle.”WhilememoriesofLandau stilllinger,sotoodoesfearoftherealnumbers. Inmyopinion,theproblemwithLandau’sbookisnotsomuchtherigor(though it is excessive), but the lack of background, history, examples, and explanatory remarks.Also,thefactthathedoesnothingwiththerealnumbersexceptconstruct them.Inshort,itcouldbeanentirelydifferentstoryifitwereexplainedthatthereal numbersareinteresting!ThisiswhatIhavetriedtodointhepresentbook. In fact the real numbers perfectly exemplify the saying of Carl Ludwig Siegel thatthemathematicaluniverseisinhabitednotonlybyimportantspeciesb√utalsoby interestingindividuals.Thereareinterestingindividualnumbers(suchas 2,e,and π),interestingsetsofrealnumbers(suchastheCantorset,Vitali’snonmeasurable vii viii Preface set),andeveninterestingsetsofwhichnointerestingmemberisknown(suchasthe set of normalnumbers).All of these exampleswere knownin 1930,but in recent decadesthey have been joined by many new exotic sets arising from the study of fractals,chaos,anddynamicalsystems. Theexoticsetsarisingfromdynamicalsystemsareonereason,Ibelieve,toshift the emphasis of analysis somewhat from functions to sets. Of course, we are still interestedinsequencesofnumbersandsequencesoffunctions,andtheirlimits.But nowitseemsequallyreasonabletostudysequencesofsets,sincemanyinteresting sets,suchastheCantorset,ariseastheirlimits.Anotherreasonissimplythegreat advancesmade by set theory itself in recent decades, many of them motivated by the desire to betterunderstandthe realnumbers.These advancesare too technical forustodiscussindetail,buttheyresultfromthefundamentalfactthatanalysisis basedonuncountablesetsandthestruggletounderstandthisfact. The set of real numbers is the first, and still the most interesting, example of an uncountableset. The second exampleis the set of countableordinals. Itis less familiartomostmathematicians,butalsoofgreatimportanceinanalysis.Ifanalysis istakentobethestudyoflimitprocesses,thencountableordinalsarethenumbers that measure the complexity of functions and sets defined as limits of sequences. In particular, we assign the lowest level of complexity (zero) to the continuous functions,thenextlevelofcomplexity(one)tothefunctionsthatarenotcontinuous butarelimitsofcontinuousfunctions,complexityleveltwotofunctionsthatarenot oflevelonebutarelimitsoffunctionsoflevelone,andsoon.Itturnsoutthatthere arefunctionsofalllevels0,1,2,3,...andbeyond,becauseonecanfindasequence offunctions f , f , f ,...,respectivelyoflevels0,1,2,...,whoselimitisnotatany 0 1 2 ofthese levels. Thiscalls fora transfinitenumber,called ω, to labelthe first level beyond0,1,2,.... The transfinite numbersneededto label the levels of complexityobtainable by limit processes not only make up an uncountable set: in fact they make up the smallest uncountable set. Thus, the raw materials of analysis—real numbers and limits—leadustotwouncountablesetsthatareseeminglyverydifferent.Whether thesetwosetsareactuallyrelated—specifically,whetherthereisabijectionbetween thetwo—isthefundamentalproblemaboutrealnumbers:thecontinuumproblem. ThecontinuumproblemwasnumberoneonHilbert’sfamouslistofmathematical problemsof 1900, and it still has not been solved. However,it has had enormous influenceonthedevelopmentofsettheoryandanalysis. The above train of thoughtexplains, I hope, why the present book is aboutset theoryandanalysis.Thetwosubjectsaretoocloselyrelatedtobetreatedseparately, eventhoughtheusualundergraduatecurriculumtriestodoso.Thetypicalsettheory course fails to explain how set conceptsare relevantto analysis—even seemingly abstruseonessuchasdifferentaxiomsofchoiceandlargecardinals.Andthetypical realanalysiscoursefailstoaddressthesetissuesthatariseinevitablyfromthereal numbers,andfrommeasuretheoryinparticular.Whenthetwosubjectsaretreated togetheronegets(almost)twocoursesforthepriceofone. The book expands some of the material in my semi-popular book Roads to Infinity(Stillwell2010)intextbookformat,withmorecompleteproofs,exercisesto Preface ix reinforcethem,andstrengthenedconnectionswithanalysis.Thehistoricalremarks, inparticular,explainhowtheconceptsofrealnumberandinfinitydevelopedtomeet theneedsofanalysisfromancienttimestothelatetwentiethcentury. Inwritingthebook,Ihadinmindanaudienceofseniorundergraduateswhohave studiedcalculusandotherbasicmathematics.ButIexpectitwillalsobeusefulto graduatestudentsandprofessionalmathematicianswhountilnowhavebeencontent to “assume” the real numbers. I would not go as far as Landau (“please forget everything you have learned in school; for you haven’t learned it”) but I believe itisenlightening,andfun,tolearnsomethingnewabouttherealnumbers. My thanks go to José Ferreiros and anonymous reviewers at Springer for corrections and helpful comments, and to my wife Elaine for her usual tireless proofreading.IalsothanktheUniversityofSanFranciscoandMonashUniversity fortheirsupportwhileIwasresearchingandwritingthebook. SanFrancisco,CA,USA JohnStillwell
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