Can Mathematics Be Proved Consistent?: Gödel's Shorthand Notes & Lectures on Incompleteness

271 Pages·2020·1.603 MB·English

by  Jan von Plato

Sources and Studies in the History of Mathematics
and Physical Sciences
Jan von Plato
Can Mathematics
Be Proved
Consistent?
Gödel s Shorthand Notes &
ʹ
Lectures on Incompleteness
Sources and Studies in the History of Mathematics
and Physical Sciences
ManagingEditor
JedZ.Buchwald
AssociateEditors
A.Jones
J.Lu¨tzen
J.Renn
AdvisoryBoard
C.Fraser
T.Sauer
A.Shapiro
Sources and Studies in the History of Mathematics and Physical Sciences was inaugu-
ratedastwoseriesin1975withthepublicationinStudiesofOttoNeugebauer’sseminal
three-volume History of Ancient Mathematical Astronomy, which remains the central
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Burning Mirrors. The two series were eventually amalgamated under a single editorial
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theforemosthistoriansofmodernandancientphysicalscience.Thegoalofthejointse-
ries, as of its two predecessors, is to publish probing histories and thorough editions of
technical developments in mathematics and physics, broadly construed. Its scope cov-
ers all relevant work from pre-classical antiquity through the last century, ranging from
BabylonianmathematicstothescientificcorrespondenceofH.A.Lorentz.Booksinthis
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physicists,engineers,andanyonewhoseekstounderstandthehistoricalunderpinningsof
themodernphysicalsciences.
Moreinformationaboutthisseriesathttp://www.springer.com/series/4142
Jan von Plato
Can Mathemati cs
Be Proved Consistent?
Gödel’s Shorthand Notes & Lectures
on Incompleteness
Jan von Plato
Department of Philosophy
University of Helsinki
Helsinki, Finland

ISSN 2196-8810 ISSN 2196-8829 (electronic )
Sources and Studies in the History of Mathematics and Physical Sciences
ISBN 978-3-030-50875-3 ISBN 978-3-030-50876-0 (eB ook)
https://doi.org/10.1007/978-3-030-50876-0

Mathematics Subject Classification (2010): 03F40, 01Axx

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Preface
This book contains all that is found in Go¨del’s preserved shorthand note-
books on his research that led to the famous incompleteness theorems of
formal systems. The notes are followed by the original version of his ar-
ticle, before a dramatic change just a few days after it was handed in for
publication, and six lectures and seminars in consequence of his celebra-
ted result published in 1931. The notebooks and one of the lectures were
writteninGermanGabelsbergershorthandthatIhavetranslatedintoEng-
lish,usuallyfromanintermediatetranscriptionintoGerman,butatplaces
directly. I thank Tim Lethen for his help in the reading of many difficult
shorthandpassages,andMariaHa¨meen-Anttilaforhersupport,especially
atthetroublesomemomentwhenIdiscoveredGo¨del’strickychangeofhis
manuscript after it had been submitted for publication. Marcia Tucker of
the Institute for Advanced Study was very helpful during my visit to the
Firestone Library of Princeton University where the originals of Go¨del’s
manuscripts are kept. Finally, I recollect with affection my mother’s deci-
siontochallengeherlittleboybyenrollinghimintheGermanelementary
schoolofHelsinki,achoicewithoutwhichIwouldnothavelearnedtoread
Go¨del’smanuscripts.
JanvonPlato
v
Acknowledgment
TheKurtGo¨delPapersonincompletenessthatthisbookexploresarekept
attheFirestoneLibraryofPrincetonUniversity.Afindingaidwithdetails
abouttheircontentsisfoundattheendofthefifthvolumeofGo¨del’sCol-
lectedWorks.ThepapersweredividedbytheircataloguerJohnDawsoninto
archivalboxesandwithinboxesintofolders.Folderscanhaveathirddivi-
sionintodocuments,witharunningdocumentnumberingsystem.Thepa-
pershavebeenmainlyaccessedthroughamicrofilmthatispubliclyavaila-
ble,butalsodirectlyinPrinceton.Referencestospecificpagesofnotebooks
usuallyrequiretheuseofthereelandframenumbersofthemicrofilmand
that is how the sources are mostly identified in this book. Passages from
Go¨del in the introductory Part I are identified in loco. The shorthand ma-
nuscript sources on which Go¨del’s 1931 article is based are described in
detailinPartII,Section2ofthisbook.Thetypewrittensourcesforhis1930
summaryandthe1931articlearedescribedinthefollowingSection3.The
sourcesofthesixlecturesandseminarsonincompletenessaredescribedin
thelastSection4ofPartII.Thesedescriptionstogetherwiththeframeand
page numberings in Parts III–V allow the interested reader to identify the
sourcetextswiththeprecisionofanotebookpage.
The writing of this book has been financed by the European Research
CouncilAdvancedGrant GODELIANA(grantagreementNo787758).
AllworksofKurtGo¨delusedwithpermission.UnpublishedCopyright
(1906-1978) Institute for Advanced Study. All rights reserved by Institute
for Advanced Study. The papers are on deposit at Manuscripts Division,
Department of Rare Books and Special Collections, Princeton University
Library.
vii
Contents
PARTI:GO¨DEL’SSTEPSTOWARDINCOMPLETENESS ............................1
1.Thecompletenessproblem...............................................3
2.FromSkolem’sparadoxtotheKo¨nigsbergconference....................8
3.FromtheKo¨nigsbergconferencetovonNeumann’sletter...............13
4.Thesecondtheorem:“Onlyinarealmofideas”.........................24
PARTII:THESAVEDSOURCESONINCOMPLETENESS ..........................29
1.Shorthandwriting......................................................31
2.Descriptionoftheshorthandnotebooksonincompleteness..............33
3.Thetypewrittenmanuscripts............................................46
4.Lecturesandseminarsonincompleteness...............................49
PARTIII:THESHORTHANDNOTEBOOKS ......................................59
1.Undecidabilitydraft.WelayasabasisthesystemofthePrincipia........61
2.ThereareunsolvableproblemsinthePrincipiaMathematica..............68
3.Thedevelopmentofmathematicsinthedirectionofgreaterexactness...86
4.Thequestionwhethereachmathematicalproblemissolvable..........102
5.Aproofinbroadoutlinewillbesketched...............................123
6.WeproduceanundecidablepropositioninthePrincipia................126
7.Thedevelopmentofmathematicsinthedirectionofgreaterexactness..133
8.Letusturnbacktotheundecidableproposition........................162
PARTIV:THETYPEWRITTENMANUSCRIPTS..................................169
1.Somemetamathematicalresults........................................171
2.Onformallyundecidablepropositions(earlierversion).................173
PARTV:LECTURESANDSEMINARSONINCOMPLETENESS....................201
1.Lectureonundecidablepropositions(BadElster).......................203
2.Onformallyundecidablepropositions(BadElster).....................206
3.Onundecidablepropositions(Vienna).................................213
4.Ontheimpossibilityofproofsoffreedomfromcontradiction(Vienna)..226
5.Theexistenceofundecidablepropositions(NewYork).................235
6.Canmathematicsbeprovedconsistent?(Washington)..................246
IndexofnamesandlistofreferencesinGo¨del’snotes ........................ 261
ReferencesforPartsIandII ..................................................262
ix
Part I
Go¨del’s steps toward incompleteness
1
© The Editor(s) (if applicable) and The Author(s), under exclusive license to Springer Nature Switzerland AG 2020
J. von Plato, Can Mathematics Be Proved Consistent?, Sources and Studies in the History
of Mathematics and Physical Sciences, https://doi.org/10.1007/978-3-030-50876-0_1