Infinity: New Research Frontiers

327 Pages·2011·3.153 MB·English

by  Michael Heller, W. Hugh Woodin

This page intentionally left blank
Infinity
NewResearchFrontiers
Thisinterdisciplinarystudyofinfinityexplorestheconceptthroughtheprismofmathemat-
icsandthenoffersmoreexpansiveinvestigationsinareasbeyondmathematicalboundaries
toreflectthebroader,deeperimplicationsofinfinityforhumanintellectualthought.More
than a dozen world-renowned researchers in the fields of mathematics, physics, cosmol-
ogy, philosophy, and theology offer a rich intellectual exchange among various current
viewpoints,ratherthanastaticpictureofacceptedviewsoninfinity.
The book starts with a historical examination of the transformation of infinity from
a philosophical and theological study to one dominated by mathematics. It then offers
technical discussions on the understanding of mathematical infinity. Following this, the
bookconsiderstheperspectivesofphysicsandcosmology:Caninfinitybefoundinthereal
universe?Finally,thebookreturnstoquestionsofphilosophicalandtheologicalaspectsof
infinity.
Rev. Dr. Michael Heller is the Founder and Director of the Copernicus Center for Inter-
disciplinaryStudies,Cracow.HeisaProfessorofPhilosophyatthePontificalUniversity
of John Paul II, Cracow, Poland; an Adjunct Member of the Vatican Observatory; and
OrdinaryMemberofthePontificalAcademyofSciences.Helleristheauthorofmorethan
thirtybooksandnearlytwohundredscientificpapers.
Dr. W. Hugh Woodin is a Professor of Mathematics and the Chair of the Mathematics
Department at the University of California, Berkeley. Professor Woodin has published
numerous articles and books and is the managing editor of the Journal of Mathematical
Logic and editor of Mathematical Research Letters, Mathematical Logic Quarterly, and
ElectronicResearchAnnouncements(AmericanMathematicalSociety).

Infinity
New Research Frontiers
Editedby
Michael Heller
PontificalUniversityofJohnPaulII
W. Hugh Woodin
UniversityofCalifornia,Berkeley
cambridgeuniversitypress
Cambridge,NewYork,Melbourne,Madrid,CapeTown,Singapore,
Sa˜oPaulo,Delhi,Dubai,Tokyo,MexicoCity
CambridgeUniversityPress
32AvenueoftheAmericas,NewYork,NY10013-2473,USA
www.cambridge.org
Informationonthistitle:www.cambridge.org/9781107003873
(cid:2)c CambridgeUniversityPress2011
Thispublicationisincopyright.Subjecttostatutoryexception
andtotheprovisionsofrelevantcollectivelicensingagreements,
noreproductionofanypartmaytakeplacewithoutthewritten
permissionofCambridgeUniversityPress.
Firstpublished2011
PrintedintheUnitedStatesofAmerica
AcatalogrecordforthispublicationisavailablefromtheBritishLibrary.
LibraryofCongressCataloginginPublicationdata
Infinity:newresearchfrontiers/editedbyMichaelHeller,W.HughWoodin.
p. cm.
Includesindex.
ISBN978-1-107-00387-3(hardback)
1.Infinite–History. 2.Mathematics–Philosophy.
I.Heller,Michael. II.Woodin,W.H.(W.Hugh) III.Title.
QA9.I46 2011
111(cid:3).6–dc22 2010049374
ISBN978-1-107-00387-3Hardback
CambridgeUniversityPresshasnoresponsibilityforthepersistenceoraccuracyofurlsforexternal
orthird-partyInternetWebsitesreferredtointhispublicationanddoesnotguaranteethatanycontent
onsuchWebsitesis,orwillremain,accurateorappropriate.
Theinfinite!Nootherquestionhasevermovedsoprofoundlythespiritofman;noother
idea has so fruitfully stimulated his intellect; yet no other concept stands in greater
needofclarificationthanthatoftheinfinite....
DavidHilbert(1862–1943)

Contents
Contributors pageix
Preface xi
Acknowledgments xiii
Introduction 1
RudyRucker
I PerspectivesonInfinityfromHistory
1 InfinityasaTransformativeConceptinScienceandTheology 19
WolfgangAchtner
II PerspectivesonInfinityfromMathematics
2 TheMathematicalInfinity 55
EnricoBombieri
3 WarningSignsofaPossibleCollapseofContemporaryMathematics 76
EdwardNelson
III TechnicalPerspectivesonInfinityfromAdvanced
Mathematics
4 TheRealmoftheInfinite 89
W.HughWoodin
5 APotentialSubtletyConcerningtheDistinctionbetween
DeterminismandNondeterminism 119
W.HughWoodin
6 ConceptCalculus:MuchBetterThan 130
HarveyM.Friedman
vii
viii contents
IV PerspectivesonInfinityfromPhysicsandCosmology
7 SomeConsiderationsonInfinityinPhysics 167
CarloRovelli
8 CosmologicalIntimationsofInfinity 176
AnthonyAguirre
9 InfinityandtheNostalgiaoftheStars 193
MarcoBersanelli
10 InfinitiesinCosmology 218
MichaelHeller
V PerspectivesonInfinityfromPhilosophyandTheology
11 GodandInfinity:DirectionsforFutureResearch 233
GrahamOppy
12 NotesontheConceptoftheInfiniteintheHistoryofWestern
Metaphysics 255
DavidBentleyHart
13 GodandInfinity:TheologicalInsightsfromCantor’sMathematics 275
RobertJohnRussell
14 A(Partially)SkepticalResponsetoHartandRussell 290
DenysA.Turner
Index 299