A Mathematical Introduction to Logic, Second Edition

330 Pages·2001·2.011 MB·English

by  Herbert Enderton, Herbert B. Enderton

A Mathematical
Introduction to Logic
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A Mathematical
Introduction to Logic
Second Edition
Herbert B. Enderton
University of California, Los Angeles
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Contents
PREFACE ix
INTRODUCTION xi
CHAPTERZERO UsefulFactsaboutSets 1
CHAPTERONE SententialLogic 11
1.0 InformalRemarksonFormalLanguages 11
1.1 TheLanguageofSententialLogic 13
1.2 TruthAssignments 20
1.3 AParsingAlgorithm 29
1.4 InductionandRecursion 34
1.5 SententialConnectives 45
1.6 SwitchingCircuits 54
1.7 CompactnessandEffectiveness 59
CHAPTERTWO First-OrderLogic 67
2.0 PreliminaryRemarks 67
2.1 First-OrderLanguages 69
2.2 TruthandModels 80
2.3 AParsingAlgorithm 105
2.4 ADeductiveCalculus 109
2.5 SoundnessandCompletenessTheorems 131
2.6 ModelsofTheories 147
2.7 InterpretationsBetweenTheories 164
2.8 NonstandardAnalysis 173
CHAPTERTHREE Undecidability 182
3.0 NumberTheory 182
3.1 NaturalNumberswithSuccessor 187
3.2 OtherReductsofNumberTheory 193
3.3 ASubtheoryofNumberTheory 202
3.4 ArithmetizationofSyntax 224
vii
viii Contents
3.5 IncompletenessandUndecidability 234
3.6 RecursiveFunctions 247
3.7 SecondIncompletenessTheorem 266
3.8 RepresentingExponentiation 276
CHAPTERFOUR Second-OrderLogic 282
4.1 Second-OrderLanguages 282
4.2 SkolemFunctions 287
4.3 Many-SortedLogic 295
4.4 GeneralStructures 299
SUGGESTIONSFORFURTHERREADING 307
LISTOFSYMBOLS 309
INDEX 311
Preface
T
his book, like the first edition, presents the basic
conceptsandresultsoflogic:thetopicsareproofs,
truth,andcomputability.Asbefore,thepresentation
isdirectedtowardthereaderwithsomemathematicalback-
groundandinterests.Inthisrevisededition,inadditionto
numerous“local”changes,therearethree“global”waysin
whichthepresentationhasbeenchanged:
First, I have attempted to make the material more ac-
cessible to the typical undergraduate student. In the main
development,Ihavetriednottotakeforgrantedinforma-
tion or insights that might be unavailable to a junior-level
mathematicsstudent.
Second,fortheinstructorwhowantstofitthebooktohis
orhercourse,theorganizationhasbeenmademoreflexible.
Footnotesatthebeginningofmanyofthesectionsindicate
optionalpathstheinstructor—ortheindependentreader—
mightchoosetotake.
Third,theoreticalcomputersciencehasinfluencedlogic
inrecentyears,andsomeofthatinfluenceisreflectedinthis
edition. Issues of computability are taken more seriously.
Somematerialonfinitemodelshasbeenincorporatedinto
thetext.
The book is intended to serve as a textbook for an in-
troductorymathematicscourseinlogicatthejunior-senior
level.Theobjectivesaretopresenttheimportantconcepts
andtheoremsoflogicandtoexplaintheirsignificanceand
theirrelationshiptothereader’sothermathematicalwork.
Asatext,thebookcanbeusedincoursesanywherefrom
aquartertoayearinlength.Inonequarter,Igenerallyreach
thematerialonmodelsoffirst-ordertheories(Section2.6).
Theextratimeaffordedbyasemesterwouldpermitsome
glimpse of undecidability, as in Section 3.0. In a second
ix