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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’sresponsetotheparadoxRussellsownresponsetotheparadoxcamewiththedevelopmentofhistheyoftypesin1903.ItwascleartoRussellthatsomerestrictionsneededtobeplacedupontheiginalcomprehension(abstraction)axiomofnaivesetthe

10、ytheaxiomthatfmalizestheintuitionthatanycoherentconditionmaybeusedtodetermineaset(class).Russellsbasic3Part2(Paras.25):TheeffectofRussell’sparadoxonGottlobFrege’ssystem.Para.2:Russell’sparadoxdealtaheavyblowtoFrege’satte

11、mptstodevelopafoundationfallofmathematicsusingsymboliclogic.Para.3:AnillustrationofRussell’sparadoxintermsofsetsPara.4:Contradictionfoundintheset.Para.5:FregenoticedthedevastatingeffectofRussell’sparadoxonhissysteminabil

12、itytosolveit.Part3(Paras.68):SolutionsofferedbymathematicianstoRussel’sparadoxPara.6:Russell’sownresponsetotheparadoxwithhis“theyoftypes.“Para.7:ZermelossolutiontoRussellsparadoxPara.8:Whatbecameoftheeffttodevelopalogica

13、lfoundationfallofmathematicsPart4(Para.9):CrespondencebetweenRussellFregeontheparadox2.Directions:Wkinpairsdiscussthefollowingquestions.1)WhatisthebasicideaofRussell’sparadox2)HowtoexplainRussell’sparadoxintermsofsets3)C

14、anyouexplainthecontradictionfoundinthesetsrelatedtoRussell’sparadox4)IsRussell’sownresponsetotheparadoxwkable5)DoyouknowZermeloFraenkelsetthey(open)3.Directions:Readthefollowingpassagecarefullyfillintheblankswiththewdsyo

15、u’velearnedinthetext.Russellsownresponsetotheparadoxcamewiththedevelopmentofhistheyoftypesin1903.ItwascleartoRussellthatsomerestrictionsneededtobeplacedupontheiginalcomprehension(abstraction)axiomofnaivesettheytheaxiomth

16、atfmalizestheintuitionthatanycoherentconditionmaybeusedtodetermineaset(class).Russellsbasicideawasthatreferencetosetssuchasthesetofallsetsthatarenotmembersofthemselvescouldbeavoidedbyarrangingallsentencesintoahierarchybe

17、ginningwithsentencesaboutindividualsatthelowestlevelsentencesaboutsetsofindividualsatthenextlowestlevelsentencesaboutsetsofsetsofindividualsatthenextlowestlevelsoon.Usingaviciouscircleprinciplesimilartothatadoptedbythema

18、thematicianHenriPoincarhisownsocalled“noclass“theyofclassesRussellwasabletoexplainwhytheunrestrictedcomprehensionaxiomfails:propositionalfunctionssuchasthefunction“xisaset“maynotbeappliedtothemselvessinceselfapplicationw