Subsystems of Second Order Arithmetic

462 Pages·2009·1.95 MB·English

by  Stephen G. Simpson

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SubsystemsofSecondOrderArithmetic
SecondEdition
Foundations of mathematics is the study of the most basic concepts and logical
structureofmathematics,withaneyetotheunityofhumanknowledge.Almostall
oftheproblemsstudiedinthisbookaremotivatedbyanoverridingfoundational
question:Whataretheappropriateaxiomsformathematics?Throughaseriesof
casestudies,theseaxiomsareexaminedtoproveparticulartheoremsincoremathe-
maticalareassuchasalgebra,analysis,andtopology,focusingonthelanguageof
secondorderarithmetic,theweakestlanguagerichenoughtoexpressanddevelop
thebulkofmathematics.
Inmanycases,ifamathematicaltheoremisprovedfromappropriatelyweak
setexistenceaxioms,thentheaxiomswillbelogicallyequivalenttothetheorem.
Furthermore, only a few specific set existence axioms arise repeatedly in this
context,whichinturncorrespondtoclassicalfoundationalprograms.Thisisthe
theme of reverse mathematics, which dominates the first half of the book. The
second part focuses on models of these and other subsystems of second order
arithmetic.Additionalresultsarepresentedinanappendix.
StephenG.SimpsonisamathematicianandprofessoratPennsylvaniaStateUni-
versity. The winner of the Grove Award for Interdisciplinary Research Initia-
tion, Simpson specializes in research involving mathematical logic, foundations
ofmathematics,andcombinatorics.
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PERSPECTIVES IN LOGIC
Subsystems of Second Order Arithmetic
Second Edition
STEPHEN G. SIMPSON
PennsylvaniaStateUniversity
association for symbolic logic
v
CAMBRIDGEUNIVERSITYPRESS
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Cambridge University Press
The Edinburgh Building, Cambridge CB2 8RU, UK
Published in the United States of America by Cambridge University Press, New York
www.cambridge.org
Information on this title: www.cambridge.org/9780521884396
© The Association for Symbolic Logic 2009
This publication is in copyright. Subject to statutory exception and to the
provision of relevant collective licensing agreements, no reproduction of any part
may take place without the written permission of Cambridge University Press.
First published in print format 2009
ISBN-13 978-0-511-57985-1 eBook (EBL)
ISBN-13 978-0-521-88439-6 Hardback
Cambridge University Press has no responsibility for the persistence or accuracy
of urls for external or third-party internet websites referred to in this publication,
and does not guarantee that any content on such websites is, or will remain,
accurate or appropriate.
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CONTENTS
ListofTables................................................... xi
Preface.......................................................... xiii
Acknowledgments.............................................. xv
ChapterI. Introduction...................................... 1
I.1. TheMainQuestion.................................... 1
I.2. SubsystemsofZ ...................................... 2
2
I.3. TheSystemACA ..................................... 6
0
I.4. MathematicswithinACA ............................. 9
0
I.5. Π1-CA andStrongerSystems.......................... 16
1 0
I.6. MathematicswithinΠ1-CA ........................... 19
1 0
I.7. TheSystemRCA ..................................... 23
0
I.8. MathematicswithinRCA ............................. 27
0
I.9. ReverseMathematics.................................. 32
I.10. TheSystemWKL ..................................... 35
0
I.11. TheSystemATR ...................................... 38
0
I.12. TheMainQuestion,Revisited.......................... 42
I.13. OutlineofChaptersIIthroughX....................... 43
I.14. Conclusions........................................... 60
PartA. DevelopmentofMathematicswithinSubsystemsofZ
2
ChapterII. RecursiveComprehension......................... 63
II.1. TheFormalSystemRCA .............................. 63
0
II.2. FiniteSequences...................................... 65
II.3. PrimitiveRecursion................................... 69
II.4. TheNumberSystems.................................. 73
II.5. CompleteSeparableMetricSpaces..................... 78
II.6. ContinuousFunctions................................. 84
II.7. MoreonCompleteSeparableMetricSpaces............ 88
II.8. MathematicalLogic................................... 92
II.9. CountableFields...................................... 96
vii
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viii Contents
II.10. SeparableBanachSpaces.............................. 99
II.11. Conclusions...........................................103
ChapterIII. ArithmeticalComprehension.....................105
III.1. TheFormalSystemACA ..............................105
0
III.2. SequentialCompactness...............................106
III.3. StrongAlgebraicClosure.............................. 110
III.4. CountableVectorSpaces...............................112
III.5. MaximalIdealsinCountableCommutativeRings.......115
III.6. CountableAbelianGroups............................ 118
III.7. Ko¨nig’sLemmaandRamsey’sTheorem................ 121
III.8. Conclusions...........................................125
ChapterIV. WeakKo¨nig’sLemma............................. 127
IV.1. TheHeine/BorelCoveringLemma.....................127
IV.2. PropertiesofContinuousFunctions....................133
IV.3. TheGo¨delCompletenessTheorem.....................139
IV.4. FormallyRealFields.................................. 141
IV.5. UniquenessofAlgebraicClosure.......................144
IV.6. PrimeIdealsinCountableCommutativeRings..........146
IV.7. FixedPointTheorems................................. 149
IV.8. OrdinaryDifferentialEquations........................154
IV.9. TheSeparableHahn/BanachTheorem.................160
IV.10. Conclusions...........................................165
ChapterV. ArithmeticalTransfiniteRecursion..............167
V.1. CountableWellOrderings;AnalyticSets................167
V.2. TheFormalSystemATR ..............................173
0
V.3. BorelSets.............................................178
V.4. PerfectSets;Pseudohierarchies.........................185
V.5. Reversals............................................. 189
V.6. ComparabilityofCountableWellOrderings............ 195
V.7. CountableAbelianGroups............................ 199
V.8. Σ0and∆0Determinacy................................203
1 1
V.9. TheΣ0and∆0RamseyTheorems...................... 210
1 1
V.10. Conclusions...........................................215
ChapterVI. Π1 Comprehension................................217
1
VI.1. PerfectKernels........................................217
VI.2. CoanalyticUniformization ............................ 221
VI.3. CoanalyticEquivalenceRelations...................... 225
VI.4. CountableAbelianGroups............................ 230
VI.5. Σ0 Π0Determinacy..................................232
1∧ 1
VI.6. The∆0RamseyTheorem.............................. 236
2
VI.7. StrongerSetExistenceAxioms.........................239
VI.8. Conclusions...........................................240
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