One hundred years of Russell's paradox: mathematics, logic, philosophy

672 Pages·2004·2.221 MB·English

by  Godehard Link

de Gruyter Series in Logic and Its Applications 6
Editors: W.A. Hodges (London) · R. Jensen (Berlin)
S. Lempp (Madison) · M. Magidor (Jerusalem)

One Hundred Years of Russell’s Paradox
Mathematics, Logic, Philosophy
Editor
Godehard Link
≥
Walter de Gruyter
Berlin · New York
Editor
GodehardLink
Philosophie-Department
Ludwig-Maximilians-UniversitätMünchen
Ludwigstr.31/I
80539München
Germany
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SchoolofMathematicalSciences InstitutfürMathematik
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OnehundredyearsofRussell’sparadox:mathematics,logic,phi-
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1.Paradox. 2.Liarparadox. I.Title:100yearsofRussell’s
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Preface
InJune2001aninternationalconferencewasheldattheUniversityofMunich,Ger-
many,tocommemoratethecentenaryofthediscoveryofRussell’sparadox. Itbrought
togetherleadingscholarsfromthefollowingfields:
(cid:127) RussellStudies
(cid:127) MathematicalLogic
(cid:127) SetTheory
(cid:127) PhilosophyofMathematics
The aims of the event were twofold, historical and systematic. One focus was
on Russell’s logic and logical philosophy, which were shaped so much by his own
andrelatedparadoxes,inparticularsincenewmaterialhasbecomeavailableinrecent
years from the rich sources of the Russell Archives through the publication of the
CollectedPapersofBertrandRussell. Butthesecondaimwasofequalimportance:
topresentoriginalresearchinthebroadrangeoffoundationalstudiesthatdrawson
bothcurrentconceptionsandrecenttechnicaladvancesintheabove-mentionedfields.
It was hoped to contribute this way to the well-established body of mathematical
philosophyinitiatedtoalargeextentbyRussellahundredyearsago.
Theconferencefeaturedplenarysessionswithdistinguishedinvitedspeakers,sec-
tionmeetingswithbothinvitedandcontributedpapers,andasspecialeventstwopanel
discussions, “RussellinContext”and“TheMeaningofSetTheory”, aswellastwo
symposia,oneon“PropositionalFunctions”andtheotheronthe“FiniteMathematics
ofSetTheory”. TheRussellpanel,chairedbyA.Irvine,wascomposedofN.Griffin,
P.Hylton,D.Kaplan,andA.Urquhart.Y.MoschovakischairedthepanelonSetThe-
ory,withS.Feferman,H.Friedman,D.C.McCarty,andW.H.Woodinaspanelists.
The symposiasts of the first symposium, chaired by R. Wahl, were G. Landini and
B.Linsky,thoseofthesecondS.LavineandK.-G.Niebergall,chairedbytheeditor.
Thepaperscollectedinthisvolumerepresentthemainbodyofresearchemanating
fromtheproceedingsoftheconference. Allauthorsdeliveredatalkattheconference
exceptJ.Mycielski,whowasunabletoattendbutsenthisinvitedpaper,andH.Field,
whoattendedtheconferenceasadiscussantandwasinvitedbytheeditortocontribute
his present paper. G. Jäger and H. Schwichtenberg were each joined by a co-author
for their paper included here. A number of invited speakers delivered a talk, but
wereunabletocontributeapapertothevolumebecauseofothercommitments;these
are W. Buchholz, P. Hylton, H. Kamp, P. Martin-Löf,Y. Moschovakis, C. Parsons,
G.Priest,andA.Urquhart.
vi Preface
Thepaperswerealloriginallywrittenforthisvolume,withtwoexceptions. The
originalplaceofpublicationofH.Field’spaper“TheConsistencyoftheNaiveTheory
of Properties” is The Philosophical Quarterly, volume 54, No. 214, January 2004.
It is reprinted here by kind permission of Blackwell Publishing and the author. A
condensed version of Hazen’s paper appeared as A. P. Hazen and J. M. Davoren,
Russell’s 1925 logic, in: Australasian Journal of Philosophy, volume 78, No. 4,
pp.534–556,December2000. Reprintofthismaterialaspartofthepresentpaperby
kindpermissionofOxfordUniversityPressandtheco-author.
The occasion to publish this volume would never have arisen without the essen-
tialfinancialsupportfromanumberofinstitutionsthatmadetheconferencepossible
in the first place. It is therefore fitting to express my deep appreciation here to the
DeutscheForschungsgemeinschaft(DFG)forcoveringthemainbulkoftheconfer-
encecoststhroughfundsmadeavailabletotheinterdisciplinaryDFGgraduateprogram
Graduiertenkolleg “Sprache, Information, Logik” (SIL) at Munich University, and
specialconferencefunds;furthermoretothefollowinginstitutionsandprivatesponsors
for filling the remaining financial gaps: the Bayerisches Staatsministerium fürWis-
senschaft, Forschung und Kunst, the Philosophy Department, University of Munich,
theGesellschaftvonFreundenundFörderernderUniversitätMünchen(Münchener
Universitätsgesellschaft e.V.), the University of the German Federal Armed Forces
Munich,AppleComputerGermany,andAlpinaBurkardBovensiepenGmbH&Co.
Many people helped in various ways to organize the conference and to prepare
this volume. I wish to thank all of them, in particular Solomon Feferman for his
encouragement, and Andrew Irvine for generous advice, in the early stages of the
conferenceproject. UlrichAlbertcarriedthemainorganizationalburdenofthecon-
ference as “front liner” on the internet and at the conference site; Sebastian Paasch
was responsible for the technical part of the preparation of the volume. My thanks
tobothofthemfortheirefficiencyandtheircommitmentfarbeyondthecallofduty.
UlrichandSebastianwereassistedatdifferentstagesbyMarie-NicoleEhlers,Martin
Fischer, Roland Kastler, Uwe Lück, Michaela Paivarinta, Hannes Petermair, Marek
Polanski, Christian Tapp, Mai-Lan Thai, Julia Zink, and Mauricio Zuluaga. Daniel
MookhelpedwithcheckingtheEnglishofnon-nativespeakers.
IoweaspecialdebttoKarl-GeorgNiebergall,whowasalwayscloseathandfor
any advise on matters of content pertaining to the contributions of the volume, and
whosephilosophicalseriousness,enthusiasmandunfailingtechnicaljudgmentIhave
hadtheprivilegetoenjoyinmanyyearsofjointseminarsandprivatediscussions.
Finally, I wish to extend my thanks to Dr. Manfred Karbe of Walter de Gruyter
Publishers for supporting the decision to publish the volume, and for his sustained
encouragement,advice,andpatienceduringthephaseofitspreparation.
May2004 G.L.
Table of Contents
Preface v
GodehardLink
Introduction. BertrandRussell—TheInventionofMathematicalPhilosophy 1
W.HughWoodin
SetTheoryafterRussell: TheJourneyBacktoEden 29
HarveyM.Friedman
AWayOut 49
SyD.Friedman
CompletenessandIterationinModernSetTheory 85
KaiHauser
Wassindundwassollen(neue)Axiome? 93
GerhardJägerandDieterProbst
Iterating(cid:1) OperationsinAdmissibleSetTheorywithoutFoundation:
AFurtherAspectofMetapredicativeMahlo 119
SolomonFeferman
TypicalAmbiguity: TryingtoHaveYourCakeandEatItToo 135
Karl-GeorgNiebergall
IsZFFinitisticallyReducible? 153
TobiasHürter
InconsistencyintheRealWorld 181
MichaelRathjen
Predicativity,Circularity,andAnti-Foundation 191
JohnL.Bell
Russell’sParadoxandDiagonalizationinaConstructiveContext 221
viii TableofContents
PeterSchusterandHelmutSchwichtenberg
ConstructiveSolutionsofContinuousEquations 227
KaiF.Wehmeier
Russell’sParadoxinConsistentFragmentsofFrege’s
GrundgesetzederArithmetik 247
AndreaCantini
OnaRussellianParadoxaboutPropositionsandTruth 259
HartryField
TheConsistencyoftheNaiveTheoryofProperties 285
UlrichBlau
TheSignificanceoftheLargestandSmallestNumbers
fortheOldestParadoxes 311
NicholasGriffin
ThePrehistoryofRussell’sParadox 349
GregoryLandini
Logicism’s‘Insolubilia’andTheirSolutionbyRussell’sSubstitutionalTheory 373
PhilippedeRouilhan
SubstitutionandTypes: Russell’sIntermediateTheory 401
FranciscoRodríguez-Consuegra
PropositionalOntologyandLogicalAtomism 417
BernardLinsky
ClassesofClassesandClassesofFunctionsinPrincipiaMathematica 435
AllenP.Hazen
A“Constructive”ProperExtensionofRamifiedTypeTheory(TheLogicof
PrincipiaMathematica,SecondEdition,AppendixB) 449
AndrewD.Irvine
RussellonMethod 481
VolkerPeckhaus
ParadoxesinGöttingen 501
TableofContents ix
DavidCharlesMcCarty
DavidHilbertandPaulduBois-Reymond: LimitsandIdeals 517
JanMycielski
Russell’sParadoxandHilbert’s(muchForgotten)ViewofSetTheory 533
ShaughanLavine
Objectivity: TheJustificationforExtrapolation 549
GeoffreyHellman
Russell’sAbsolutismvs.(?) Structuralism 561
RobertS.D.Thomas
MathematiciansandMathematicalObjects 577
HolgerSturm
Russell’sParadoxandOurConceptionofProperties,or: WhySemantics
IsnoProperGuidetotheNatureofProperties 591
VannMcGee
TheManyLivesofEbenezerWilkesSmith 611
AlbertVisser
WhatMakesExpressionsMeaningful? AReflectiononContextsandActions 625
ListofContributors 645
NameIndex 649
SubjectIndex 653