An_introduction_to_Set_Theory

《An_introduction_to_Set_Theory》由会员上传分享，免费在线阅读，更多相关内容在学术论文-天天文库。

1、ANINTRODUCTIONTOSETTHEORYProfessorWilliamA.R.WeissOctober2,20082Contents0Introduction71LOST112FOUND193TheAxiomsofSetTheory234TheNaturalNumbers315TheOrdinalNumbers416RelationsandOrderings537Cardinality598ThereIsNothingRealAboutTheRealNumbers659TheUniverse7334CONT

2、ENTS10Reflection7911ElementarySubmodels8912Constructibility10113Appendices117.1TheAxiomsofZFC........................117.2TentativeAxioms.........................118CONTENTS5PrefaceThesenotesforagraduatecourseinsettheoryareontheirwaytobe-comingabook.Theyoriginate

3、dashandwrittennotesinacourseattheUniversityofTorontogivenbyProf.WilliamWeiss.CynthiaChurchpro-ducedthefirstelectroniccopyinDecember2002.JamesTalmageAdamsproducedthecopyhereinFebruary2005.Chapters1to9areclosetofi-nalform.Chapters10,11,and12arequitereadable,butshoul

4、dnotbeconsideredasafinaldraft.Onemorechapterwillbeadded.6CONTENTSChapter0IntroductionSetTheoryisthetruestudyofinfinity.Thisaloneassuresthesubjectofaplaceprominentinhumanculture.Butevenmore,SetTheoryisthemilieuinwhichmathematicstakesplacetoday.Assuch,itisexpectedto

5、provideafirmfoundationfortherestofmathematics.Anditdoes—uptoapoint;wewillprovetheoremssheddinglightonthisissue.BecausethefundamentalsofSetTheoryareknowntoallmathemati-cians,basicproblemsinthesubjectseemelementary.Herearethreesimplestatementsaboutsetsandfunctions.

6、Theylookliketheycouldappearonahomeworkassignmentinanundergraduatecourse.1.ForanytwosetsXandY,eitherthereisaone-to-onefunctionfromXintoYoraone-to-onefunctionfromYintoX.2.Ifthereisaone-to-onefunctionfromXintoYandalsoaone-to-onefunctionfromYintoX,thenthereisaone-to

7、-onefunctionfromXontoY.3.IfXisasubsetoftherealnumbers,theneitherthereisaone-to-onefunctionfromthesetofrealnumbersintoXorthereisaone-to-onefunctionfromXintothesetofrationalnumbers.Theywon’tappearonanassignment,however,becausetheyarequitedif-78CHAPTER0.INTRODUCTIO

8、Nficulttoprove.Statement(2)istrue;itiscalledtheSchroder-BernsteinTheorem.Theproof,ifyouhaven’tseenitbefore,isquitetrickybutnever-thelessusesonlystandardideasfromthenin