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1、1Unit1MathematicsPartIESTReadingReading1(article.cfmid=whatisrussellsparadoxSectionAPrereadingTaskWarmupQuestions:Wkinpairsdiscussthefollowingquestions.1.WhoisBertrRussellBertrArthurWilliamRussell(b.1872–d.1970)wasaBriti

2、shphilosopherlogicianessayistsocialcriticbestknownfhiswkinmathematicallogicanalyticphilosophy.Hismostinfluentialcontributionsincludehisdefenseoflogicism(theviewthatmathematicsisinsomeimptantsensereducibletologic)hisrefin

3、ingofthepredicatecalculusintroducedbyGottlobFrege(whichstillfmsthebasisofmostcontemparylogic)hisdefenseofneutralmonism(theviewthatthewldconsistsofjustonetypeofsubstancethatisneitherexclusivelymentalnexclusivelyphysical)h

4、istheiesofdefinitedeionslogicalatomism.Russellisgenerallyrecognizedasoneofthefoundersofmodernanalyticphilosophyisregularlycreditedwithbeingoneofthemostimptantlogiciansofthetwentiethcentury.2.WhatisRussell’sParadoxRussell

5、discoveredtheparadoxthatbearshisnamein1901whilewkingonhisPrinciplesofMathematics(1903).Theparadoxarisesinconnectionwiththesetofallsetsthatarenotmembersofthemselves.Suchasetifitexistswillbeamemberofitselfifonlyifitisnotam

6、emberofitself.Theparadoxissignificantsinceusingclassicallogicallsentencesareentailedbyacontradiction.Russellsdiscoverythuspromptedalargeamountofwkinlogicsettheythephilosophyfoundationsofmathematics.3.WhateffectdidRussell

7、’sParadoxhaveonGottlobFregg’ssystemAtfirstFregeobservedthattheconsequencesofRussell’sparadoxarenotimmediatelyclear.Fexample“Isitalwayspermissibletospeakoftheextensionofaconceptofaclassifnothowdowerecognizetheexceptionalc

8、asesCanwealwaysinferfromtheextensionofoneconcept’scoincidingwiththatofasecondthateveryobjectwhichfallsunderthefirstconceptalsofallsunderthesecondBecauseofthesekindsofwriesFregeeventuallyfeltfcedtoabonmanyofhisviews.4.Wha

9、tisRussell’sresponsetotheparadoxRussellsownresponsetotheparadoxcamewiththedevelopmentofhistheyoftypesin31.Directions:Wkonyourownfillintheblankswiththemainidea.Part1(Para.1):BriefintroductiontoRussell’sparadoxPart2(Paras.

10、25):TheeffectofRussell’sparadoxonGottlobFrege’ssystem.Para.2:Russell’sparadoxdealtaheavyblowtoFrege’sattemptstodevelopafoundationfallofmathematicsusingsymboliclogic.Para.3:AnillustrationofRussell’sparadoxintermsofsetsPar

11、a.4:Contradictionfoundintheset.Para.5:FregenoticedthedevastatingeffectofRussell’sparadoxonhissysteminabilitytosolveit.Part3(Paras.68):SolutionsofferedbymathematicianstoRussel’sparadoxPara.6:Russell’sownresponsetotheparad

12、oxwithhis“theyoftypes.“Para.7:ZermelossolutiontoRussellsparadoxPara.8:WhatbecameoftheeffttodevelopalogicalfoundationfallofmathematicsPart4(Para.9):CrespondencebetweenRussellFregeontheparadox2.Directions:Wkinpairsdiscusst

13、hefollowingquestions.1)WhatisthebasicideaofRussell’sparadox2)HowtoexplainRussell’sparadoxintermsofsets3)CanyouexplainthecontradictionfoundinthesetsrelatedtoRussell’sparadox4)IsRussell’sownresponsetotheparadoxwkable5)Doyo

14、uknowZermeloFraenkelsetthey(open)3.Directions:Readthefollowingpassagecarefullyfillintheblankswiththewdsyou’velearnedinthetext.Russellsownresponsetotheparadoxcamewiththedevelopmentofhistheyoftypesin1903.ItwascleartoRussel

15、lthatsomerestrictionsneededtobeplacedupontheiginalcomprehension(abstraction)axiomofnaivesettheytheaxiomthatfmalizestheintuitionthatanycoherentconditionmaybeusedtodetermineaset(class).Russellsbasicideawasthatreferencetose

16、tssuchasthesetofallsetsthatarenotmembersofthemselvescouldbeavoidedbyarrangingallsentencesintoahierarchybeginningwithsentencesaboutindividualsatthelowestlevelsentencesaboutsetsofindividualsatthenextlowestlevelsentencesabo

17、utsetsofsetsofindividualsatthenextlowestlevelsoon.UsingaviciouscircleprinciplesimilartothatadoptedbythemathematicianHenriPoincarhisownsocalled“noclass“theyofclassesRussellwasabletoexplainwhytheunrestrictedcomprehensionax