Reverse mathematics: proofs from the inside out

199 Pages·2018·2.8 MB·English

by  John Stillwell

REVERSE
MATHEMATICS
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REVERSE
MATHEMATICS
proofs from the inside out
■■■■■
John Stillwell
princeton universit y press
princeton and oxford
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Title:Reversemathematics:proofsfromtheinsideout/JohnStillwell.
Description:Princeton:PrincetonUniversityPress,[]|Includes
bibliographicalreferencesandindex.
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Subjects:LCSH:Reversemathematics.|BISAC:MATHEMATICS/History&
Philosophy.|MATHEMATICS/General.|MATHEMATICS/Logic.|SCIENCE/History.
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To Elaine
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Contents
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Preface xi
 HistoricalIntroduction 
. EuclidandtheParallelAxiom 
. SphericalandNon-EuclideanGeometry 
. VectorGeometry 
. Hilbert’sAxioms 
. Well-orderingandtheAxiomofChoice 
. LogicandComputability 
 ClassicalArithmetization 
. FromNaturaltoRationalNumbers 
. FromRationalstoReals 
. CompletenessPropertiesofR 
. FunctionsandSets 
. ContinuousFunctions 
. ThePeanoAxioms 
. TheLanguageofPA 
. ArithmeticallyDefinableSets 
. LimitsofArithmetization 
 ClassicalAnalysis 
. Limits 
. AlgebraicPropertiesofLimits 
. ContinuityandIntermediateValues 
. TheBolzano-WeierstrassTheorem 
. TheHeine-BorelTheorem 
. TheExtremeValueTheorem 
. UniformContinuity 
. TheCantorSet 
. TreesinAnalysis 
 Computability 
. ComputabilityandChurch’sThesis 
viii ■ CONTENTS
. TheHaltingProblem 
. ComputablyEnumerableSets 
. ComputableSequencesinAnalysis 
. ComputableTreewithNoComputablePath 
. ComputabilityandIncompleteness 
. ComputabilityandAnalysis 
 ArithmetizationofComputation 
. FormalSystems 
. Smullyan’sElementaryFormalSystems 
. NotationsforPositiveIntegers 
. Turing’sAnalysisofComputation 
. OperationsonEFS-GeneratedSets 
. GeneratingΣSets 

. EFSforΣRelations 

. ArithmetizingElementaryFormalSystems 
. ArithmetizingComputableEnumeration 
. ArithmetizingComputableAnalysis 
 ArithmeticalComprehension 
. TheAxiomSystemACA 
. ΣandArithmeticalComprehension 

. CompletenessPropertiesinACA 
. ArithmetizationofTrees 
. TheKőnigInfinityLemma 
. RamseyTheory 
. SomeResultsfromLogic 
. PeanoArithmeticinACA 
 RecursiveComprehension 
. TheAxiomSystemRCA 
. RealNumbersandContinuousFunctions 
. TheIntermediateValueTheorem 
. TheCantorSetRevisited 
. FromHeine-BoreltoWeakKőnigLemma 
. FromWeakKőnigLemmatoHeine-Borel 
. UniformContinuity 
. FromWeakKőnigtoExtremeValue 
. TheoremsofWKL 
. WKL,ACA,andBeyond 
 ABiggerPicture 
. ConstructiveMathematics 
. PredicateLogic 
. VarietiesofIncompleteness 
CONTENTS ■ ix
. Computability 
. SetTheory 
. Conceptsof“Depth” 
Bibliography 
Index 