MathematicalLogic
OnNumbers,Sets,Structures,andSymmetry
RomanKossak
CityUniversityofNewYork
NewYork,NY,USA
SpringerGraduateTextsinPhilosophy
ISBN978-3-319-97297-8ISBN978-3-319-97298-5(eBook) https://doi.org/10.1007/978-3-319-97298-5
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Preface
Logicandsermonsneverconvince, Thedampofthenightdrivesdeeperintomysoul. (Onlywhatprovesitselftoeverymanandwomanisso, Onlywhatnobodydeniesisso.)
WaltWhitman,LeavesofGrass
In WhyIsTherePhilosophyofMathematicsatAll? [8],IanHackingwrites:
Yetalthoughmostmembersofourspecieshavesomecapacityforgeometricaland numericalconcepts,veryfewhumanbeingshavemuchcapacityfordoingoreven understandingmathematics.Thisisoftenheldtobetheconsequenceofbadeducation,but althougheducationcansurelyhelp,thereisnoevidencethatvastdisparityoftalent,oreven interestin,mathematics,isaresultofbadpedagogy...Aparadox:wearethemathematical animal.Afewofushavemadeastonishingmathematicaldiscoveries,afewmoreofus canunderstandthem.Mathematicalapplicationshaveturnedouttobeakeytounlockand disciplinenaturetoconformtosomeofourwishes.Butthesubjectrepelsmosthuman beings.
Itallringstruetoanyonewhohasevertaughtthesubject.WhenIteach undergraduates,Isometimessay,“Mathdoesnotmakeanysense,right?”towhich Ihearinunison“Right.”Myownschoolexperiencehasnotbeenmuchdifferent. EventhoughIwasconsidered“goodatmath,”itonlymeantthatIcouldfollow instructionsanddohomeworktoasatisfyingresult.OnlyoccasionallyI’dhave momentsIwasproudof.Inmyfirsthighschoolyear,Ihadamathematicsprofessor aboutwhomlegendsweretold.Everyoneknewthatinordertosurviveinhisclass, onehadto understand.Atanytimeyoucouldbecalledtotheblackboardtobe askedapenetratingquestion.Wewitnessedhumiliatingmomentswhensomeone’s ignorancewasruthlesslyrevealed.Once,whenthetopicwassquarerootsofreal numbers,Iwascalledandasked:
–Doeseverypositivenumberhaveasquareroot?
–Yes.
–Whatisthesquarerootoffour?
–Two.
–Whatisthesquarerootoftwo?
–Thesquarerootoftwo.
–Good.Whatisthesquarerootofthesquarerootoftwo.
–Thesquarerootofthesquarerootoftwo.
–Good.Sitdown.
Theprofessorsmiled.Helikedmyanswers,andIwashappywiththemtoo. ItseemedIhadunderstoodsomethingandthatthisunderstandingwassomehow complete.Thetruthwassimple,andonceyougraspedit,therewasnothingmore toit,justplainsimpletruth.Perhapssuchraresatisfyingmomentswerethereason Idecidedtostudymathematics.
InmyfreshmenyearattheUniversityofWarsaw,ItookAnalysis,Abstract Algebra,Topology,MathematicalLogic,andIntroductiontoComputerScience.I wascompletelyunpreparedforthelevelofabstractionofthesecourses.Ithought thatwewouldjustcontinuewiththekindofmathematicswestudiedinhigh school.Iexpectedmoreofthesamebutmorecomplicated.Perhapssomeadvanced formulasforsolvingalgebraicandtrigonometricequationsandmoreelaborate constructionsinplaneandthree-dimensionalgeometry.Instead,thecoursein analysisbeganwith defining realnumbers.Somethingwetookforgrantedandwe thoughtweknewwellnowneededadefinition!Myanswersattheblackboardthat Ihadbeensoproudofturnedouttoberathernaive.Thetruthwasnotsoplain andsimpleafterall.Andtomakemattersworse,thecoursefollowedwithaproof ofexistenceanduniquenessoftheexponentialfunction f(x) = e x ,andinthe process,wehadtolearnaboutcomplexnumbersandthefundamentaltheoremof algebra.Insteadofadvancedapplicationsofmathematicswehavealreadylearned, weweregoingback,askingmoreandmorefundamentalquestions.Itwasrather unexpected.Thealgebraandtopologycourseswereevenharder.Insteadofthe alreadyfamiliarplanes,spheres,cones,andcylinders,nowwewereexposedto generalalgebraicsystemsandtopologicalspaces.Insteadofconcreteobjectsone couldtrytovisualize,westudiedinfinitespaceswithsurprisinggeneralproperties expressedintermsofalgebraicsystemsthatwereassociatedwiththem.Ilikedthe prospectoflearningallthat,andinparticularthepromisethatintheend,after allthishigh-levelabstractstuffgotsortedout,therewouldbeareturntomore down-to-earthapplications.Butwhatwasevenmoreattractivewastheattemptto gettothebottomofthings,tounderstandcompletelywhatthiselaborateedificeof mathematicswasfoundedupon.Idecidedtospecializeinmathematicallogic.
Myfirstencounterswithmathematicallogicweretraumatic.Whilestillinhigh school,IstartedreadingAndrzejGrzegorczyk’s OutlineofMathematicalLogic, 1 atextbookwhosesubtitlepromised FundamentalResultsandNotionsExplained withAllDetails.Grzegorczykwasaprominentmathematicianwhomadeimportant contributionstothemathematicaltheoryofcomputabilityandlatermovedto philosophy.ThankstoGoogleBooks,wecannowseealistofthewordsandphrases mostfrequentlyusedinthebook:axiomschema,computablefunctions,concept, emptydomain,existentialquantifier,false,finitenumber,freevariable,andmany
1 AnEnglishtranslationis[17].
more.Ifoundthisallattractive,butIdidnotunderstandanyofit,anditwasnot Grzegorczyk’sfault.Intheintroduction,theauthorwrites:
RecentyearshaveseentheappearanceofmanyEnglish-languagehandbooksoflogicand numerousmonographsontopicaldiscoveriesinthefoundationsofmathematics.These publicationsonthefoundationsofmathematicsasawholeareratherdifficultforthe beginnersorreferthereadertootherhandbooksandvariouspiecemealcontributionsand alsosometimestolargelyconceived‘mathematicalfolklore’ofunpublishedresults.As distinctfromthese,thepresentbookisaseasyaspossiblesystematicexpositionofthe nowclassicalresultsinthefoundationsofmathematics.Hencethebookmaybeuseful especiallyforthosereaderswhowanttohavealltheproofscarriedoutinfullandallthe conceptsexplainedindetail.Inthissensethebookisself-contained.Thereader’sability toguessisnotassumed,andtheauthor’sambitionwastoreducetheuseofsuchwordsas evidentandobviousinproofstoaminimum.Thisiswhythebook,itisbelieved,maybe helpfulinteachingorlearningthefoundationofmathematicsinthosesituationsinwhich thestudentcannotrefertoaparallellectureonthesubject.
NowthatIknowwhatGrzegorczykistalkingabout,Itendtoagree.WhenIwas readingthebookthen,Ifounditalmostincomprehensible.Itisnotbadlywritten,it isjustthatthematerial,despiteitsdeceptivesimplicity,ishard.
Inmyfreshmenyear,AndrzejZarach,wholaterbecameadistinguishedset theorist,wasfinishinghisdoctoraldissertationandhadtheratherunusualideaof conductingaseminarforfreshmenonsettheoryandGödel’saxiomofconstructibility.Thisisanadvancedtopicthatrequiressolidunderstandingofformalmethods thatcannotbeexpectedfrombeginners.Zarachgaveusafewlecturesonaxiomatic settheory,andtheneachofuswasgivenanassignmentforaclasspresentation. MinewastheLöwenheim-Skolemtheorem.TheLöwenheim-Skolemtheoremis oneoftheearlyresultsinmodeltheory.ModeltheoryiswhatIlistnowasmy researchspecialty.Fortheseminar,myjobwastopresenttheproofasgiveninthe thenrecentlypublishedbook ConstructibleSetswithApplications [23],byanother prominentPolishlogicianAndrzejMostowski.Thetheoremisnotdifficulttoprove. Incoursesinmodeltheory,aproofisusuallygivenearly,asitdoesnotrequiremuch preparation.InMostowski’sbook,theprooftakesaboutone-thirdofthepage.Iwas readingitandreadingit,andthenreadingitagain,andIdidnotunderstand.Not onlydidInotunderstandtheideaoftheproof;asfarasIcanrecallnow,Ididnot understandasinglesentenceinit.Eventually,Imemorizedtheproofandreproduced itattheseminarinthewaythatclearlyexposedmyignorance.Itwasahumiliating experience.
Ibroughtupmyearlylearningexperienceshereforjustonereason:Ireally knowwhatitisnottounderstand.Iamfamiliarwithnotunderstanding.Atthe sametime,Iamalsofamiliarwiththoseextremelysatisfyingmomentswhenone doesfinallyunderstand.Itisaveryindividualandprivateprocess.Sometimes momentsofunderstandingcomewhensmallpieceseventuallyadduptothepoint whenonegraspsageneralidea.Sometimes,itworkstheotherwayaround.An understandingofageneralconceptcancomefirst,andthenitshedsbrightlighton anarrayofsmallerrelatedissues.Therearenosimplerecipesforunderstanding. Inmathematics,sometimestheonlygoodadviceisstudy,study,study...,but thisisnotwhatIrecommendforreadingthisbook.Thereareattractiveareasof
mathematicsanditsapplicationsthatcannotbefullyunderstoodwithoutsufficient technicalknowledge.Itishard,forexample,tounderstandmodernphysicswithout asolidgraspofmanyareasofmathematicalanalysis,topology,andalgebra.HereI willtrytodosomethingdifferent.Mygoalistotrytoexplainacertainapproachto thetheoryofmathematicalstructures.Thismaterialisalsotechnical,butitsnature isdifferent.Therearenoprerequisites,otherthansomegenuinecuriosityaboutthe subject.Nopriormathematicalexperienceisnecessary.Somewhatparadoxically,to followthelineofthought,itmaybehelpfultoforgetsomemathematicsonelearns inschool.Everythingwillbebuiltupfromscratch,butthisisnottosaythatthe subjectiseasy.
Muchofthematerialinthisbookwasdevelopedinconversationswithmywife, Wanda,andfriends,whoarenotmathematicians,butwerekindandcuriousenough tolistentomyexplanationsofwhatIdoforaliving.Ihopeitwillshedsomelight onsomeareasofmodernmathematics,butexplainingmathematicsisnottheonly goal.Iwanttopresentamethodologicalframeworkthatpotentiallycouldbeapplied outsidemathematics,theclosestareasIcanthinkofbeingarchitectureandvisual arts.Afterall,everythingisorhasastructure.
IamverygratefultoBethCaspar,AndrewMcInerney,PhilipOrding,Robert Tragesser,TonyWeaver,JimSchmerl,andJanZwickywhohavereadpreliminary versionsofthisbookandhaveprovidedinvaluableadviceandeditorialhelp.
AbouttheContent
Thecentraltopicofthisbookisfirst-orderlogic,thelogicalformalismthathas broughtmuchclarityintothestudyofclassicalmathematicalnumbersystemsand isessentialinthemodernaxiomaticapproachtomathematics.Therearemany booksthatconcentrateonthematerialleadingtoGödel’sfamousincompleteness theorems,andonresultsaboutdecidabilityandundecidabilityofformalsystems. Theapproachinthisbookisdifferent.Wewillseehowfirst-orderlogicserves asalanguageinwhichsalientfeaturesofclassicalmathematicalstructurescanbe describedandhowstructurescanbecategorizedwithrespecttotheircomplexity,or lackthereof,thatcanbemeasuredbythecomplexityoftheirfirst-orderdescriptions.
Allkindsofgeometric,combinatorial,andalgebraicobjectsarecalledstructures, butforustheword“structure”willhaveastrictmeaningdeterminedbyaformal definition.Part I ofthebookpresentsaframeworkinwhichsuchformaldefinitions canbegiven.Theexpositioninthispartiswrittenforthereaderforwhomthis materialisentirelynew.Allnecessarybackgroundisprovided,sometimesina repetitivefashion.
Theroleofexercisesistogivethereaderachancetorevisitthemainideas presentedineachchapter.Newlylearnedconceptsbecomemeaningfulonlyafter we“internalize”them.Onlythen,canonequestiontheirsoundness,lookfor alternatives,andthinkofexamplesandsituationswhentheycanbeapplied,and, sometimesmoreimportantly,whentheycannot.Internalizingtakestime,soonehas
tobepatient.Exercisesshouldhelp.Tothereaderwhohasnopriorpreparationin abstractmathematicsormathematicallogic,theexercisesmaylookintimidating, buttheyaredifferent,andmucheasier,thaninamathematicstextbook.Mostof themonlyrequirecheckingappropriatedefinitionsandfacts,andmostofthemhave pointersandhints.Theexercisesthataremarkedbyasterisksareformoreadvanced readers.
Allinstructorswillhavetheirownwayofintroducingthematerialcoveredin Part I.AselectionfromChapters 1 through 6 canbechosenforindividualreading, andexercisescanbeassignedbasedonhowadvancedthestudentsintheclassare. MysuggestionistonotskipChapter 2,wheretheideaof logicalseeing isfirst introduced.Thattermisoftenusedinthesecondpartofthebook.Iwouldalso recommendnottoskipthedevelopmentofaxiomaticsettheory,whichisdiscussed inChapter 6.Itisdonethererigorouslybutinalesstechnicalfashionthanone usuallyseesintextbooksonmathematicallogic.
HereisabriefoverviewofthechaptersinPart I.Allchaptersinbothpartshave moreextensiveintroductions:
•Chapter 1 beginswithadetaileddiscussionofaformalizationofthestatement “thereareinfinitelymanyprimenumbers,”followedbyanintroductionofthe fullsyntaxoffirst-orderlogicandAlfredTarski’sdefinitionoftruth.
•Chapter 2 introducesthemodel-theoreticconceptofsymmetry(automorphism) usingsimplefinitegraphsasexamples.Theideaof“logicalseeing”isdiscussed.
•ShortChapter 3 isdevotedtotheelusiveconceptofnaturalnumber.
•InChapter 4,buildinguponthestructureofthenaturalnumbers,adetailedformal reconstructionofthearithmeticstructuresoftheintegers(wholenumbers)and therationalnumbers(fractions)intermsoffirst-orderlogicisgiven.Thischapter isimportantforfurtherdevelopments.
•Chapter 5 providesmotivationforgroundingtherestofthediscussionin axiomaticsettheory.Itaddressesimportantquestions:Whatisarealnumber, andhowcanacontinuousreallinebemadeofpoints?
•Chapter 6 isashortintroductiontotheaxiomsofZermelo-Fraenkelsettheory.
Part II ismoreadvanced.Itsaimistogiveagentleintroductiontomodeltheory andtoexplainsomeclassicalandsomerecentresultsontheclassificationoffirstorderstructures.Afewdetailedproofsareincluded.Undoubtedly,thispartwillbe morechallengingforthereaderwhohasnopriorknowledgeofmathematicallogic; nevertheless,itiswrittenwithsuchareaderinmind.
•Chapter 7 formallyintroducesorderedpairs,Cartesianproducts,relations,and first-orderdefinability.Itconcludeswithanexampleofavarietyofstructureson aone-elementdomainandanimportantstructurewithatwo-elementdomain.
•Chapter 8 isdevotedtoadetaileddiscussionofdefinableelementsand,in particular,definabilityofnumbersinthefieldofrealnumbers.
•InChapter 9,typesandsymmetriesaredefinedforarbitrarystructures.The conceptsofminimalityandorder-minimalityareillustratedbyexamplesof orderingrelationsonsetsofnumbers.
•Chapter 10 introducestheconceptofgeometryofdefinablesetsmotivatedby theexampleofgeometryofconicsectionsintheorderedfieldofrealnumbers. ThechapterendswithadiscussionofDiophantineequationsandHilbert’s10th problem.
•InChapter 11,itisshownhowthefundamentalcompactnesstheoremisusedto constructelementaryextensionsofstructures.
•Chapter 12 isdevotedtoelementaryextensionsadmittingsymmetries.Aproof ofminimalityoftheorderedsetofnaturalnumbersisgiven.
•InChapter 13,formalargumentsaregiventoshowwhythefieldsofrealand complexnumbersareconsideredtameandwhythefieldofrationalnumbersis wild.
•Chapter 14 includesafurtherdiscussionoffirst-orderdefinabilityandabrief sketchofdefinabilityinhigher-orderlogics.Well-orderingsandtheMandelbrot setareusedasexamples.
•Chapter 15 isanextendedsummaryofPart II.Thereaderwhoisfamiliarwith first-orderlogicmaywanttoreadthischapterfirst.Thechapterisfollowedby suggestionsforfurtherreading.
InAppendix A,thereaderwillfindcompleteproofsofirrationalityofthesquare rootoftwo(Tennenbaum’sproof),Cantor’stheoremonnon-denumerabilityofthe setofallrealnumbers,first-ordercategoricityofstructureswithfinitedomains, existenceofproperelementaryextensionsofstructureswithinfinitedomains,anda “nonstandard”proofoftheInfiniteRamsey’stheoremforpartitionsofsetsofpairs. Appendix B containsabriefdiscussionofHilbert’sprogramforfoundationsof mathematics.
NewYork,USARomanKossak
PartILogic,Sets,andNumbers
1First-OrderLogic
2LogicalSeeing
2.1FiniteGraphs
3WhatIsaNumber?
3.1HowNaturalAretheNaturalNumbers?
3.1.1ArithmeticOperationsandtheDecimalSystem
3.1.2HowManyNumbersAreThere?
3.1.3Zero .........................................................38
3.1.4TheSetofNaturalNumbers
4SeeingtheNumberStructures
4.1WhatIstheStructureoftheNaturalNumbers?
4.1.1SetsandSetNotation
4.1.2LanguageofFormalArithmetic
4.1.3LinearlyOrderedSets
4.1.4TheOrderingoftheNaturalNumbers
4.2TheArithmeticStructureoftheNaturalNumbers
4.3TheArithmeticStructureoftheIntegers
4.3.1NaturalNumbersAreIntegers,Naturally
4.4Fractions!
4.5TheArithmeticStructureoftheRationals
4.5.1EquivalenceRelationsandtheRationals
4.5.2DefiningAdditionandMultiplicationofthe RationalNumbersFormally ................................54
4.5.3DenseOrderingoftheRationals
5Points,Lines,andtheStructureof R
5.1DensityofRationalNumbers
5.2WhatAreRealNumbers,Really?
5.3DedekindCuts
5.3.1DedekindCompleteOrderings
5.3.2Summary ....................................................64
5.4DangerousConsequences ............................................66
5.5InfiniteDecimals .....................................................68 6SetTheory
6.1WhattoAssumeAboutInfiniteSets?
PartIIRelations,Structures,Geometry
7Relations
7.1OrderedPairs
7.3WhatIsaRelationonaSet?
7.4Definability:NewRelationsfromOld
7.5HowManyDifferentStructuresAreThere?
7.5.1AVerySmallComplexStructure
8DefinableElementsandConstants .......................................97
8.2Databases,Oracles,andtheTheoryofaStructure
8.3DefiningRealNumbers ..............................................100
8.4DefinabilityWithandWithoutParameters
9MinimalandOrder-MinimalStructures ................................105
9.1Types,Symmetries,andMinimalStructures
9.2TrivialStructures .....................................................107
9.3TheOrderingoftheNaturalNumbers
9.4TheOrderingoftheIntegers .........................................109
9.5TheAdditiveStructureoftheIntegers
9.6TheOrderingoftheRationalNumbers
10GeometryofDefinableSets ................................................115 10.1BooleanCombinations ...............................................115
10.2HigherDimensions ...................................................117
10.2.1EuclideanSpaces ...........................................119
10.3ShadowsandComplexity ............................................121
10.3.1DiophantineEquationsandHilbert’s10thProblem
10.3.2TheRealsvs.TheRationals ................................127
11WhereDoStructuresComeFrom?
12ElementaryExtensionsandSymmetries
13Tamevs.Wild
14First-OrderProperties
14.2Well-OrderedSetsandSecond-OrderLogic
15SymmetriesandLogicalVisibilityOneMoreTime
First-OrderLogic
Howevertreacherousagroundmathematicallogic,strictly interpreted,maybeforanamateur,philosophyproperisa subject,ononehandsohopelesslyobscure,ontheotherso astonishinglyelementary,thatthereknowledgehardlycounts.If onlyaquestionbesufficientlyfundamental,theargumentsfor anyanswermustbecorrespondinglycrudeandsimple,andall menmaymeettodiscussitonmoreorlessequalterms.
G.H.Hardy
MathematicalProof [10]
Abstract Thisbookisaboutaformalapproachtomathematicalstructures.Formal methodsarebytheirverynatureformal.Whenstudyingmathematicallogic, initiallyoneoftenhastogritonesteethandabsorbcertainpreliminarydefinitionson faith.Conceptsaregivenprecisedefinitions,andtheirmeaningisrevealedlaterafter onehasachancetoseetheirutility.Wewilltrytofollowadifferentroute.Beforeall formalitiesareintroduced,inthischapter,wewilltakeadetourtoseeexamplesof mathematicalstatementsandsomeelementsofthelanguagethatisusedtoexpress them.
Keywords Arithmetic·Euclid’stheorem·Formalization·Vocabularyof first-order-logic·Booleanconnectives·Quantifiers·Truthvalues·Trivial structures
1.1WhatWeTalkAboutWhenWeTalkAboutNumbers
Thenaturalnumbersare0,1,2,3,....1 Anaturalnumberis prime ifitislarger than1andisnotequaltoaproductoftwosmallernaturalnumbers.Forexample, 11and13areprime,but15isnot,because15 = 3 · 5.Proposition20inBookIXof Euclid’s Elements states:“Primenumbersaremorethananyassignedmultitudeof
1 Accordingtosomeconventions,zeroisnotanaturalnumber.Forreasonsthatwillbeexplained later,wewillcountzeroamongthenaturalnumbers.
©SpringerInternationalPublishingAG,partofSpringerNature2018 R.Kossak, MathematicalLogic,SpringerGraduateTextsinPhilosophy3, https://doi.org/10.1007/978-3-319-97298-5_1
primenumbers.”Inotherwords,thereareinfinitelymanyprimenumbers.Thisisthe celebratedEuclid’stheorem.Whatisthistheoremabout?Inthebroadestsense,itis astatementabouttheworldinwhichsomeobjectsareidentifiedasnaturalnumbers, aboutaparticularpropertyofthosenumbers—primeness,andaboutinexhaustibility ofthenumberswiththatproperty.Weunderstandwhatthetheoremsays,because weunderstanditscontext.Weknowwhatnaturalnumbersare,andwhatitmeans thatthereareinfinitelymanyofthem.However,noneofitisentirelyobvious,and wewilltakeacloserlookatbothissueslater.Concerningtheinfinitudeofprimes, itoccurredtomeoncewhenIwasabouttoshowtheproofofEuclid’stheoremin myclass,toaskstudentswhattheythoughtaboutasimplertheorem:“Thereare infinitelymanynaturalnumbers.”Itwasnotafairquestion,asitimmediatelytakes usawayfromthesolidgroundofmathematicsintothemurkywatersofphilosophy. Thestudentswerebemused,andIwasnotsurprised.
WewillformalizeEuclid’stheoreminaparticularway,andtodothiswewill havetosignificantlynarrowdownitscontext.Inaradicalapproach,thecontextwill bereducedtoabareminimum.Wewillbetalkingaboutcertaindomainsofobjects, andinthecaseofEuclid’stheoremthedomainisthesetofallnaturalnumbers.Once thedomainofdiscourseisspecified,weneedtodecidewhatfeaturesofitselements wewanttoconsider.Inschoolwefirstlearnhowtoaddandhowtomultiply naturalnumbers;andwewillfollowthatpath.WewillexpressEuclid’stheorem asastatementaboutadditionandmultiplicationinthedomainofnaturalnumbers.
Wewilltalkaboutadditionandmultiplicationusingexpressions,called formulas, inaveryrestrictedvocabulary.Wewilluse variables,twooperationsymbols: + and ·,andthesymbol = forequality.Thevariableswillbelowercaseletters x , y , z,....Forexample, x + y = z isaformulaexpressingthattheresultofaddinga number x toanumber y issomenumber z.Thisexpressionbyitselfcarriesno truth value.Itcanbeneithertruenorfalse,sincewedonotassignanyspecificvalues tothevariables.Laterwewillseewaysinwhichwecanspeakaboutindividual elementsofadomain,butfornowwewillonlyhavetheoptionof quantifying over theelementsofthedomain,andthatmeansstatingthateithersomethingholdsfor allelements,orthatsomethingholdsforsome.Forexample:
(1.1)
Thesentenceaboveexpressesthattheresultdoesnotdependontheorderinwhich thenumbersareadded.Itisanexampleofa universal statement;itdeclaresthat somethingholdsforallelementsinthedomain.
Andhereisanexampleofan existential statement,itdeclaresthatobjectswitha certainpropertyexistinthedomain:
x suchthat x + x = x.
(1.2)
Thisstatementisalsotrue.Thereisanelementinthedomainofnaturalnumbers thathastherequiredproperty.Inthiscasethereisonlyonesuchelement,zero.But ingeneral,therecanbemoreelementsthatwitnesstruthofanexistentialstatement.
Forall x andall y,x + y = y + x.
Thereisan
Forexample,
Thereisan x suchthat x x = x
isatrueexistentialstatementaboutthenaturalnumbers,andtherearetwowitnesses toitsveracity,zero,andone.
Interestingstatementsaboutnumbersofteninvolvecomparisonsoftheirsizes.To expresssuchstatements,wecanenlargeourvocabularybyaddingarelationsymbol, forexample <,andinterpretexpressionsoftheform x<y as“somenumber x is lessthansomenumber y .”Hereisanexampleofatruestatementaboutnatural numbersinthisextendedlanguage.
Forall x,y, and z, if x<y, then x + z<y + z. (1.3)
Noticethegrammaticalform“if...then....”
Thenextexampleisaboutmultiplication.Itisanexpressionwithouta truthvalue.
1 <x andforall y and z, if x = y z, then x = y or x = z. (1.4)
Instatements(1.1),(1.2),and(1.3),allvariableswere quantified byaprefix,either “forall”or“thereexists.”In(1.4)thevariable x isnotquantified,itisleft free;it doesnotassumeanyspecificvalue.
Becauseofthepresenceofafreevariable,(1.4)doesnothaveatruthvalue, neverthelessitservesapurpose.It defines thepropertyofbeingaprimenumberin termsofmultiplicationandtherelation <.Letmeexplainhowitworks.
Thinkofaprimenumber,say7,asavalueof x .IfItellyouthat7 = y z,for somenaturalnumbers y and z,withouttellingyouwhatthesenumbersare,then youknowthatoneofthemmustbe7andtheotheris1,becauseonecannotbreak downsevenintoaproductofsmallernumbers.Itistrue“forall y and z,”because forallbutacoupleofthemitisnottruethat7 = y · x ,andinsuchcasesitdoesnot matterwhattherestoftheformulasays.Weonlyconsiderthe“then”partifindeed 7 = y · z.Ifthevalueof x isnotprime,say6,then6 = 2 · 3,sowhenyouthinkof y as2and z as3,itistruethat6 = y · z,butneither y nor z isequalto6,hencethe propertydescribedin(1.4)doesnothold“forall y and z.”
Ifyouarefamiliarwithformallogic,Iamexplainingtoomuch,butifyouarenot, itisworthwhiletomakesurethatyouseehowtheformula(1.4)definesprimeness. Chosesomeothercandidatesfor x andseehowitworks.Also,noticethreenew additionstothevocabulary:thesymbol1forthenumberone;andtwoconnectives “and”and“or.”
Withtheaidof(1.4)wecannowwritethefullstatementofEuclid’stheorem: Forall w ,thereisan x suchthat w<x ,andforall y and z,if x = y · z,then x = y or x = z.
Whatisthedifferencebetweenthestatementaboveandtheoriginal“Thereare infinitelymanyprimenumbers.”?Firstofall,thenewformulationincludesthe
definitionofprimenessinthestatement.Secondly,whatismoreimportant,the directreferencetoinfinityiseliminated.Instead,wejustsaythatforeverynumber w thereisaprimenumbergreaterthanitwithsuchandsuchproperties,soitfollows thatsincethereareinfinitelymanynaturalnumbers,theremustbeinfinitelymany primenumbersaswell.Themostimportanthoweveristhatwemanagedtoexpress animportantfactaboutnumberswithmodestmeans,justvariables,thesymbols · and <,theprefixes“forevery”and“thereis,”andtheconnectives:“and,”“or”,and “if...then....”
Wehavemadethefirststeptowardsformalizingmathematics,andwedid thisinformally.Thepointwastowriteastatementrepresentingameaningful mathematicalfactinalanguagethatisasunambiguousaspossible.Wesucceeded, byreducingthevocabularytoafewbasicelements.Thiswillguideusinoursecond step,inwhichwewillformallydefineacertainformallanguageanditsgrammar. Wewillcarefullyspecifythewayinwhichexpressionsinthislanguagecanbe formed.Someofthoseexpressionswillbestatementsthatcanbeassignedtruth values—trueorfalse—wheninterpretedinparticularstructures.Theevaluationof thosetruthvalueswillalsobepreciselydefined.Someotherexpressions,those thatcontainfreevariables,willserveasdefinitionsofpropertiesofelementsin structures,andwillplayanimportantrole.Allthoseproperlyformedexpressions willbereferredtoas formulas.Mydictionaryexplainsthataformulais“a mathematicalrelationshiporruleexpressedinsymbols.”Themeaninginthisbook isdifferent.Wewilltalkaboutrelationships,andwewillusesymbols,butformulas willalwaysrepresentstatements.Forexample,theexpression b · b 4a · c isa computationalrulewritteninsymbols,butitisnotaformulainoursense,sinceit isnotastatementaboutthenumbers a , b ,and c .Incontrast, d = b · b 4a · c isa formula.Itstatesthatifwemultiply b byitselfandsubtractfromittheproductof fourtimes a times c ,theresultis d .
1.1.1HowtoChooseaVocabulary?
Intheprevioussection,weformulatedanimportantfactaboutnumbers—Euclid’s theorem—usingsymbolsformultiplicationaandtheordering(<).Thisisjustone example,buthowdoesitworkingeneral?Whatpropertiesofnumbersdowewant totalkabout?Whatbasicoperationsorrelationscanwechoose?Theanswersare verymuchdrivenbyapplicationsandparticularneedsandtrendsinmathematics.In thecaseofnumbertheory,thedisciplinethatdealswithfundamentalpropertiesof naturalnumbers,itturnsoutthatalmostanyimportantresultcanbeformulatedina formallanguageinwhichonerefersonlytoadditionandmultiplication.2 Number
2 Inourexamplewealsousedtheorderingrelation <,butinthedomainofthenaturalnumbers, therelation x<y canbedefinedintermsofaddition,sinceforallnaturalnumbers x and y , x is lessthan y ifandonlyifthereisanaturalnumber z suchthat z isnot0and x + z = y
theorymaybethemostdifficultandmysteriousbranchofmathematics.Proofs ofmanycentralresultsareimmenselycomplex,andtheyoftenusemathematical machinerythatreacheswellbeyondthenaturalnumbers.Still,abitsurprisingly, aformalsystemwithafewsymbolsinitsvocabularysufficestoexpressalmost alltheoremsofnumbertheory.Itissimilarinotherbranchesofmathematics.The mathematicalstructures,andthefactsaboutthemarecomplex,butthevocabulary andthegrammaroftheformalsystemthatwewilldiscussinthisbookaremuch simpler.
The realnumbers willbedefinedpreciselylater.Forthemoment,youcan thinkofthemasallnumbersrepresentinggeometricdistancesandtheirnegative opposites.Thefollowingstatementiswritteninarigorous,butinformallanguage ofmathematics.Itinvolvestheconceptofone-to-onecorrespondence.Aone-to-one correspondencebetweentwosets A and B isamatchingthattoeveryelementof A assignsexactlyonelementof B insuchawaythateveryelementof B hasamatch.
Let A beaninfinitesetofrealnumbers.Theneitherthereisaone-to-onecorrespondence between A andthesetofallnaturalnumbers,orthereisaone-to-onecorrespondence A andthesetofallrealnumbers.
Thisisavariantofwhatisknownasthe ContinuumHypothesis.TheContinuum Hypothesiscanalsobestatedintermsofsizesofinfinitesets.Inthe1870s,Georg Cantorfoundawaytomeasuresizesofinfinitesetsbyassigningtothemcertain infiniteobjects,whichhecalled cardinalnumbers.Thesmallestinfinitecardinal numberis ℵ0 anditisthesizeofthesetofallnaturalnumbers.ItwasCantor’s greatdiscoverythatthesizeofthesetofallrealnumbers,denotedby c,islarger that ℵ0 .AnotherwaytostatetheContinuumHypothesisis:if A isaninfiniteset ofrealnumbers,thenthecardinalityof A iseither ℵ0 or c.Thehypothesiswas proposedbyGeorgCantorinthe1870s,andDavidHilbertputitprominentlyat thetopofhislistofopenproblemsinmathematicspresentedtotheInternational CongressofMathematiciansinParisin1900.TheContinuumHypothesisisabout numbers,butitisnotaboutarithmetic.Itisaboutinfinitesets,andaboutoneto-onecorrespondencesbetweenthem.Whatarethoseobjects,andhowcanwe knowanythingaboutthem?Whatisanappropriatelanguageinwhichfactsabout infiniteobjectscanbeexpressed?Whatprinciplescanbeusedinproofs?Precisely suchquestionsledDavidHilberttotheideaofformalizingandaxiomatizing mathematics.ThereisashorthistoricalnoteaboutHilbert’sprogramforfoundations ofmathematicsinAppendix B.
TheContinuumHypothesisisastatementaboutsetsorrealnumbersandtheir correspondences.Toexpressitformallyoneneedstoconsideralargedomain inwhichallrealnumbers,theirsets,andmatchingsbetweenthemareelements. Remarkably,itturnedoutthatthevocabularyofaformalsysteminwhichonecan talkaboutallthosedifferentelements,andmuchmore,canbereducedtological symbolsofthekindweusedforthedomainofthenaturalnumbers,andjustone symbolforthesetmembershiprelation ∈.Allthatwillbediscussedindetailin Chap. 6.
Sofornowwejusthavetwoexamplesofvocabularies,onewithsymbols+and forarithmetic,andtheotherwithjustonesymbol ∈ forsettheory.Wewillsee moreexampleslater,andourfocuswillbeonthenumberstructures.Ingeneral,for everymathematicalstructure,andforeverycollectionofmathematicalstructuresof aparticularkind,thereisachoiceofsymbolsthatissometimesnaturalandobvious, andsometimesarrivedatwithagreateffort.
Adigression:Onceformalized,mathematicalproofsbecomestringsofsymbols thataremanipulatedaccordingtowell-definedsyntacticrules.Inthisform,they themselvesbecomesubjectsofmathematicalinquiry.Onecanaskwhethersuch andsuchformalstatementcanbederivedformallyfromagivensetofpremises.The wholedisciplineknownasprooftheorydealswithsuchquestionswithremarkable successes.In1940,KurtGödelprovedthattheContinuumHypothesiscannotbe disprovedonthebasisofour,suitablyformalizedandcommonlyacceptedaxioms ofsettheory,andin1963,PaulCohenprovedthatitcannotbeprovedfrom thoseaxiomseither.Thisisallremarkable,andwasaresultofagreateffortin foundationalstudies.
1.2SymbolicLogic
Mathematicallogicissometimescalled symboliclogic,sinceinlogicalformulas ordinaryexpressionsarereplacedwithformalsymbols.Wewillintroducethose symbolsinthenextsection.HenriPoincaré,thegreatFrenchmathematician,who wasstronglyopposedtoformalmethodsinmathematics,wrotein1908:“Itis difficulttoadmitthattheword if acquires,whenwritten ⊃,avirtueitdidnot possesswhenwritten if.”3 Poincaréwasright.Nothingisgainedconceptuallyby justreplacingwordswithsymbols,buttheintroductionofsymbolsisjustafirststep. Themoreimportantfeatureisaprecisedefinitionofthegrammaroftheformalized language.Wearegoingtopaycloseattentiontotheshapeoflogicalformulas,and thelogicalsymbolswillhelp.Itisverymuchasinalgebra: x + y ,simplymeans x plus y ,and x · y means x times y ,butifyouthoughtthatourformalizedexpression forEuclid’stheoremwascomplicated,thinkhowcomplicateditwouldhavebeenif wedidnotuse + and · .
Therearemanyadvantagesofthesymbolicnotation.Itispreciseandconcise. Onenotonlysavesspacebyusingsymbols;sometimessymbolicnotationallows onetoexpresscomplexstatementsthatwouldbehardtounderstandinthenatural language.Themostcommonsymbolicsystemofmathematicsiscalledfirst-order logic.Itwillbedefinedinthissectionanditwillbeextensivelyusedintherestof thebook.
Themathematicalnotationwithallitssymbolsandabbreviationsisthelanguage ofmodernmathematicsthathastobelearnedasanyotherlanguage,andlearninga
3 InPoincaré’stime, ⊃ wasusedtodenoteimplication.
languagetakestime.Iwilltrytolimitnotationtoaminimum,butyouhavetobear withme.Thelanguagethatyouwilllearnservescommunicationamongmathematicianswell,butthefactthatsomuchofmathematicsrequiresitcreatesproblemsin writingaboutmathematicsfornon-mathematicians.Moreover,mathematicianshave createdtheirlanguageinaratherchaotichistoricalprocesswithoutparticularregard totheneedsofbeginners.AlexandreBorovikwrote[6]:“Whyarewesosurethatthe alphabet ofmathematics,asweteachit—allthatcorpusofterminology,notation, symbolism—isnatural?...Wehavenothingtocompareourmathematicallanguage with.Howdoweknowthatitisoptimal?”
Intheprevioussectionwesawhowmathematicalstatementsarewrittenusing logicalconnectivesandquantifiers.Nowwewillbewritingthemusing logical symbols.Thoseare ∧, ∨,and ¬,representing and, or,and not,respectively.The connectiveswillbeusedtogrouptogetherstatementsaboutrelations,andthose statementswillbecomposedofvariablesand relationsymbols.Theprefixesofthe form“forall x ...”and“thereisan x suchthat...”arethe quantifiers,thefirstis called universal anditwillbewritten ∀x... ;thesecondiscalled existential,andit willbewritten ∃x... (thinkof ∀ll,and ∃xist).
Therearetwowaysofintroducingrelationsymbols.Onecouldfirstdefinean infinitecollectionofsymbols,andthenforeachstructurechooseonlyparticular symbolsspecifictothestructure.Thiswouldgiveusa“onelanguage—allstructures”model.Alternatively,onecanfirstmakeachoiceofsymbolsforaparticular structure,oraclassofstructures,anduseonlythose.Thelatter“onekindof structures—onelanguage”modeldoesnotneedsomeofthesmalltechnicalities thattheformerrequires,sowewilladoptit.Sinceatfirstwewanttodiscuss numberstructures,wechoosethefollowingthreerelationsymbols: A foraddition, M formultiplication,and L forthe“lessthan”relation.Bythestandardconvention, regardlessofthechoiceofotherrelationsymbols,theequalityrelationsymbol=is alsoalwaysincludedinthevocabulary.
Foranimportanttechnicalreason,wewillneedinfinitelymanyvariables.We willindexthembynaturalnumbers: x0 , x1 , x2 , x3 ,andsoon.Eachformulawill onlyusefinitelymanyvariables,butthereisnolimitonthenumberofvariables thatcanbeused.Thisisanimportantfeatureoffirst-orderlogicsowehavetokeep allthoseinfinitelymanyvariablesinmind,andfromtimetotimetherewillbea needtorefertoallofthem.Tosimplifynotation,wewilloftendropthesubscripts, andwewilluseotherlettersaswell.
Weareusedtothinkingofadditionandmultiplicationasfunctions,oroperations onnumbers.NowIwillaskyoutothinkofthemasrelations.
Infullgenerality,thelanguageoffirst-orderlogicincludesrelationsymbols andfunctionsymbols,buttoavoidsometechnicalitieswewillnotusefunction symbols.Theword“technicalities”isoneofthosetreacherousexpressionsthat oftenhidessomeimportantissuesthattheauthoristryingtosweepundertherug,so letmeofferanexplanation.Inmathematicsonestudiesbothfunctionsandrelations. Weusemathematicalfunctionstomodelprocessesandoperations.Metaphorically speaking,afunction“takes”anobjectasinputand“produces”anotherobjectasan output.Additionisatwoargumentfunction,theinputisapairofnumbers,say1
and3,andtheoutputistheirsum4.Functionsareusefulwhenchangeisinvolved; when,forexample,somequantitychangesasafunctionoftime.Relationsaremore likedatabases—theyrecordrelationships.Bothconceptshavetheirformalizations, andinmathematicalpracticethedistinctionbetweenthemisnotsharp.Arelation canevolveintime;afunctioncanbeconsideredasarelation,relatinginputsto outputs.Thetechnicalitieshintedatabovearetherulesthatmustbeobeyedwhen wecomposefunctions,i.e.whenweapplyonefunctionafteranotherinaspecified order.Thoserulesarenotcomplicated,butatthislevelofexpositiontheywould requireamorecarefultreatment.Wewillnotdothat,and,sinceeveryfunctioncan berepresentedasarelation,thepricethatwillbepaidwillnotbegreat.
Letusseehowadditionandmultiplicationcanberepresentedasrelations.As wenotedearlier,additionofnaturalnumbersisafunction.Toeachpairofnatural numbers,thefunctionassignsavaluethatistheirsum.Theinputsarepairsof numbers m, n,andtheoutputsarethesums m + n.Letusnamethisfunction f . Usingfunctionnotation,wecanwrite2 + 2 = 4as f(2, 2) = 4,and100 + 0 = 100 as f(100, 0) = 100.Wearenotconcernedherewithanyactualprocessofadding numbers,wethinkof f asadevicethatinstantlyprovidesacorrectanswerineach case.Butadditionalsodeterminesarelationshipbetweenorderedtriplesofnumbers asfollows.Wecansaythatthenumbers k , m,and n arerelatedifandonlyif4 f(k,m) = n,or,inotherwords, k + m = n.Inthissense,thenumbers2,3,and5 arerelated,andsoare100,0,and100,but0,0,and1arenot.Noticethatwemust becarefulabouttheorderinwhichwelistthenumbers.Forexample,2,2,and4 arerelated,but2,4,and2arenot.Additionasarelationcarriesexactlythesame informationasthefunction f does.
Togofurther,wemustnowdefinetherulesthatgenerateallformulasoffirstorderlogic.Aformulaisaformalexpressionthatcanbegenerated(constructed)in aprocessthatstartswithbasicformulas,accordingtopreciserules.Thedefinition itselfisanexampleofaformalmathematicaldefinition.Itisan inductivedefinition. Inaninductivedefinition,onefirstdefinesabasiccollectionofobjects,andthen describestherulesbywhichnewobjectscanbeconstructedfromthoseobjectswe alreadyhaveconstructed.Thedefinitionalsodeclaresthatonlyobjectsobtainedthis wayqualify.
Hereisanexampleofasimpleinductivedefinition.Everyoneknowswhata finitesequenceof0’sand1’sis.Itisenoughtoseeanexampleortwo.Hereis one:100011101.Hereisanother:1111111.Werecognizesuchsequenceswhenwe seethem,butnoticethatthisrestsonanintuitiveunderstandingoftheconceptof finitesequence.Theinductivedefinitionwillnotmakeanyexplicitreferencesto finiteness,insteadthefinitecharacteroftheconceptwillbebuiltintothedefinition. Thisaspectisnotjustaphilosophicalnicety,ithaspracticalconsequences.We useinductivedefinitioninaspecificwayproveresultsaboutthedefinedconcepts. Theadvantageofinductivedefinitionsisthattheygiveaninsightintotheinternal
4 Thephrase“ifandonlyif”iscommonlyusedinmathematicstoconnecttwoequivalent statements.
1.2SymbolicLogic11 structureoftheobjectstheydefine.Theyshowushowtheyaremadeinastep-bystepprocess.
Letusnowdefinesequencesof0’sand1’sinductively.Webeginbysayingthat 0and1arefinitesequences.5 Theseareourbasicobjects.Thencomestheinductive rule:if s isafinitesequencethensoare s 0and s 1.Finally,wedeclarethatthefinite sequencesof0’sand1’sareonlythoseobjectsthatareobtainedfromthebasic sequences0and1byapplyingtheinductiverule(overandoveragain).
Nowwegobacktoformulas.Recallthatwechose A, M ,and L forrelation symbols. A and M are ternary—theybindthreevariables,and L is binary—itbinds twovariables.“Binding”isatechnicalterm,andyoushouldnotputmoremeaning toitbeyondwhatiswritteninthefollowingdefinitionofbasicformulas,which wewillcall atomic.Atomicformulasareallexpressionsoftheform A(xi ,xj ,xk ), M(xi ,xj ,xk ), L(xi ,xj ),and xi = xj ,where i , j ,and k arearbitrarynatural numbers.Forexample, A(x0 ,x1 ,x2 ), M(x5 ,x3 ,x1 ), M(x1 ,x0 ,x0 ), L(x1 ,x2 ) are atomicformulas.Inotherwords,anatomicformulaisarelationsymbolfollowedby alistofthreearbitraryvariables,inthecaseof A and M ,ortwoarbitraryvariables, inthecaseof L.Noticethatsincethereareinfinitelymanyvariables,therearealso infinitelymanyatomicformulas.
Whenwediscussparticularexamplesofformulas,forgreaterreadabilitywewill usuallydropthesubscriptsanduseotherlettersforvariables,soexpressionssuchas A(x,y,z) or L(y,x) (notatypo)willalsobeconsideredatomicformulas,although inthestrictsense,accordingtothedefinitiontheyarenot.
Alongwithformalexpressions,definedaccordingtostrictrules,wewillalso useothercommonmathematicalexpressionsandthosewillalsooftenusesymbols. Thetwodifferentkindsofsymbolsshouldnotbeconfused.Wewillneednames formanydifferentobjects,includingformulasandsentencesoffirst-orderlogic. Wewillseethisinthedefinitionbelow.TheGreekcharacters ϕ (phi), ψ (psi),and other,areusedasnamesforformulasofthelanguagewedefine.Theyarenotapart oftheformalism.
Definition1.1
1.Everyatomicformulaisaformula.
2.If ϕ and ψ areformulas,thensoare (ϕ) ∧ (ψ), (ϕ) ∨ (ψ),and ¬(ϕ).
3.If ϕ isaformula,then,foreach n, ∃xn (ϕ) and ∀xn (ϕ),areformulas.
4.Therearenootherformulas.
Noticetheuseofparentheses.Theyplayanimportantrole.Theyguaranteethat everyfirst-orderformulacanbereadinonlyoneway.6
5 Wecouldactuallystartonelevellower.Wecouldsaythattheemptysequence,withnosymbols atall,isafinitesequenceof0’sand1’s.
6 Thisuniquereadabilityoffirst-orderformulasisnotanobviousfactandrequiresaproof,which isnotdifficult,butwewillnotpresentithere.
LetusseehowDefinition 1.1 works.Perclause(1.1),theexpressions x = y and A(x,z,y) (notyposhere)areformulas,becausetheyareatomicformulas.7 Per rule(1.2), ¬(x = y) and ∃z(A(x,y,z)) areformulas.Byapplying(1.2),weseethat (¬(x = y)) ∧ (∃z(A(x,y,z)))
isaformulaaswell.Letuscallthisformula ϕ(x,y) 8 Theonlydisplayedvariables are x and y ,becausetheyarefreein ϕ(x,y).Thethirdvariable z isboundbythe existentialquantifier ∃z.Thinkof A astheadditionrelationofthenaturalnumbers. Forwhatvaluesof x and y does ϕ(x,y) becomeatruestatements.Firstofallthey mustbedifferent,asdeclaredbythefirstcomponentof ϕ(x,y),butalso x mustbe lessthan y ,becauseonlythenthereisanaturalnumber z suchthat x + z = y Animportantcaveat.Accordingtotherules,wearefreetochooseanyvariables weliketoformatomicformulas,soforexample A(x,z,z),and L(x,x) arewellformedformulas.Ifthesamefreevariableisusedindifferentplacesinaformula, whenweinterprettheformulainastructure,thatvariablewillalwaysbeevaluated bythesameelement,butthisdoesnotmeanthatifthevariablesaredifferentthat theyrepresentdifferentobjects.Wearefreetoevaluateanyfreevariablebyany object,inparticularwecanusethesameobjectfordifferentvariables.
Letusrecapitulate.Thelistofsymbolsofthefirst-orderlogicis: ∧, ∨, ¬ , ∃, ∀, and =,andthenforeachparticularstructure,inaddition,itincludesacollectionof relationsymbols.Inourcase,wechose A, M ,and L.Eachrelationsymbolhasa prescribed arity whichisgiveninthedefinitionoftheatomicformulas.Thesymbols A and M areofaritythree,and L isofaritytwo.Thismeansthat,forexample, A(x0 ,x1 ,x2 ) isawell-formedatomicformula,but A(x0 ,x1 ) and L(x0 ,x1 ,x2 ) are not,becausethenumberofvariablesdoesnotmatchthearityofthesymbol.
Theattentivereaderwillask:Butwhataboutallthoseparenthesesandcommas? Yes,theyarealsoformalsymbolsofourlanguage,andtheiruseisentirely determinedbyDefinition 1.1.Therulesforcommasarehiddeninclause(1.1). Onecoulddowithoutthem.Forexample Ax0 x1 x2 alsorepresentsauniquely recognizablestringofsymbols,andthisisallwewant,butforgreaterreadability, andtoavoidadditionalconventionsthatwewouldhavetointroduceintheabsence ofparentheses,weuseparenthesesandcommas.Oftenwewillalsouse“[,”and“],” andsometimes“{”and“}.”Theyallhavethesamestatusas“(,”and“).”
7 Thisisanexampleofmathematicalpedantry.Ofcourse,youwouldsay,theyareformulas.They areevenatomicformulas!Butwhenwedefinedatomicformulas,wedefinedaspecialkindof expression,andcalledexpressionsofthiskind“atomicformulas.”Whenwedidthat,theformal conceptofformulahadnotbeendefinedyet.Toknowwhataformulaoffirst-orderlogicisonehas towaitforaformaldefinitionofthekindwegavehere.Toavoidthiswholediscussionwecould havecalledatomicformulasatoms.Ifwedidthat,thenclause(1.1)ofthedefinitionabovewould say“Everyatomisaformula,”butsincetheterm“atomicformula”iscommonlyused,wedidnot havethatchoice.
8 Thisisanotherexampleofaninformalabbreviation.
Eachapplicationofrules(1.2)and(1.3)introducesanewlayerofparentheses. Ifwecontinuethisway,formulasquicklybecomeunreadable,butthisformalism isnotdesignedforthehumaneye.Wesacrificeeasyreading,butwegainmuchin return.Onebonusisthatitisnoweasytocheckandcorrectgrammar.Theonly grammaticalrulesarethoseinDefinition 1.1.Inparticular,ineveryformulathe numberofleftparenthesesmustbeequaltothenumberofrightparentheses.Ifitis not,thesequenceofsymbolsisnotproperlyformedanditisnotaformula.
ThemostimportantaspectofDefinition 1.1 isthatitshowshowallformulasare generatedinastep-by-stepprocessinwhichmoreandmorecomplexformulasare generated.Thisisacrucialfeature,thatopensthedoortoinvestigationsofformal languagesbymathematicalmeans.Anotheressentialfeatureisthatthesetofall formulasisgeneratedwithoutanyregardtowhatthoseformulasmayexpress.In fact,mostformulasdonotexpressanythinginterestingatall.Forexample
((x = x) ∨ (x = x) ∧ (x = x))
isaproper,butuninterestingformula,andsois ∃xL(y,z)
Whatisthepointofallowingmeaninglessformulas?Whatweareafterare formulasandsentencesthatexpresssalientpropertiesofstructuresandtheir elements,butwewouldbeatalosstryingtogiveamathematicaldefinitionof ameaningfulformula.Itismucheasiertoacceptthemall,whatevertheymay beexpressing.Thereissomethingprofoundintreatingallformulasthisway. Meaningfulnessisavagueconcept.Asentenceofnointeresttoday,mayturnoutto bemostimportanttomorrow,soitwouldmakenosensetoeliminateanyofthemin advance,butthisisnotthemainpoint.Mostmechanicallyformedformulasarenot onlyuninteresting,theyactuallymakenosenseatall.Stillwewanttokeepthem in,becauseitisthepricetopayfortheclarityofthedefinition.Moreover,thereare alsosomeunexpectedtechnicalapplications.If ϕ isaformula,then,accordingto rule(1.2),soare (ϕ) ∧ (ϕ) and (ϕ) ∧ ((ϕ) ∧ (ϕ)),and (ϕ) ∧ ((ϕ) ∧ ((ϕ) ∧ (ϕ))),andso on.Nothingnewisexpressed,buttherearesomeimportantresultsinmathematical logicthatdependinanessentialwayonexistenceofsuchstatements.
Thedefinitionofthesyntaxoffirst-orderlogiciscompleted.Nowitistime todefinethe semantics,i.e.theprocedurethatgivesmeaningandtruthvaluesto formulaswheninterpretedinastructure.Wealreadydidthatinformally,when wetalkedaboutEuclid’stheoremandinterpretationsofformulasinthenatural numbers.Fulldefinitionofsemanticsforfirst-orderlogicisbasedonAlfredTarski’s famous definitionoftruth from1933[34].Itisformulatedinaset-theoreticsetting thatwewilldiscusslater.Fornow,wewillshowhowitallworksusingexamples. Forafullformaldefinitionconsultanytextbookonmathematicallogic.Agood sourceonlineistheStanfordEncyclopediaofPhilosophy[13].
Wewillinterpretformulasinthedomainofthenaturalnumbers.Tobeginwith, foranythreenumbers m, n,and k ,weneedtoassigntruthvalues(trueorfalse)to allatomicformulas A(x,y,z) and M(x,y,z), L(x,y),and x = y ,when x , y ,and z areinterpretedas m, n,and k respectively.Wedeclare A(x,y,z) tobetrueifand onlyif m + n = k , M(x,y,z) tobetrue,ifandonlyif m · n = k , L(x,y) tobetrue
ifandonlyif m islessthan n,andfinally x = y tobetrueifandonlyif m equals n.Weareexceedinglypedantichere,andforagoodreason.Wejustdescribedthe definitionoftruthfortheatomicformulas.
Whatmakesnotationcomplicatedintheexplanationsaboveisthereference toevaluationofthevariables.Tosimplifymatters,oneistemptedtoassigntruth valuesdirectlytoexpressionssuchas A(m,n,k).Thereisaproblemwiththat.The expression A(x,y,z) isaformula.Itisjustastringofsymbolsofthelanguageof first-orderlogic.Theexpression A(m,n,k) isnotaformula.Theletters m, n,and k ,asusedhereareinformalnamesfornumbers.NoruleinDefinition 1.1 allows insertingnamesofobjectsintoformulas.Intheexpression A(m,n,k) twoworlds aremixed.Therelationsymbol A,theparenthesesandcommas,comefromthe worldofsyntax; m, n,and k arenotsymbolsoftheformallanguage,theyare informalnamesofelementsofthedomainofthestructure.
Whatisthedifferencebetweenthestatement“A(x,y,z) istrue,when x , y and z areinterpretedas m, n,and k ”andthestatement“m + n = k ”?Theformerstates thatacertain truthvalue isassignedtoacertainformulaundercertainconditions. Thelatterisastatementaboutthestateofaffairsinacertainstructure.Whilethe definitionistellingusunderwhatconditionscertainstatementsaretrue,ithas nothingtodowithwhetherwecanactuallycheckifthoseconditionsaresatisfied. Inthecaseofcheckingwhether m plus n equals k ,thinkofnumberssoincredibly largethatthereisnotenoughspacetowritethemdown.Wearenottalkingofany practicalaspectsofcomputationhere.Still,itmakessensetodefinethetruthvalues ofinterpretationsofformulasthisway.Thedefinitionisprecise,anditisexactlythis definitionthatmakesabridgebetweenthesyntaxandtheworldofmathematical objectsinwhichitisinterpreted.
Oncethedefinitionoftruthvaluesforatomicformulasisestablished,truth valuesformorecomplexformulasaredeterminedinawayparalleltotherules forgeneratingformulasinDefinition 1.1.Forexample, ¬(ϕ) istrueifandonlyif ϕ isfalse; (ϕ) ∧ (ψ) istrue,ifandonlyifboth ϕ and ψ aretrue;and ∃x(ϕ) istrueif andonlyifthereisanevaluationofthevariable x underwhich ϕ becomestrue.Here wetakeadvantageoftheinductiveformofDefinition 1.1.Inthesamewayinwhich themorecomplexformulasareinductivelybuiltfromsimplerones,thetruthvalues ofmorecomplexformulasareinductivelydeterminedbythetruthvaluesassigned totheirsimplercomponents,withtheatomiccaseasthebase.Aswasmentioned earlier,thefullformaldefinitionofthisprocessissomewhattechnical,andwewill omitit.
InourdiscussionofEuclid’stheorem,weincluded“if ...then...”amongthe logicconnectives.Conditionalstatementsoftheform“if ϕ then ψ ”abbreviatedby (ϕ) ⇒ (ψ),areessentialinmathematics,butDefinition 1.1 hasnoprovisionfor them.Onecouldaddanotherclausethereexplaininghow (ϕ) ⇒ (ψ) istobe interpreted,butthisisnotnecessary.Inclassicallogic,theformula (ϕ) ⇒ (ψ) is definedasanabbreviationof ¬(ϕ) ∨ (ψ),henceitsinterpretationisalreadycovered bytheDefinition 1.1.Letusseeitonanexamplewealreadydiscussed.
Theformula(1.4)definingprimenumbersintheprevioussectionincludedthe followingconditionalstatement:
Forall y andall z, if x = y z, then x = y or x = z.
Itssymbolicversionis
whichinturnisequivalentto
Convinceyourself(1 4 )istrueonlyif x isinterpretedaseither1oraprimenumber. Anothercommonlogicalconnectiveis“ifandonlyif.”Thesymbolofitis ⇐⇒, and (ϕ) ⇐⇒ (ψ) isdefinedasanabbreviationfor ((ϕ) ⇒ (ψ)) ∧ ((ψ) ⇒ (ϕ)) Foranexample,seeExercise 1.5
Afirst-orderpropertyisapropertythatcanbeexpressedinfirst-orderlogic, whichmeansitcanbedefinedbyaformulaintheformalismwejustdescribed. Thiswholebookisaboutmathematicalstructuresandtheirfirst-orderproperties. Notallpropertiesarefirst-order.Forexample,hereisapropertyofnaturalnumbers thatisnotdefinedinafirst-orderway
Everysetofnaturalnumbershasaleastelement.(1.5)
In(1.5)wequantifyoversetsofnumbers,andthatmakesthisstatement secondorder.Infirst-orderlogicwecanonlyquantifyoverindividualelementsofdomains, butwecannotquantifyoversetsofelements.Quantificationoversetsisallowedin second-orderlogicwithitsspecialsyntaxandsemantics.Thereisathird-orderlogic thatallowsquantificationoversetsofsetsofelements.Therearehigher-orderlogics, eachwithstrongerexpressivepowers.ThereismoreaboutthisinChap. 14
Wewillsticktofirst-orderlogicfortworeasons.Oneisthatevenwithits restrictions,first-orderlogicisastrongenoughformalframeworkforasubstantive analysisofmathematicalstructuresingeneral,butthereisalsoanotherappealing reason.First-orderlogicisbasedonrudimentaryprinciples.Itonlyusessimple connectives“and,”“or,”and“not,”andthequantificationonlyallowsustoask whethersomepropertyholdsforallelementsinadomain(∀),orifthereisan elementinadomainwithagivenproperty(∃).Inotherwords,itisaformalization ofthemostbasicelementsoflogic,andonecouldarguethatitalsocapturessome basicfeaturesofperception.Let’sthinkofcollectionsofelementsandsomeoftheir propertiesthatcanbevisuallyrecognized.IfIseeasetofelementshavingaproperty ϕ(x),Ialsoseeitscomplementconsistingoftheelementsthatdonothavethat property,whichisthesameashavingtheproperty ¬ϕ(x).Ifsomeelementshavea property ϕ(x),andsomehaveanotherproperty ψ(x),thenIcanseethecollection ofelementswithbothproperties,i.e.thesetofdefinedbyproperty ϕ(x) ∧ ψ(x)
SimilarlyIcanseetheelementshavingonepropertyortheother: ϕ(x) ∨ ψ(x).If allelementshaveaproperty ϕ(x),Iseethat ∀xϕ(x).Toseethatitisnotthecase, itisenoughtonoticeoneelementthatdoesnothavetheproperty,soitisenough toseethat ∃x ¬ϕ(x).Thefirst-orderapproachprovidesabasicframeworkforwhat Iwillcall logicalvisibility.Equippedwiththisframework,wewilltrytofindout whatcan,andwhatcannotbeseeninstructuresthroughtheeyesoflogic.
Hereisaroughoutlineofwhatwewilldonext.Todefineastructure,westart withacollectionofindividualobjectssharingcertainfeatures.Ineachstructure,the objectsinthecollectionarerelatedtooneanotherinvariousways.Wewillgive thoserelationsnames,andthenwewilltrytoseewhatpropertiesofthestructure anditsindividualelementsarefirst-order.Ananalysisofthecomplexityofformulas andsentencesoffirst-orderlogicwillallowustoapplygeometricintuitions,andto seegeometricpatternsinthestructure.Inthissense,theformalismwillallowusto gobacktomorenatural,unformalizedwaysofthinkingaboutthestructure,andto “logicallysee”someofitsfeatures,thatotherwisemighthavestayedinvisible.This workswellinmathematicsandwewillexaminesomeexamples.
1.2.1TrivialStructures
Thesimpleststructuresaredomainswithnorelationsonthem.Thinkofadomain withfiveobjects.Ifthoseobjectsarenotrelatedtooneanotherinanyway,this isanexampleofa trivial structure.Whatcanbesaidaboutit?Notmuchmore thanwhatwehavesaidalready,butitisgoodtokeeptrivialstructuresinmindfor furtherdiscussion.Theyareagoodsourceofexamplesandcounterexamples.Due toourconvention,theequalityrelationisalwaysamongtherelationsymbolsfor anystructure.Hence,eventhoughatrivialstructurehasnorelationsofitsown,it stillhastheequalityrelation,anditallowsustoexpressspecificfactsaboutitinthe first-orderway.Considerthesentence:
Itsaysthattherearetwodistinctelementsandanyelementinthestructuremustbe oneofthem.Inotherwords,itexpressesthatthestructurehasexactlytwoelements. Inasimilarway,foranynumber n,onecanwriteafirst-ordersentenceexpressing thatthestructurehasexactly n elements.
Itisaninterestingfact,thatfollowsfromthecompactnesstheoremforfirstorderlogic,thatwhileforeachnumber n,havingexactly n elementsisafirst-order property,havingafinitenumberofelementsisnot.Thecompactnesstheoremand anargumentshowingwhyfinitenessisnotafirst-orderpropertyarepresentedin Chap. 11.
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