Dedekind Cut

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1、Dedekindcutsrmilsony2013-03-2114:05:58ThepurposeofDedekindcutsistoprovideasoundlogicalfoundationfortherealnumbersystem.Dedekind'smotivationbehindthisprojectistonoticethatarealnumber,intuitively,iscompletelydeterminedbytherationalsstrictlysmallerthanandthosestrictlylargerthan.Concerningthecom

2、pletenessorcontinuityoftherealline,Dedekindnotesin[?]thatIfallpointsofthestraightlinefallintotwoclassessuchthateverypointoftherstclassliestotheleftofeverypointofthesecondclass,thenthereexistsoneandonlyonepointwhichproducesthisdivisionofallpointsintotwoclasses,thisseveringofthestraightlineint

3、otwoportions.Dedekinddenesapointtoproducethedivisionofthereallineifthispointiseithertheleastorgreatestelementofeitheroneoftheclassesmentionedabove.Hefurthernotesthatthecompletenessproperty,ashejustphrasedit,isdecientintherationals,whichmotivatesthedenitionofrealsascutsofrationals.Becauseal

4、lrationalsgreaterthanarereallyjustexcessbaggage,weprefertoswaysomewhatfromDedekind'soriginaldenition.Instead,weadoptthefollowingdenition.Denition.ADedekindcutisasubsetoftherationalnumbersQthatsatisestheseproperties:1.isnotempty.2.Qnisnotempty.3.containsnogreatestelement4.Forx;y2Q,ifx2andy

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6、.1Dedekindcutsareparticularlyappealingfortworeasons.First,theymakeitveryeasytoprovethecompleteness,orcontinuityoftherealline.Also,theymakeitquiteplaintodistinguishtherationalsfromtheirrationalsontherealline,andputthelatteronarmlogicalfoundation.IntheconstructionoftherealnumbersfromDedekindcu

7、ts,wemakethefollowingdenition:Denition.ArealnumberisaDedekindcut.WedenotethesetofallrealnumbersbyRandweorderthembyset-theoreticinclusion,thatistosay,forany;2R,<ifandonlyifwheretheinclusionisstrict.Wefurtherdene=asrealnumbersifandareeq