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GdelsProofofIncompletenessEnglishTranslation

ThisisanEnglishtranslationofGdelsProofofIncompletenessandwhichisbasedonbasedonMeltzersEnglishtranslationoftheoriginalGerman

berformalunentscheidbareStzederPrincipiaMathematicaundverwandterSystemeI.

Note:Headingsinitalicsenclosedinsquarebracketsareadditionaltotheoriginaltext,theseareincludedforconvenience,e.g.,[Recursion]

Contents

Part1

Part2DescriptionoftheformalsystemPTheaxiomsofthesystemPTherulesofinferenceofthesystemPTheGdelnumberingsystemRecursion

PropositionsIIVTheRelations146

1 2 3 4 5 6 7 8 9 1011 12 13 14 15 16 17 18 19 2021 22 23 24 25 26 27 28 29 3031 32 33 34 35 36 37 38 39 4041 42 43 44 45 46

PropositionVPropositionVI

Part3PropositionVIIPropositionVIIIPropositionIXPropositionX

Part4PropositionXI

ONFORMALLYUNDECIDABLEPROPOSITIONSOFPRINCIPIAMATHEMATICAANDRELATEDSYSTEMS1

byKurtGdel,Vienna

1Thedevelopmentofmathematicsinthedirectionofgreaterexactnesshasasiswellknownledtolargetractsofitbecomingformalized,sothatproofscanbecarriedoutaccordingtoafewmechanicalrules.Themostcomprehensiveformalsystemsyetsetupare,ontheonehand,thesystemofPrincipiaMathematica(PM) and,ontheother,theaxiomsystemforsettheoryofZermeloFraenkel(laterextendedbyJ.v.Neumann). Thesetwosystemsaresoextensivethatallmethodsofproofusedinmathematicstodayhavebeenformalizedinthem,i.e.reducedtoafewaxiomsandrulesofinference.Itmaythereforebesurmisedthattheseaxiomsandrulesofinferencearealsosufficienttodecideallmathematicalquestionswhichcaninanywayatallbeexpressedformallyinthesystemsconcerned.Itisshownbelowthatthisisnotthecase,andthatinboththesystemsmentionedthereareinfactrelativelysimpleproblemsinthetheoryofordinarywholenumbers whichcannotbedecidedfromtheaxioms.Thissituationisnotdueinsomewaytothespecialnatureofthesystemssetup,butholdsforaveryextensiveclassofformalsystems,including,inparticular,allthosearisingfromtheadditionofafinitenumberofaxiomstothetwosystemsmentioned, providedthattherebynofalsepropositionsofthekinddescribedinfootnote4becomeprovable.

Beforegoingintodetails,weshallfirstindicatethemainlinesoftheproof,naturallywithoutlayingclaimtoexactness.TheformulaeofaformalsystemwerestrictourselvesheretothesystemPMare,lookedatfromoutside,finiteseriesofbasicsigns(variables,logicalconstantsandbracketsorseparationpoints),anditiseasytostatepreciselyjustwhichseriesofbasicsignsaremeaningfulformulaeandwhicharenot. Proofs,fromtheformalstandpoint,arelikewisenothingbutfiniteseriesofformulae(withcertainspecifiablecharacteristics).Formetamathematicalpurposesitisnaturallyimmaterialwhatobjectsaretakenasbasicsigns,andweproposetousenaturalnumbers forthem.Accordingly,then,aformulaisafiniteseriesofnaturalnumbers, andaparticularproofschemaisafiniteseriesoffiniteseriesofnaturalnumbers.Metamathematicalconceptsandpropositionstherebybecomeconceptsandpropositionsconcerningnaturalnumbers,orseriesofthem, andthereforeatleastpartiallyexpressibleinthesymbolsofthesystemPMitself.Inparticular,itcanbeshownthattheconcepts,"formula","proofschema","provableformula"aredefinableinthesystemPM,i.e.onecangive aformulaF(v)ofPMforexamplewithonefreevariablev(ofthetypeofaseriesofnumbers),suchthatF(v)interpretedastocontentstates:visaprovableformula.WenowobtainanundecidablepropositionofthesystemPM,i.e.apropositionA,forwhichneitherAnornotAareprovable,inthefollowingmanner:

AformulaofPMwithjustonefreevariable,andthatofthetypeofthenaturalnumbers(classofclasses),weshalldesignateaclasssign.Wethinkoftheclasssignsasbeingsomehowarrangedinaseries, anddenotethen onebyR(n)andwenotethattheconcept"classsign"aswellastheorderingrelationRaredefinableinthesystemPM.Letbeanyclasssignby[n]wedesignatethatformulawhichisderivedonreplacingthefreevariableintheclasssignbythesignforthenaturalnumbern.Thethreetermrelationx=[yz]alsoprovestobedefinableinPM.WenowdefineaclassKofnaturalnumbers,asfollows:

nK~(Bew[R(n)n])

(whereBewxmeans:xisaprovableformula).SincetheconceptswhichappearinthedefinitionsarealldefinableinPM,sotooistheconceptKwhichisconstitutedfromthem,i.e.thereisaclasssignS, suchthattheformula[Sn]interpretedastoitscontentstatesthatthenaturalnumbernbelongstoK.S,beingaclasssign,isidenticalwithsomedeterminateR(q),i.e.

S=R(q)

holdsforsomedeterminatenaturalnumberq.Wenowshowthattheproposition[R(q)q] isundecidableinPM.Forsupposingtheproposition[R(q)q]wereprovable,itwouldalsobecorrectbutthatmeans,ashasbeensaid,thatqwouldbelongtoK,i.e.accordingto(1),~(Bew[R(q)q])wouldholdgood,incontradictiontoourinitialassumption.If,onthecontrary,thenegationof[R(q)q]wereprovable,then~(nK),i.e.Bew[R(q)q]wouldholdgood.[R(q)q]wouldthusbeprovableatthesametimeasitsnegation,whichagainisimpossible.

TheanalogybetweenthisresultandRichardsantinomyleapstotheeyethereisalsoacloserelationshipwiththe"liar"antinomy, sincetheundecidableproposition[R(q)q]statespreciselythatqbelongstoK,i.e.accordingto(1),that[R(q)q]isnotprovable.Wearethereforeconfrontedwithapropositionwhichassertsitsownunprovability. Themethodofproofjustexhibitedcanclearlybeappliedtoeveryformalsystemhavingthefollowingfeatures:firstly,interpretedastocontent,itdisposesofsufficientmeansofexpressiontodefinetheconceptsoccurringintheaboveargument(inparticulartheconcept"provableformula")secondly,everyprovableformulainitisalsocorrectasregardscontent.Theexactstatementoftheaboveproof,whichnowfollows,willhaveamongothersthetaskofsubstitutingforthesecondoftheseassumptionsapurelyformalandmuchweakerone.

Fromtheremarkthat[R(q)q]assertsitsownunprovability,itfollowsatoncethat[R(q)q]iscorrect,since[R(q)q]iscertainlyunprovable(becauseundecidable).Sothepropositionwhichisundecidableinthesystem PMyetturnsouttobedecidedbymetamathematicalconsiderations.Thecloseanalysisofthisremarkablecircumstanceleadstosurprisingresultsconcerningproofsofconsistencyofformalsystems,whicharedealtwithinmoredetailinSection4(PropositionXI).

Cf.thesummaryoftheresultsofthiswork,publishedinAnzeigerderAkad.d.Wiss.inWien(math.naturw.Kl.)1930,No.19.A.WhiteheadandB.Russell,PrincipiaMathematica,2ndedition,Cambridge1925.Inparticular,wealsoreckonamongtheaxiomsofPMtheaxiomofinfinity(intheform:thereexistdenumerablymanyindividuals),andtheaxiomsofreducibilityandofchoice(foralltypes).Cf.A.Fraenkel,'ZehnVorlesungenberdieGrundlegungderMengenlehre',Wissensch.u.Hyp.,Vol.XXXIJ.v.Neumann,'DieAxiomatisierungderMengenlehre',Math.Zeitschr.27,1928,Journ.f.reineu.angew.Math.154(1925),160(1929).Wemaynotethatinordertocompletetheformalization,theaxiomsandrulesofinferenceofthelogicalcalculusmustbeaddedtotheaxiomsofsettheorygivenintheabovementionedpapers.TheremarksthatfollowalsoapplytotheformalsystemspresentedinrecentyearsbyD.Hilbertandhiscolleagues(sofarasthesehaveyetbeenpublished).Cf.D.Hilbert,Math.Ann.88,Abh.ausd.math.Sem.derUniv.HamburgI(1922),VI(1928)P.Bernays,Math.Ann.90J.v.Neumann,Math.Zeitsehr.26(1927)W.Ackermann,Math.Ann.93.I.e.,moreprecisely,thereareundecidablepropositionsinwhich,besidesthelogicalconstants~(not),(or),(x)(forall)and=(identicalwith),therearenootherconceptsbeyond+(addition)and.(multiplication),bothreferredtonaturalnumbers,andwheretheprefixes(x)canalsoreferonlytonaturalnumbers.Inthisconnection,onlysuchaxiomsinPMarecountedasdistinctasdonotarisefromeachotherpurelybychangeoftype.Hereandinwhatfollows,weshallalwaysunderstandtheterm"formulaofPM"tomeanaformulawrittenwithoutabbreviations(i.e.withoutuseofdefinitions).Definitionsserveonlytoabridgethewrittentextandarethereforeinprinciplesuperfluous.I.e.wemapthebasicsignsinonetoonefashiononthenaturalnumbers(asisactuallydoneon).I.e.acoveringofasectionofthenumberseriesbynaturalnumbers.(Numberscannotinfactbeputintoaspatialorder.)Inotherwords,theabovedescribedprocedureprovidesanisomorphicimageofthesystemPMinthedomainofarithmetic,andallmetamathematicalargumentscanequallywellbeconductedinthisisomorphicimage.Thisoccursinthefollowingoutlineproof,i.e."formula","proposition","variable",etc.arealwaystobeunderstoodasthecorrespondingobjectsoftheisomorphicimage.Itwouldbeverysimple(thoughratherlaborious)actuallytowriteoutthisformula.Perhapsaccordingtotheincreasingsumsoftheirtermsand,forequalsums,inalphabeticalorder.Thebarsignindicatesnegation.[Replacedwith~.]AgainthereisnottheslightestdifficultyinactuallywritingouttheformulaS.Notethat"[R(q)q]"(orwhatcomestothesamething"[Sq]")ismerelyametamathematicaldescriptionoftheundecidableproposition.ButassoonasonehasascertainedtheformulaS,onecannaturallyalsodeterminethenumberq,andtherebyeffectivelywriteouttheundecidablepropositionitself.Everyepistemologicalantinomycanlikewisebeusedforasimilarundecidabilityproof.Inspiteofappearances,thereisnothingcircularaboutsuchaproposition,sinceitbeginsbyassertingtheunprovabilityofawhollydeterminateformula(namelytheq inthealphabeticalarrangementwithadefinitesubstitution),andonlysubsequently(andinsomewaybyaccident)doesitemergethatthisformulaispreciselythatbywhichthepropositionwasitselfexpressed.

2

[DescriptionoftheformalsystemP]

Weproceednowtotherigorousdevelopmentoftheproofsketchedabove,andbeginbygivinganexactdescriptionoftheformalsystemP,forwhichweseektodemonstratetheexistenceofundecidablepropositions.PisessentiallythesystemobtainedbysuperimposingonthePeanoaxiomsthelogicofPM (numbersasindividuals,relationofsuccessorasundefinedbasicconcept).

ThebasicsignsofthesystemParethefollowing:

I. Constants:"~"(not),""(or),""(forall),"0"(nought),"f"(thesuccessorof),"(",")"(brackets).

II. Variablesoffirsttype(forindividuals,i.e.naturalnumbersincluding0):"x ","y ","z ",Variablesofsecondtype(forclassesofindividuals):"x ","y ","z ",Variablesofthirdtype(forclassesofclassesofindividuals):"x ","y ","z ",

andsoonforeverynaturalnumberastype.

Note:Variablesfortwotermedandmanytermedfunctions(relations)aresuperfluousasbasicsigns,sincerelationscanbedefinedasclassesoforderedpairsandorderedpairsagainasclassesofclasses,e.g.theorderedpaira,bby((a),(a,b)),where(x,y)meanstheclasswhoseonlyelementsarexandy,and(x)theclasswhoseonlyelementisx.

Byasignoffirsttypeweunderstandacombinationofsignsoftheform:

a,fa,ffa,fffaetc.

whereaiseither0oravariableoffirsttype.Intheformercasewecallsuchasignanumbersign.Forn>1weunderstandbyasignofn typethesameasvariableofn type.

Combinationsofsignsoftheforma(b),wherebisasignofn andaasignof(n+1) type,wecallelementaryformulae.Theclassofformulaewedefineasthesmallestclass containingallelementaryformulaeand,also,alongwithanyaandbthefollowing:~(a),(a)(b),x(a)(wherexisanygivenvariable). Weterm(a)(b)thedisjunctionofaandb,~(a)thenegationand(a)(b)ageneralizationofa.Aformulainwhichthereisnofreevariableiscalledapropositionalformula(freevariablebeingdefinedintheusualway).Aformulawithjustnfreeindividualvariables(andotherwisenofreevariables)wecallannplacerelationsignandforn=1alsoaclasssign.

BySubsta(v|b)(whereastandsforaformula,vavariableandbasignofthesametypeasv)weunderstandtheformuladerivedfroma,whenwereplacevinit,whereveritisfree,byb. Wesaythataformulaaisatypeliftofanotheroneb,ifaderivesfromb,whenweincreasebythesameamountthetypeofallvariablesappearinginb.

[AxiomsoftheformalsystemP]

Thefollowingformulae(IV)arecalledaxioms(theyaresetoutwiththehelpofthecustomarilydefinedabbreviations:.,,,(x),= andsubjecttotheusualconventionsaboutomissionofbrackets):

I.1. ~(fx =0)2. fx =fy x =y3. x (0).x (x (x )x (fx ))x (x (x ))

II.Everyformuladerivedfromthefollowingschematabysubstitutionofanyformulaeforp,qandr.

1. ppp2. ppq3. pqqp4. (pq)(rprq)

III.Everyformuladerivedfromthetwoschemata

1.v(a)Substa(v|c)2. v(ba)bv(a)

bymakingthefollowingsubstitutionsfora,v,b,c(andcarryingoutinItheoperationdenotedby"Subst"):foraanygivenformula,forvanyvariable,forbanyformulainwhichvdoesnotappearfree,forcasignofthesametypeasv,providedthatccontainsnovariablewhichisboundinaataplacewherevisfree.

IV.Everyformuladerivedfromtheschema

1. (u)(v(u(v)a))

onsubstitutingforvoruanyvariablesoftypesnorn+1respectively,andforaaformulawhichdoesnotcontainufree.Thisaxiomrepresentstheaxiomofreducibility(theaxiomofcomprehensionofsettheory).

V.Everyformuladerivedfromthefollowingbytypelift(andthisformulaitself):

1. x (x (x )y (x ))x =y

Thisaxiomstatesthataclassiscompletelydeterminedbyitselements.

[RulesofinferenceoftheformalsystemP]

Aformulaciscalledanimmediateconsequenceofaandb,ifaistheformula(~(b))(c),andanimmediateconsequenceofa,ifcistheformulav(a),wherevdenotesanygivenvariable.Theclassofprovableformulaeisdefinedasthesmallestclassofformulaewhichcontainstheaxiomsandisclosedwithrespecttotherelation"immediateconsequenceof".

[TheGdelnumberingsystem]

ThebasicsignsofthesystemParenoworderedinonetoonecorrespondencewithnaturalnumbers,asfollows:

0"1"f"3"~"5""7""9"("11")"13

Furthermore,variablesoftypenaregivennumbersoftheformp (wherepisaprimenumber>13).Hence,toeveryfiniteseriesofbasicsigns(andsoalsotoeveryformula)therecorresponds,onetoone,afiniteseriesofnaturalnumbers.Thesefiniteseriesofnaturalnumberswenowmap(againinonetoonecorrespondence)ontonaturalnumbers,bylettingthenumber2 ,3 p correspondtotheseriesn ,n ,n ,wherep denotesthek primenumberinorderofmagnitude.Anaturalnumberistherebyassignedinonetoonecorrespondence,notonlytoeverybasicsign,butalsotoeveryfiniteseriesofsuchsigns.Wedenoteby(a)thenumbercorrespondingtothebasicsignorseriesofbasicsignsa.SupposenowoneisgivenaclassorrelationR(a ,a ,a )ofbasicsignsorseriesofsuch.Weassigntoitthatclass(orrelation)R'(x ,x ,x )ofnaturalnumbers,whichholdsforx ,x ,x whenandonlywhenthereexista ,a ,a suchthatx =(a )(i=1,2,n)andR(a ,a ,a )holds.Werepresentbythesamewordsinitalicsthoseclassesandrelationsofnaturalnumberswhichhavebeenassignedinthisfashiontosuchpreviouslydefinedmetamathematicalconceptsas"variable","formula","propositionalformula","axiom","provableformula",etc.ThepropositionthatthereareundecidableproblemsinthesystemPwouldthereforeread,forexample,asfollows:Thereexistpropositionalformulaeasuchthatneitheranorthenegationofaareprovableformulae.

[Recursion]

WenowintroduceaparentheticconsiderationhavingnoimmediateconnectionwiththeformalsystemP,andfirstputforwardthefollowingdefinition:Anumbertheoreticfunction (x ,x ,x )issaidtoberecursivelydefinedbythenumbertheoreticfunctions(x ,x ,x )and(x ,x ,x ),ifforallx ,x ,k thefollowinghold:

(0,x ,x )=(x ,x )(k+1,x ,x )=(k,(k,x ,x ),x ,x ).

Anumbertheoreticfunctioniscalledrecursive,ifthereexistsafiniteseriesofnumbertheoreticfunctions , , whichendsinandhasthepropertythateveryfunction oftheseriesiseitherrecursivelydefinedbytwooftheearlierones,orisderivedfromanyoftheearlieronesbysubstitution, or,finally,isaconstantorthesuccessorfunctionx+1.Thelengthoftheshortestseriesof ,whichbelongstoarecursivefunction,istermeditsdegree.ArelationR(x ,x ,x )amongnaturalnumbersiscalledrecursive, ifthereexistsarecursivefunction(x ,x ,x )suchthatforallx ,x ,x

R(x ,x ,x )[(x ,x ,x )=0] .

[PropositionsIIV]

Thefollowingpropositionshold:

I.Everyfunction(orrelation)derivedfromrecursivefunctions(orrelations)bythesubstitutionofrecursivefunctionsinplaceofvariablesisrecursivesoalsoiseveryfunctionderivedfromrecursivefunctionsbyrecursivedefinitionaccordingtoschema(2).

II.IfRandSarerecursiverelations,thensoalsoare~R,RS(andthereforealsoR&S).

III.Ifthefunctions()and()arerecursive,soalsoistherelation:()=().

IV.Ifthefunction()andtherelationR(x,)arerecursive,soalsothenaretherelationsS,T

S(,)~(x)[x()&R(x,)]T(,)~(x)[x()R(x,)]

andlikewisethefunction

(,)=x[x()&R(x,)]

wherexF(x)means:thesmallestnumberxforwhichF(x)holdsand0ifthereisnosuchnumber.

PropositionIfollowsimmediatelyfromthedefinitionof"recursive".PropositionsIIandIIIarebasedonthereadilyascertainablefactthatthenumbertheoreticfunctionscorrespondingtothelogicalconcepts~,,=

(x),(x,y),(x,y)namely

(0)=1(x)=0forx0(0,x)=(x,0)=0(x,y)=1,ifx,ybothO

(x,y)=0,ifx=y(x,y)=1,ifxy

arerecursive.TheproofofPropositionIVisbrieflyasfollows:Accordingtotheassumptionthereexistsarecursive(x,)suchthat

R(x,)[(x,)=0].

Wenowdefine,accordingtotherecursionschema(2),afunction(x,)inthefollowingmanner:

(0,)=0(n+1,)=(n+1).a+(n,).(a)

where

a=[((0,))].[(n+1,)].[(n,)].

(n+1,)isthereforeeither=n+1(ifa=1)or=(n,)(ifa=0). Thefirstcaseclearlyarisesifandonlyifalltheconstituentfactorsofaare1,i.e.if

~R(O,)&R(n+1,)&[(n,)=0].

Fromthisitfollowsthatthefunction(n,)(consideredasafunctionofn)remains0uptothesmallestvalueofnforwhichR(n,)holds,andfromthenonisequaltothisvalue(ifR(0,)isalreadythecase,thecorresponding(x,)isconstantand=0).Therefore:

(,)=C((),)S(,)R[(,)),)]

TherelationTcanbereducedbynegationtoacaseanalogoustoS,sothatPropositionIVisproved.

[TheRelations146]

Thefunctionsx+y,x.y,x ,andalsotherelationsx1

xisaprimenumber.

0Prx0(n+1)Prxy[yx&Prim(y)&x/y&y>nPrx]

nPrxisthen (inorderofmagnitude)primenumbercontainedinx.

0!1(n+1)!(n+1).n!

Pr(0)0Pr(n+1)y[y{Pr(n)}!+1&Prim(y)&y>Pr(n)]

Pr(n)isthen primenumber(inorderofmagnitude).

nGlxy[yx&x/(nPrx) &~x/(nPrx) ]

nGlxisthen termoftheseriesofnumbersassignedtothenumberx(forn>0andnnotgreaterthanthelengthofthisseries).

l(x)y[yx&yPrx>0&(y+1)Prx=0]

l(x)isthelengthoftheseriesofnumbersassignedtox.

x*yz[z[Pr{l(x)+l(y)}] &(n)[nl(x)nGlz=nGlx] &(n)[00).

E(x)R(11)*x*R(13)

E(x)correspondstotheoperationof"bracketing"[11and13areassignedtothebasicsigns"("and")"].

nVarx(z)[131 &(v){vx&nVarv&x=R(v)}]

xisasignofn type.

Elf(x)(y,z,n)[y,z,nx &Typ (y)&Typ (z)&x=z*E(y)]

xisanelementaryformula.

Op(x,y,z)x=Neg(y)x=yDisz (v)[vx&Var(v)&x=vGeny]

FR(x)(n){0

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GdelsProofofIncompletenessEnglishTranslation

ThisisanEnglishtranslationofGdelsProofofIncompletenessandwhichisbasedonbasedonMeltzersEnglishtranslationoftheoriginalGerman

berformalunentscheidbareStzederPrincipiaMathematicaundverwandterSystemeI.

Note:Headingsinitalicsenclosedinsquarebracketsareadditionaltotheoriginaltext,theseareincludedforconvenience,e.g.,[Recursion]

Contents

Part1

Part2DescriptionoftheformalsystemPTheaxiomsofthesystemPTherulesofinferenceofthesystemPTheGdelnumberingsystemRecursion

PropositionsIIVTheRelations146

1 2 3 4 5 6 7 8 9 1011 12 13 14 15 16 17 18 19 2021 22 23 24 25 26 27 28 29 3031 32 33 34 35 36 37 38 39 4041 42 43 44 45 46

PropositionVPropositionVI

Part3PropositionVIIPropositionVIIIPropositionIXPropositionX

Part4PropositionXI

ONFORMALLYUNDECIDABLEPROPOSITIONSOFPRINCIPIAMATHEMATICAANDRELATEDSYSTEMS1

byKurtGdel,Vienna

1Thedevelopmentofmathematicsinthedirectionofgreaterexactnesshasasiswellknownledtolargetractsofitbecomingformalized,sothatproofscanbecarriedoutaccordingtoafewmechanicalrules.Themostcomprehensiveformalsystemsyetsetupare,ontheonehand,thesystemofPrincipiaMathematica(PM) and,ontheother,theaxiomsystemforsettheoryofZermeloFraenkel(laterextendedbyJ.v.Neumann). Thesetwosystemsaresoextensivethatallmethodsofproofusedinmathematicstodayhavebeenformalizedinthem,i.e.reducedtoafewaxiomsandrulesofinference.Itmaythereforebesurmisedthattheseaxiomsandrulesofinferencearealsosufficienttodecideallmathematicalquestionswhichcaninanywayatallbeexpressedformallyinthesystemsconcerned.Itisshownbelowthatthisisnotthecase,andthatinboththesystemsmentionedthereareinfactrelativelysimpleproblemsinthetheoryofordinarywholenumbers whichcannotbedecidedfromtheaxioms.Thissituationisnotdueinsomewaytothespecialnatureofthesystemssetup,butholdsforaveryextensiveclassofformalsystems,including,inparticular,allthosearisingfromtheadditionofafinitenumberofaxiomstothetwosystemsmentioned, providedthattherebynofalsepropositionsofthekinddescribedinfootnote4becomeprovable.

Beforegoingintodetails,weshallfirstindicatethemainlinesoftheproof,naturallywithoutlayingclaimtoexactness.TheformulaeofaformalsystemwerestrictourselvesheretothesystemPMare,lookedatfromoutside,finiteseriesofbasicsigns(variables,logicalconstantsandbracketsorseparationpoints),anditiseasytostatepreciselyjustwhichseriesofbasicsignsaremeaningfulformulaeandwhicharenot. Proofs,fromtheformalstandpoint,arelikewisenothingbutfiniteseriesofformulae(withcertainspecifiablecharacteristics).Formetamathematicalpurposesitisnaturallyimmaterialwhatobjectsaretakenasbasicsigns,andweproposetousenaturalnumbers forthem.Accordingly,then,aformulaisafiniteseriesofnaturalnumbers, andaparticularproofschemaisafiniteseriesoffiniteseriesofnaturalnumbers.Metamathematicalconceptsandpropositionstherebybecomeconceptsandpropositionsconcerningnaturalnumbers,orseriesofthem, andthereforeatleastpartiallyexpressibleinthesymbolsofthesystemPMitself.Inparticular,itcanbeshownthattheconcepts,"formula","proofschema","provableformula"aredefinableinthesystemPM,i.e.onecangive aformulaF(v)ofPMforexamplewithonefreevariablev(ofthetypeofaseriesofnumbers),suchthatF(v)interpretedastocontentstates:visaprovableformula.WenowobtainanundecidablepropositionofthesystemPM,i.e.apropositionA,forwhichneitherAnornotAareprovable,inthefollowingmanner:

AformulaofPMwithjustonefreevariable,andthatofthetypeofthenaturalnumbers(classofclasses),weshalldesignateaclasssign.Wethinkoftheclasssignsasbeingsomehowarrangedinaseries, anddenotethen onebyR(n)andwenotethattheconcept"classsign"aswellastheorderingrelationRaredefinableinthesystemPM.Letbeanyclasssignby[n]wedesignatethatformulawhichisderivedonreplacingthefreevariableintheclasssignbythesignforthenaturalnumbern.Thethreetermrelationx=[yz]alsoprovestobedefinableinPM.WenowdefineaclassKofnaturalnumbers,asfollows:

nK~(Bew[R(n)n])

(whereBewxmeans:xisaprovableformula).SincetheconceptswhichappearinthedefinitionsarealldefinableinPM,sotooistheconceptKwhichisconstitutedfromthem,i.e.thereisaclasssignS, suchthattheformula[Sn]interpretedastoitscontentstatesthatthenaturalnumbernbelongstoK.S,beingaclasssign,isidenticalwithsomedeterminateR(q),i.e.

S=R(q)

holdsforsomedeterminatenaturalnumberq.Wenowshowthattheproposition[R(q)q] isundecidableinPM.Forsupposingtheproposition[R(q)q]wereprovable,itwouldalsobecorrectbutthatmeans,ashasbeensaid,thatqwouldbelongtoK,i.e.accordingto(1),~(Bew[R(q)q])wouldholdgood,incontradictiontoourinitialassumption.If,onthecontrary,thenegationof[R(q)q]wereprovable,then~(nK),i.e.Bew[R(q)q]wouldholdgood.[R(q)q]wouldthusbeprovableatthesametimeasitsnegation,whichagainisimpossible.

TheanalogybetweenthisresultandRichardsantinomyleapstotheeyethereisalsoacloserelationshipwiththe"liar"antinomy, sincetheundecidableproposition[R(q)q]statespreciselythatqbelongstoK,i.e.accordingto(1),that[R(q)q]isnotprovable.Wearethereforeconfrontedwithapropositionwhichassertsitsownunprovability. Themethodofproofjustexhibitedcanclearlybeappliedtoeveryformalsystemhavingthefollowingfeatures:firstly,interpretedastocontent,itdisposesofsufficientmeansofexpressiontodefinetheconceptsoccurringintheaboveargument(inparticulartheconcept"provableformula")secondly,everyprovableformulainitisalsocorrectasregardscontent.Theexactstatementoftheaboveproof,whichnowfollows,willhaveamongothersthetaskofsubstitutingforthesecondoftheseassumptionsapurelyformalandmuchweakerone.

Fromtheremarkthat[R(q)q]assertsitsownunprovability,itfollowsatoncethat[R(q)q]iscorrect,since[R(q)q]iscertainlyunprovable(becauseundecidable).Sothepropositionwhichisundecidableinthesystem PMyetturnsouttobedecidedbymetamathematicalconsiderations.Thecloseanalysisofthisremarkablecircumstanceleadstosurprisingresultsconcerningproofsofconsistencyofformalsystems,whicharedealtwithinmoredetailinSection4(PropositionXI).

Cf.thesummaryoftheresultsofthiswork,publishedinAnzeigerderAkad.d.Wiss.inWien(math.naturw.Kl.)1930,No.19.A.WhiteheadandB.Russell,PrincipiaMathematica,2ndedition,Cambridge1925.Inparticular,wealsoreckonamongtheaxiomsofPMtheaxiomofinfinity(intheform:thereexistdenumerablymanyindividuals),andtheaxiomsofreducibilityandofchoice(foralltypes).Cf.A.Fraenkel,'ZehnVorlesungenberdieGrundlegungderMengenlehre',Wissensch.u.Hyp.,Vol.XXXIJ.v.Neumann,'DieAxiomatisierungderMengenlehre',Math.Zeitschr.27,1928,Journ.f.reineu.angew.Math.154(1925),160(1929).Wemaynotethatinordertocompletetheformalization,theaxiomsandrulesofinferenceofthelogicalcalculusmustbeaddedtotheaxiomsofsettheorygivenintheabovementionedpapers.TheremarksthatfollowalsoapplytotheformalsystemspresentedinrecentyearsbyD.Hilbertandhiscolleagues(sofarasthesehaveyetbeenpublished).Cf.D.Hilbert,Math.Ann.88,Abh.ausd.math.Sem.derUniv.HamburgI(1922),VI(1928)P.Bernays,Math.Ann.90J.v.Neumann,Math.Zeitsehr.26(1927)W.Ackermann,Math.Ann.93.I.e.,moreprecisely,thereareundecidablepropositionsinwhich,besidesthelogicalconstants~(not),(or),(x)(forall)and=(identicalwith),therearenootherconceptsbeyond+(addition)and.(multiplication),bothreferredtonaturalnumbers,andwheretheprefixes(x)canalsoreferonlytonaturalnumbers.Inthisconnection,onlysuchaxiomsinPMarecountedasdistinctasdonotarisefromeachotherpurelybychangeoftype.Hereandinwhatfollows,weshallalwaysunderstandtheterm"formulaofPM"tomeanaformulawrittenwithoutabbreviations(i.e.withoutuseofdefinitions).Definitionsserveonlytoabridgethewrittentextandarethereforeinprinciplesuperfluous.I.e.wemapthebasicsignsinonetoonefashiononthenaturalnumbers(asisactuallydoneon).I.e.acoveringofasectionofthenumberseriesbynaturalnumbers.(Numberscannotinfactbeputintoaspatialorder.)Inotherwords,theabovedescribedprocedureprovidesanisomorphicimageofthesystemPMinthedomainofarithmetic,andallmetamathematicalargumentscanequallywellbeconductedinthisisomorphicimage.Thisoccursinthefollowingoutlineproof,i.e."formula","proposition","variable",etc.arealwaystobeunderstoodasthecorrespondingobjectsoftheisomorphicimage.Itwouldbeverysimple(thoughratherlaborious)actuallytowriteoutthisformula.Perhapsaccordingtotheincreasingsumsoftheirtermsand,forequalsums,inalphabeticalorder.Thebarsignindicatesnegation.[Replacedwith~.]AgainthereisnottheslightestdifficultyinactuallywritingouttheformulaS.Notethat"[R(q)q]"(orwhatcomestothesamething"[Sq]")ismerelyametamathematicaldescriptionoftheundecidableproposition.ButassoonasonehasascertainedtheformulaS,onecannaturallyalsodeterminethenumberq,andtherebyeffectivelywriteouttheundecidablepropositionitself.Everyepistemologicalantinomycanlikewisebeusedforasimilarundecidabilityproof.Inspiteofappearances,thereisnothingcircularaboutsuchaproposition,sinceitbeginsbyassertingtheunprovabilityofawhollydeterminateformula(namelytheq inthealphabeticalarrangementwithadefinitesubstitution),andonlysubsequently(andinsomewaybyaccident)doesitemergethatthisformulaispreciselythatbywhichthepropositionwasitselfexpressed.

2

[DescriptionoftheformalsystemP]

Weproceednowtotherigorousdevelopmentoftheproofsketchedabove,andbeginbygivinganexactdescriptionoftheformalsystemP,forwhichweseektodemonstratetheexistenceofundecidablepropositions.PisessentiallythesystemobtainedbysuperimposingonthePeanoaxiomsthelogicofPM (numbersasindividuals,relationofsuccessorasundefinedbasicconcept).

ThebasicsignsofthesystemParethefollowing:

I. Constants:"~"(not),""(or),""(forall),"0"(nought),"f"(thesuccessorof),"(",")"(brackets).

II. Variablesoffirsttype(forindividuals,i.e.naturalnumbersincluding0):"x ","y ","z ",Variablesofsecondtype(forclassesofindividuals):"x ","y ","z ",Variablesofthirdtype(forclassesofclassesofindividuals):"x ","y ","z ",

andsoonforeverynaturalnumberastype.

Note:Variablesfortwotermedandmanytermedfunctions(relations)aresuperfluousasbasicsigns,sincerelationscanbedefinedasclassesoforderedpairsandorderedpairsagainasclassesofclasses,e.g.theorderedpaira,bby((a),(a,b)),where(x,y)meanstheclasswhoseonlyelementsarexandy,and(x)theclasswhoseonlyelementisx.

Byasignoffirsttypeweunderstandacombinationofsignsoftheform:

a,fa,ffa,fffaetc.

whereaiseither0oravariableoffirsttype.Intheformercasewecallsuchasignanumbersign.Forn>1weunderstandbyasignofn typethesameasvariableofn type.

Combinationsofsignsoftheforma(b),wherebisasignofn andaasignof(n+1) type,wecallelementaryformulae.Theclassofformulaewedefineasthesmallestclass containingallelementaryformulaeand,also,alongwithanyaandbthefollowing:~(a),(a)(b),x(a)(wherexisanygivenvariable). Weterm(a)(b)thedisjunctionofaandb,~(a)thenegationand(a)(b)ageneralizationofa.Aformulainwhichthereisnofreevariableiscalledapropositionalformula(freevariablebeingdefinedintheusualway).Aformulawithjustnfreeindividualvariables(andotherwisenofreevariables)wecallannplacerelationsignandforn=1alsoaclasssign.

BySubsta(v|b)(whereastandsforaformula,vavariableandbasignofthesametypeasv)weunderstandtheformuladerivedfroma,whenwereplacevinit,whereveritisfree,byb. Wesaythataformulaaisatypeliftofanotheroneb,ifaderivesfromb,whenweincreasebythesameamountthetypeofallvariablesappearinginb.

[AxiomsoftheformalsystemP]

Thefollowingformulae(IV)arecalledaxioms(theyaresetoutwiththehelpofthecustomarilydefinedabbreviations:.,,,(x),= andsubjecttotheusualconventionsaboutomissionofbrackets):

I.1. ~(fx =0)2. fx =fy x =y3. x (0).x (x (x )x (fx ))x (x (x ))

II.Everyformuladerivedfromthefollowingschematabysubstitutionofanyformulaeforp,qandr.

1. ppp2. ppq3. pqqp4. (pq)(rprq)

III.Everyformuladerivedfromthetwoschemata

1.v(a)Substa(v|c)2. v(ba)bv(a)

bymakingthefollowingsubstitutionsfora,v,b,c(andcarryingoutinItheoperationdenotedby"Subst"):foraanygivenformula,forvanyvariable,forbanyformulainwhichvdoesnotappearfree,forcasignofthesametypeasv,providedthatccontainsnovariablewhichisboundinaataplacewherevisfree.

IV.Everyformuladerivedfromtheschema

1. (u)(v(u(v)a))

onsubstitutingforvoruanyvariablesoftypesnorn+1respectively,andforaaformulawhichdoesnotcontainufree.Thisaxiomrepresentstheaxiomofreducibility(theaxiomofcomprehensionofsettheory).

V.Everyformuladerivedfromthefollowingbytypelift(andthisformulaitself):

1. x (x (x )y (x ))x =y

Thisaxiomstatesthataclassiscompletelydeterminedbyitselements.

[RulesofinferenceoftheformalsystemP]

Aformulaciscalledanimmediateconsequenceofaandb,ifaistheformula(~(b))(c),andanimmediateconsequenceofa,ifcistheformulav(a),wherevdenotesanygivenvariable.Theclassofprovableformulaeisdefinedasthesmallestclassofformulaewhichcontainstheaxiomsandisclosedwithrespecttotherelation"immediateconsequenceof".

[TheGdelnumberingsystem]

ThebasicsignsofthesystemParenoworderedinonetoonecorrespondencewithnaturalnumbers,asfollows:

0"1"f"3"~"5""7""9"("11")"13

Furthermore,variablesoftypenaregivennumbersoftheformp (wherepisaprimenumber>13).Hence,toeveryfiniteseriesofbasicsigns(andsoalsotoeveryformula)therecorresponds,onetoone,afiniteseriesofnaturalnumbers.Thesefiniteseriesofnaturalnumberswenowmap(againinonetoonecorrespondence)ontonaturalnumbers,bylettingthenumber2 ,3 p correspondtotheseriesn ,n ,n ,wherep denotesthek primenumberinorderofmagnitude.Anaturalnumberistherebyassignedinonetoonecorrespondence,notonlytoeverybasicsign,butalsotoeveryfiniteseriesofsuchsigns.Wedenoteby(a)thenumbercorrespondingtothebasicsignorseriesofbasicsignsa.SupposenowoneisgivenaclassorrelationR(a ,a ,a )ofbasicsignsorseriesofsuch.Weassigntoitthatclass(orrelation)R'(x ,x ,x )ofnaturalnumbers,whichholdsforx ,x ,x whenandonlywhenthereexista ,a ,a suchthatx =(a )(i=1,2,n)andR(a ,a ,a )holds.Werepresentbythesamewordsinitalicsthoseclassesandrelationsofnaturalnumberswhichhavebeenassignedinthisfashiontosuchpreviouslydefinedmetamathematicalconceptsas"variable","formula","propositionalformula","axiom","provableformula",etc.ThepropositionthatthereareundecidableproblemsinthesystemPwouldthereforeread,forexample,asfollows:Thereexistpropositionalformulaeasuchthatneitheranorthenegationofaareprovableformulae.

[Recursion]

WenowintroduceaparentheticconsiderationhavingnoimmediateconnectionwiththeformalsystemP,andfirstputforwardthefollowingdefinition:Anumbertheoreticfunction (x ,x ,x )issaidtoberecursivelydefinedbythenumbertheoreticfunctions(x ,x ,x )and(x ,x ,x ),ifforallx ,x ,k thefollowinghold:

(0,x ,x )=(x ,x )(k+1,x ,x )=(k,(k,x ,x ),x ,x ).

Anumbertheoreticfunctioniscalledrecursive,ifthereexistsafiniteseriesofnumbertheoreticfunctions , , whichendsinandhasthepropertythateveryfunction oftheseriesiseitherrecursivelydefinedbytwooftheearlierones,orisderivedfromanyoftheearlieronesbysubstitution, or,finally,isaconstantorthesuccessorfunctionx+1.Thelengthoftheshortestseriesof ,whichbelongstoarecursivefunction,istermeditsdegree.ArelationR(x ,x ,x )amongnaturalnumbersiscalledrecursive, ifthereexistsarecursivefunction(x ,x ,x )suchthatforallx ,x ,x

R(x ,x ,x )[(x ,x ,x )=0] .

[PropositionsIIV]

Thefollowingpropositionshold:

I.Everyfunction(orrelation)derivedfromrecursivefunctions(orrelations)bythesubstitutionofrecursivefunctionsinplaceofvariablesisrecursivesoalsoiseveryfunctionderivedfromrecursivefunctionsbyrecursivedefinitionaccordingtoschema(2).

II.IfRandSarerecursiverelations,thensoalsoare~R,RS(andthereforealsoR&S).

III.Ifthefunctions()and()arerecursive,soalsoistherelation:()=().

IV.Ifthefunction()andtherelationR(x,)arerecursive,soalsothenaretherelationsS,T

S(,)~(x)[x()&R(x,)]T(,)~(x)[x()R(x,)]

andlikewisethefunction

(,)=x[x()&R(x,)]

wherexF(x)means:thesmallestnumberxforwhichF(x)holdsand0ifthereisnosuchnumber.

PropositionIfollowsimmediatelyfromthedefinitionof"recursive".PropositionsIIandIIIarebasedonthereadilyascertainablefactthatthenumbertheoreticfunctionscorrespondingtothelogicalconcepts~,,=

(x),(x,y),(x,y)namely

(0)=1(x)=0forx0(0,x)=(x,0)=0(x,y)=1,ifx,ybothO

(x,y)=0,ifx=y(x,y)=1,ifxy

arerecursive.TheproofofPropositionIVisbrieflyasfollows:Accordingtotheassumptionthereexistsarecursive(x,)suchthat

R(x,)[(x,)=0].

Wenowdefine,accordingtotherecursionschema(2),afunction(x,)inthefollowingmanner:

(0,)=0(n+1,)=(n+1).a+(n,).(a)

where

a=[((0,))].[(n+1,)].[(n,)].

(n+1,)isthereforeeither=n+1(ifa=1)or=(n,)(ifa=0). Thefirstcaseclearlyarisesifandonlyifalltheconstituentfactorsofaare1,i.e.if

~R(O,)&R(n+1,)&[(n,)=0].

Fromthisitfollowsthatthefunction(n,)(consideredasafunctionofn)remains0uptothesmallestvalueofnforwhichR(n,)holds,andfromthenonisequaltothisvalue(ifR(0,)isalreadythecase,thecorresponding(x,)isconstantand=0).Therefore:

(,)=C((),)S(,)R[(,)),)]

TherelationTcanbereducedbynegationtoacaseanalogoustoS,sothatPropositionIVisproved.

[TheRelations146]

Thefunctionsx+y,x.y,x ,andalsotherelationsx1

xisaprimenumber.

0Prx0(n+1)Prxy[yx&Prim(y)&x/y&y>nPrx]

nPrxisthen (inorderofmagnitude)primenumbercontainedinx.

0!1(n+1)!(n+1).n!

Pr(0)0Pr(n+1)y[y{Pr(n)}!+1&Prim(y)&y>Pr(n)]

Pr(n)isthen primenumber(inorderofmagnitude).

nGlxy[yx&x/(nPrx) &~x/(nPrx) ]

nGlxisthen termoftheseriesofnumbersassignedtothenumberx(forn>0andnnotgreaterthanthelengthofthisseries).

l(x)y[yx&yPrx>0&(y+1)Prx=0]

l(x)isthelengthoftheseriesofnumbersassignedtox.

x*yz[z[Pr{l(x)+l(y)}] &(n)[nl(x)nGlz=nGlx] &(n)[00).

E(x)R(11)*x*R(13)

E(x)correspondstotheoperationof"bracketing"[11and13areassignedtothebasicsigns"("and")"].

nVarx(z)[131 &(v){vx&nVarv&x=R(v)}]

xisasignofn type.

Elf(x)(y,z,n)[y,z,nx &Typ (y)&Typ (z)&x=z*E(y)]

xisanelementaryformula.

Op(x,y,z)x=Neg(y)x=yDisz (v)[vx&Var(v)&x=vGeny]

FR(x)(n){0

(1)

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GdelsProofofIncompletenessEnglishTranslation

ThisisanEnglishtranslationofGdelsProofofIncompletenessandwhichisbasedonbasedonMeltzersEnglishtranslationoftheoriginalGerman

berformalunentscheidbareStzederPrincipiaMathematicaundverwandterSystemeI.

Note:Headingsinitalicsenclosedinsquarebracketsareadditionaltotheoriginaltext,theseareincludedforconvenience,e.g.,[Recursion]

Contents

Part1

Part2DescriptionoftheformalsystemPTheaxiomsofthesystemPTherulesofinferenceofthesystemPTheGdelnumberingsystemRecursion

PropositionsIIVTheRelations146

1 2 3 4 5 6 7 8 9 1011 12 13 14 15 16 17 18 19 2021 22 23 24 25 26 27 28 29 3031 32 33 34 35 36 37 38 39 4041 42 43 44 45 46

PropositionVPropositionVI

Part3PropositionVIIPropositionVIIIPropositionIXPropositionX

Part4PropositionXI

ONFORMALLYUNDECIDABLEPROPOSITIONSOFPRINCIPIAMATHEMATICAANDRELATEDSYSTEMS1

byKurtGdel,Vienna

1Thedevelopmentofmathematicsinthedirectionofgreaterexactnesshasasiswellknownledtolargetractsofitbecomingformalized,sothatproofscanbecarriedoutaccordingtoafewmechanicalrules.Themostcomprehensiveformalsystemsyetsetupare,ontheonehand,thesystemofPrincipiaMathematica(PM) and,ontheother,theaxiomsystemforsettheoryofZermeloFraenkel(laterextendedbyJ.v.Neumann). Thesetwosystemsaresoextensivethatallmethodsofproofusedinmathematicstodayhavebeenformalizedinthem,i.e.reducedtoafewaxiomsandrulesofinference.Itmaythereforebesurmisedthattheseaxiomsandrulesofinferencearealsosufficienttodecideallmathematicalquestionswhichcaninanywayatallbeexpressedformallyinthesystemsconcerned.Itisshownbelowthatthisisnotthecase,andthatinboththesystemsmentionedthereareinfactrelativelysimpleproblemsinthetheoryofordinarywholenumbers whichcannotbedecidedfromtheaxioms.Thissituationisnotdueinsomewaytothespecialnatureofthesystemssetup,butholdsforaveryextensiveclassofformalsystems,including,inparticular,allthosearisingfromtheadditionofafinitenumberofaxiomstothetwosystemsmentioned, providedthattherebynofalsepropositionsofthekinddescribedinfootnote4becomeprovable.

Beforegoingintodetails,weshallfirstindicatethemainlinesoftheproof,naturallywithoutlayingclaimtoexactness.TheformulaeofaformalsystemwerestrictourselvesheretothesystemPMare,lookedatfromoutside,finiteseriesofbasicsigns(variables,logicalconstantsandbracketsorseparationpoints),anditiseasytostatepreciselyjustwhichseriesofbasicsignsaremeaningfulformulaeandwhicharenot. Proofs,fromtheformalstandpoint,arelikewisenothingbutfiniteseriesofformulae(withcertainspecifiablecharacteristics).Formetamathematicalpurposesitisnaturallyimmaterialwhatobjectsaretakenasbasicsigns,andweproposetousenaturalnumbers forthem.Accordingly,then,aformulaisafiniteseriesofnaturalnumbers, andaparticularproofschemaisafiniteseriesoffiniteseriesofnaturalnumbers.Metamathematicalconceptsandpropositionstherebybecomeconceptsandpropositionsconcerningnaturalnumbers,orseriesofthem, andthereforeatleastpartiallyexpressibleinthesymbolsofthesystemPMitself.Inparticular,itcanbeshownthattheconcepts,"formula","proofschema","provableformula"aredefinableinthesystemPM,i.e.onecangive aformulaF(v)ofPMforexamplewithonefreevariablev(ofthetypeofaseriesofnumbers),suchthatF(v)interpretedastocontentstates:visaprovableformula.WenowobtainanundecidablepropositionofthesystemPM,i.e.apropositionA,forwhichneitherAnornotAareprovable,inthefollowingmanner:

AformulaofPMwithjustonefreevariable,andthatofthetypeofthenaturalnumbers(classofclasses),weshalldesignateaclasssign.Wethinkoftheclasssignsasbeingsomehowarrangedinaseries, anddenotethen onebyR(n)andwenotethattheconcept"classsign"aswellastheorderingrelationRaredefinableinthesystemPM.Letbeanyclasssignby[n]wedesignatethatformulawhichisderivedonreplacingthefreevariableintheclasssignbythesignforthenaturalnumbern.Thethreetermrelationx=[yz]alsoprovestobedefinableinPM.WenowdefineaclassKofnaturalnumbers,asfollows:

nK~(Bew[R(n)n])

(whereBewxmeans:xisaprovableformula).SincetheconceptswhichappearinthedefinitionsarealldefinableinPM,sotooistheconceptKwhichisconstitutedfromthem,i.e.thereisaclasssignS, suchthattheformula[Sn]interpretedastoitscontentstatesthatthenaturalnumbernbelongstoK.S,beingaclasssign,isidenticalwithsomedeterminateR(q),i.e.

S=R(q)

holdsforsomedeterminatenaturalnumberq.Wenowshowthattheproposition[R(q)q] isundecidableinPM.Forsupposingtheproposition[R(q)q]wereprovable,itwouldalsobecorrectbutthatmeans,ashasbeensaid,thatqwouldbelongtoK,i.e.accordingto(1),~(Bew[R(q)q])wouldholdgood,incontradictiontoourinitialassumption.If,onthecontrary,thenegationof[R(q)q]wereprovable,then~(nK),i.e.Bew[R(q)q]wouldholdgood.[R(q)q]wouldthusbeprovableatthesametimeasitsnegation,whichagainisimpossible.

TheanalogybetweenthisresultandRichardsantinomyleapstotheeyethereisalsoacloserelationshipwiththe"liar"antinomy, sincetheundecidableproposition[R(q)q]statespreciselythatqbelongstoK,i.e.accordingto(1),that[R(q)q]isnotprovable.Wearethereforeconfrontedwithapropositionwhichassertsitsownunprovability. Themethodofproofjustexhibitedcanclearlybeappliedtoeveryformalsystemhavingthefollowingfeatures:firstly,interpretedastocontent,itdisposesofsufficientmeansofexpressiontodefinetheconceptsoccurringintheaboveargument(inparticulartheconcept"provableformula")secondly,everyprovableformulainitisalsocorrectasregardscontent.Theexactstatementoftheaboveproof,whichnowfollows,willhaveamongothersthetaskofsubstitutingforthesecondoftheseassumptionsapurelyformalandmuchweakerone.

Fromtheremarkthat[R(q)q]assertsitsownunprovability,itfollowsatoncethat[R(q)q]iscorrect,since[R(q)q]iscertainlyunprovable(becauseundecidable).Sothepropositionwhichisundecidableinthesystem PMyetturnsouttobedecidedbymetamathematicalconsiderations.Thecloseanalysisofthisremarkablecircumstanceleadstosurprisingresultsconcerningproofsofconsistencyofformalsystems,whicharedealtwithinmoredetailinSection4(PropositionXI).

Cf.thesummaryoftheresultsofthiswork,publishedinAnzeigerderAkad.d.Wiss.inWien(math.naturw.Kl.)1930,No.19.A.WhiteheadandB.Russell,PrincipiaMathematica,2ndedition,Cambridge1925.Inparticular,wealsoreckonamongtheaxiomsofPMtheaxiomofinfinity(intheform:thereexistdenumerablymanyindividuals),andtheaxiomsofreducibilityandofchoice(foralltypes).Cf.A.Fraenkel,'ZehnVorlesungenberdieGrundlegungderMengenlehre',Wissensch.u.Hyp.,Vol.XXXIJ.v.Neumann,'DieAxiomatisierungderMengenlehre',Math.Zeitschr.27,1928,Journ.f.reineu.angew.Math.154(1925),160(1929).Wemaynotethatinordertocompletetheformalization,theaxiomsandrulesofinferenceofthelogicalcalculusmustbeaddedtotheaxiomsofsettheorygivenintheabovementionedpapers.TheremarksthatfollowalsoapplytotheformalsystemspresentedinrecentyearsbyD.Hilbertandhiscolleagues(sofarasthesehaveyetbeenpublished).Cf.D.Hilbert,Math.Ann.88,Abh.ausd.math.Sem.derUniv.HamburgI(1922),VI(1928)P.Bernays,Math.Ann.90J.v.Neumann,Math.Zeitsehr.26(1927)W.Ackermann,Math.Ann.93.I.e.,moreprecisely,thereareundecidablepropositionsinwhich,besidesthelogicalconstants~(not),(or),(x)(forall)and=(identicalwith),therearenootherconceptsbeyond+(addition)and.(multiplication),bothreferredtonaturalnumbers,andwheretheprefixes(x)canalsoreferonlytonaturalnumbers.Inthisconnection,onlysuchaxiomsinPMarecountedasdistinctasdonotarisefromeachotherpurelybychangeoftype.Hereandinwhatfollows,weshallalwaysunderstandtheterm"formulaofPM"tomeanaformulawrittenwithoutabbreviations(i.e.withoutuseofdefinitions).Definitionsserveonlytoabridgethewrittentextandarethereforeinprinciplesuperfluous.I.e.wemapthebasicsignsinonetoonefashiononthenaturalnumbers(asisactuallydoneon).I.e.acoveringofasectionofthenumberseriesbynaturalnumbers.(Numberscannotinfactbeputintoaspatialorder.)Inotherwords,theabovedescribedprocedureprovidesanisomorphicimageofthesystemPMinthedomainofarithmetic,andallmetamathematicalargumentscanequallywellbeconductedinthisisomorphicimage.Thisoccursinthefollowingoutlineproof,i.e."formula","proposition","variable",etc.arealwaystobeunderstoodasthecorrespondingobjectsoftheisomorphicimage.Itwouldbeverysimple(thoughratherlaborious)actuallytowriteoutthisformula.Perhapsaccordingtotheincreasingsumsoftheirtermsand,forequalsums,inalphabeticalorder.Thebarsignindicatesnegation.[Replacedwith~.]AgainthereisnottheslightestdifficultyinactuallywritingouttheformulaS.Notethat"[R(q)q]"(orwhatcomestothesamething"[Sq]")ismerelyametamathematicaldescriptionoftheundecidableproposition.ButassoonasonehasascertainedtheformulaS,onecannaturallyalsodeterminethenumberq,andtherebyeffectivelywriteouttheundecidablepropositionitself.Everyepistemologicalantinomycanlikewisebeusedforasimilarundecidabilityproof.Inspiteofappearances,thereisnothingcircularaboutsuchaproposition,sinceitbeginsbyassertingtheunprovabilityofawhollydeterminateformula(namelytheq inthealphabeticalarrangementwithadefinitesubstitution),andonlysubsequently(andinsomewaybyaccident)doesitemergethatthisformulaispreciselythatbywhichthepropositionwasitselfexpressed.

2

[DescriptionoftheformalsystemP]

Weproceednowtotherigorousdevelopmentoftheproofsketchedabove,andbeginbygivinganexactdescriptionoftheformalsystemP,forwhichweseektodemonstratetheexistenceofundecidablepropositions.PisessentiallythesystemobtainedbysuperimposingonthePeanoaxiomsthelogicofPM (numbersasindividuals,relationofsuccessorasundefinedbasicconcept).

ThebasicsignsofthesystemParethefollowing:

I. Constants:"~"(not),""(or),""(forall),"0"(nought),"f"(thesuccessorof),"(",")"(brackets).

II. Variablesoffirsttype(forindividuals,i.e.naturalnumbersincluding0):"x ","y ","z ",Variablesofsecondtype(forclassesofindividuals):"x ","y ","z ",Variablesofthirdtype(forclassesofclassesofindividuals):"x ","y ","z ",

andsoonforeverynaturalnumberastype.

Note:Variablesfortwotermedandmanytermedfunctions(relations)aresuperfluousasbasicsigns,sincerelationscanbedefinedasclassesoforderedpairsandorderedpairsagainasclassesofclasses,e.g.theorderedpaira,bby((a),(a,b)),where(x,y)meanstheclasswhoseonlyelementsarexandy,and(x)theclasswhoseonlyelementisx.

Byasignoffirsttypeweunderstandacombinationofsignsoftheform:

a,fa,ffa,fffaetc.

whereaiseither0oravariableoffirsttype.Intheformercasewecallsuchasignanumbersign.Forn>1weunderstandbyasignofn typethesameasvariableofn type.

Combinationsofsignsoftheforma(b),wherebisasignofn andaasignof(n+1) type,wecallelementaryformulae.Theclassofformulaewedefineasthesmallestclass containingallelementaryformulaeand,also,alongwithanyaandbthefollowing:~(a),(a)(b),x(a)(wherexisanygivenvariable). Weterm(a)(b)thedisjunctionofaandb,~(a)thenegationand(a)(b)ageneralizationofa.Aformulainwhichthereisnofreevariableiscalledapropositionalformula(freevariablebeingdefinedintheusualway).Aformulawithjustnfreeindividualvariables(andotherwisenofreevariables)wecallannplacerelationsignandforn=1alsoaclasssign.

BySubsta(v|b)(whereastandsforaformula,vavariableandbasignofthesametypeasv)weunderstandtheformuladerivedfroma,whenwereplacevinit,whereveritisfree,byb. Wesaythataformulaaisatypeliftofanotheroneb,ifaderivesfromb,whenweincreasebythesameamountthetypeofallvariablesappearinginb.

[AxiomsoftheformalsystemP]

Thefollowingformulae(IV)arecalledaxioms(theyaresetoutwiththehelpofthecustomarilydefinedabbreviations:.,,,(x),= andsubjecttotheusualconventionsaboutomissionofbrackets):

I.1. ~(fx =0)2. fx =fy x =y3. x (0).x (x (x )x (fx ))x (x (x ))

II.Everyformuladerivedfromthefollowingschematabysubstitutionofanyformulaeforp,qandr.

1. ppp2. ppq3. pqqp4. (pq)(rprq)

III.Everyformuladerivedfromthetwoschemata

1.v(a)Substa(v|c)2. v(ba)bv(a)

bymakingthefollowingsubstitutionsfora,v,b,c(andcarryingoutinItheoperationdenotedby"Subst"):foraanygivenformula,forvanyvariable,forbanyformulainwhichvdoesnotappearfree,forcasignofthesametypeasv,providedthatccontainsnovariablewhichisboundinaataplacewherevisfree.

IV.Everyformuladerivedfromtheschema

1. (u)(v(u(v)a))

onsubstitutingforvoruanyvariablesoftypesnorn+1respectively,andforaaformulawhichdoesnotcontainufree.Thisaxiomrepresentstheaxiomofreducibility(theaxiomofcomprehensionofsettheory).

V.Everyformuladerivedfromthefollowingbytypelift(andthisformulaitself):

1. x (x (x )y (x ))x =y

Thisaxiomstatesthataclassiscompletelydeterminedbyitselements.

[RulesofinferenceoftheformalsystemP]

Aformulaciscalledanimmediateconsequenceofaandb,ifaistheformula(~(b))(c),andanimmediateconsequenceofa,ifcistheformulav(a),wherevdenotesanygivenvariable.Theclassofprovableformulaeisdefinedasthesmallestclassofformulaewhichcontainstheaxiomsandisclosedwithrespecttotherelation"immediateconsequenceof".

[TheGdelnumberingsystem]

ThebasicsignsofthesystemParenoworderedinonetoonecorrespondencewithnaturalnumbers,asfollows:

0"1"f"3"~"5""7""9"("11")"13

Furthermore,variablesoftypenaregivennumbersoftheformp (wherepisaprimenumber>13).Hence,toeveryfiniteseriesofbasicsigns(andsoalsotoeveryformula)therecorresponds,onetoone,afiniteseriesofnaturalnumbers.Thesefiniteseriesofnaturalnumberswenowmap(againinonetoonecorrespondence)ontonaturalnumbers,bylettingthenumber2 ,3 p correspondtotheseriesn ,n ,n ,wherep denotesthek primenumberinorderofmagnitude.Anaturalnumberistherebyassignedinonetoonecorrespondence,notonlytoeverybasicsign,butalsotoeveryfiniteseriesofsuchsigns.Wedenoteby(a)thenumbercorrespondingtothebasicsignorseriesofbasicsignsa.SupposenowoneisgivenaclassorrelationR(a ,a ,a )ofbasicsignsorseriesofsuch.Weassigntoitthatclass(orrelation)R'(x ,x ,x )ofnaturalnumbers,whichholdsforx ,x ,x whenandonlywhenthereexista ,a ,a suchthatx =(a )(i=1,2,n)andR(a ,a ,a )holds.Werepresentbythesamewordsinitalicsthoseclassesandrelationsofnaturalnumberswhichhavebeenassignedinthisfashiontosuchpreviouslydefinedmetamathematicalconceptsas"variable","formula","propositionalformula","axiom","provableformula",etc.ThepropositionthatthereareundecidableproblemsinthesystemPwouldthereforeread,forexample,asfollows:Thereexistpropositionalformulaeasuchthatneitheranorthenegationofaareprovableformulae.

[Recursion]

WenowintroduceaparentheticconsiderationhavingnoimmediateconnectionwiththeformalsystemP,andfirstputforwardthefollowingdefinition:Anumbertheoreticfunction (x ,x ,x )issaidtoberecursivelydefinedbythenumbertheoreticfunctions(x ,x ,x )and(x ,x ,x ),ifforallx ,x ,k thefollowinghold:

(0,x ,x )=(x ,x )(k+1,x ,x )=(k,(k,x ,x ),x ,x ).

Anumbertheoreticfunctioniscalledrecursive,ifthereexistsafiniteseriesofnumbertheoreticfunctions , , whichendsinandhasthepropertythateveryfunction oftheseriesiseitherrecursivelydefinedbytwooftheearlierones,orisderivedfromanyoftheearlieronesbysubstitution, or,finally,isaconstantorthesuccessorfunctionx+1.Thelengthoftheshortestseriesof ,whichbelongstoarecursivefunction,istermeditsdegree.ArelationR(x ,x ,x )amongnaturalnumbersiscalledrecursive, ifthereexistsarecursivefunction(x ,x ,x )suchthatforallx ,x ,x

R(x ,x ,x )[(x ,x ,x )=0] .

[PropositionsIIV]

Thefollowingpropositionshold:

I.Everyfunction(orrelation)derivedfromrecursivefunctions(orrelations)bythesubstitutionofrecursivefunctionsinplaceofvariablesisrecursivesoalsoiseveryfunctionderivedfromrecursivefunctionsbyrecursivedefinitionaccordingtoschema(2).

II.IfRandSarerecursiverelations,thensoalsoare~R,RS(andthereforealsoR&S).

III.Ifthefunctions()and()arerecursive,soalsoistherelation:()=().

IV.Ifthefunction()andtherelationR(x,)arerecursive,soalsothenaretherelationsS,T

S(,)~(x)[x()&R(x,)]T(,)~(x)[x()R(x,)]

andlikewisethefunction

(,)=x[x()&R(x,)]

wherexF(x)means:thesmallestnumberxforwhichF(x)holdsand0ifthereisnosuchnumber.

PropositionIfollowsimmediatelyfromthedefinitionof"recursive".PropositionsIIandIIIarebasedonthereadilyascertainablefactthatthenumbertheoreticfunctionscorrespondingtothelogicalconcepts~,,=

(x),(x,y),(x,y)namely

(0)=1(x)=0forx0(0,x)=(x,0)=0(x,y)=1,ifx,ybothO

(x,y)=0,ifx=y(x,y)=1,ifxy

arerecursive.TheproofofPropositionIVisbrieflyasfollows:Accordingtotheassumptionthereexistsarecursive(x,)suchthat

R(x,)[(x,)=0].

Wenowdefine,accordingtotherecursionschema(2),afunction(x,)inthefollowingmanner:

(0,)=0(n+1,)=(n+1).a+(n,).(a)

where

a=[((0,))].[(n+1,)].[(n,)].

(n+1,)isthereforeeither=n+1(ifa=1)or=(n,)(ifa=0). Thefirstcaseclearlyarisesifandonlyifalltheconstituentfactorsofaare1,i.e.if

~R(O,)&R(n+1,)&[(n,)=0].

Fromthisitfollowsthatthefunction(n,)(consideredasafunctionofn)remains0uptothesmallestvalueofnforwhichR(n,)holds,andfromthenonisequaltothisvalue(ifR(0,)isalreadythecase,thecorresponding(x,)isconstantand=0).Therefore:

(,)=C((),)S(,)R[(,)),)]

TherelationTcanbereducedbynegationtoacaseanalogoustoS,sothatPropositionIVisproved.

[TheRelations146]

Thefunctionsx+y,x.y,x ,andalsotherelationsx1

xisaprimenumber.

0Prx0(n+1)Prxy[yx&Prim(y)&x/y&y>nPrx]

nPrxisthen (inorderofmagnitude)primenumbercontainedinx.

0!1(n+1)!(n+1).n!

Pr(0)0Pr(n+1)y[y{Pr(n)}!+1&Prim(y)&y>Pr(n)]

Pr(n)isthen primenumber(inorderofmagnitude).

nGlxy[yx&x/(nPrx) &~x/(nPrx) ]

nGlxisthen termoftheseriesofnumbersassignedtothenumberx(forn>0andnnotgreaterthanthelengthofthisseries).

l(x)y[yx&yPrx>0&(y+1)Prx=0]

l(x)isthelengthoftheseriesofnumbersassignedtox.

x*yz[z[Pr{l(x)+l(y)}] &(n)[nl(x)nGlz=nGlx] &(n)[00).

E(x)R(11)*x*R(13)

E(x)correspondstotheoperationof"bracketing"[11and13areassignedtothebasicsigns"("and")"].

nVarx(z)[131 &(v){vx&nVarv&x=R(v)}]

xisasignofn type.

Elf(x)(y,z,n)[y,z,nx &Typ (y)&Typ (z)&x=z*E(y)]

xisanelementaryformula.

Op(x,y,z)x=Neg(y)x=yDisz (v)[vx&Var(v)&x=vGeny]

FR(x)(n){0

(1)

1

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29

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363738

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GdelsProofofIncompletenessEnglishTranslation

ThisisanEnglishtranslationofGdelsProofofIncompletenessandwhichisbasedonbasedonMeltzersEnglishtranslationoftheoriginalGerman

berformalunentscheidbareStzederPrincipiaMathematicaundverwandterSystemeI.

Note:Headingsinitalicsenclosedinsquarebracketsareadditionaltotheoriginaltext,theseareincludedforconvenience,e.g.,[Recursion]

Contents

Part1

Part2DescriptionoftheformalsystemPTheaxiomsofthesystemPTherulesofinferenceofthesystemPTheGdelnumberingsystemRecursion

PropositionsIIVTheRelations146

1 2 3 4 5 6 7 8 9 1011 12 13 14 15 16 17 18 19 2021 22 23 24 25 26 27 28 29 3031 32 33 34 35 36 37 38 39 4041 42 43 44 45 46

PropositionVPropositionVI

Part3PropositionVIIPropositionVIIIPropositionIXPropositionX

Part4PropositionXI

ONFORMALLYUNDECIDABLEPROPOSITIONSOFPRINCIPIAMATHEMATICAANDRELATEDSYSTEMS1

byKurtGdel,Vienna

1Thedevelopmentofmathematicsinthedirectionofgreaterexactnesshasasiswellknownledtolargetractsofitbecomingformalized,sothatproofscanbecarriedoutaccordingtoafewmechanicalrules.Themostcomprehensiveformalsystemsyetsetupare,ontheonehand,thesystemofPrincipiaMathematica(PM) and,ontheother,theaxiomsystemforsettheoryofZermeloFraenkel(laterextendedbyJ.v.Neumann). Thesetwosystemsaresoextensivethatallmethodsofproofusedinmathematicstodayhavebeenformalizedinthem,i.e.reducedtoafewaxiomsandrulesofinference.Itmaythereforebesurmisedthattheseaxiomsandrulesofinferencearealsosufficienttodecideallmathematicalquestionswhichcaninanywayatallbeexpressedformallyinthesystemsconcerned.Itisshownbelowthatthisisnotthecase,andthatinboththesystemsmentionedthereareinfactrelativelysimpleproblemsinthetheoryofordinarywholenumbers whichcannotbedecidedfromtheaxioms.Thissituationisnotdueinsomewaytothespecialnatureofthesystemssetup,butholdsforaveryextensiveclassofformalsystems,including,inparticular,allthosearisingfromtheadditionofafinitenumberofaxiomstothetwosystemsmentioned, providedthattherebynofalsepropositionsofthekinddescribedinfootnote4becomeprovable.

Beforegoingintodetails,weshallfirstindicatethemainlinesoftheproof,naturallywithoutlayingclaimtoexactness.TheformulaeofaformalsystemwerestrictourselvesheretothesystemPMare,lookedatfromoutside,finiteseriesofbasicsigns(variables,logicalconstantsandbracketsorseparationpoints),anditiseasytostatepreciselyjustwhichseriesofbasicsignsaremeaningfulformulaeandwhicharenot. Proofs,fromtheformalstandpoint,arelikewisenothingbutfiniteseriesofformulae(withcertainspecifiablecharacteristics).Formetamathematicalpurposesitisnaturallyimmaterialwhatobjectsaretakenasbasicsigns,andweproposetousenaturalnumbers forthem.Accordingly,then,aformulaisafiniteseriesofnaturalnumbers, andaparticularproofschemaisafiniteseriesoffiniteseriesofnaturalnumbers.Metamathematicalconceptsandpropositionstherebybecomeconceptsandpropositionsconcerningnaturalnumbers,orseriesofthem, andthereforeatleastpartiallyexpressibleinthesymbolsofthesystemPMitself.Inparticular,itcanbeshownthattheconcepts,"formula","proofschema","provableformula"aredefinableinthesystemPM,i.e.onecangive aformulaF(v)ofPMforexamplewithonefreevariablev(ofthetypeofaseriesofnumbers),suchthatF(v)interpretedastocontentstates:visaprovableformula.WenowobtainanundecidablepropositionofthesystemPM,i.e.apropositionA,forwhichneitherAnornotAareprovable,inthefollowingmanner:

AformulaofPMwithjustonefreevariable,andthatofthetypeofthenaturalnumbers(classofclasses),weshalldesignateaclasssign.Wethinkoftheclasssignsasbeingsomehowarrangedinaseries, anddenotethen onebyR(n)andwenotethattheconcept"classsign"aswellastheorderingrelationRaredefinableinthesystemPM.Letbeanyclasssignby[n]wedesignatethatformulawhichisderivedonreplacingthefreevariableintheclasssignbythesignforthenaturalnumbern.Thethreetermrelationx=[yz]alsoprovestobedefinableinPM.WenowdefineaclassKofnaturalnumbers,asfollows:

nK~(Bew[R(n)n])

(whereBewxmeans:xisaprovableformula).SincetheconceptswhichappearinthedefinitionsarealldefinableinPM,sotooistheconceptKwhichisconstitutedfromthem,i.e.thereisaclasssignS, suchthattheformula[Sn]interpretedastoitscontentstatesthatthenaturalnumbernbelongstoK.S,beingaclasssign,isidenticalwithsomedeterminateR(q),i.e.

S=R(q)

holdsforsomedeterminatenaturalnumberq.Wenowshowthattheproposition[R(q)q] isundecidableinPM.Forsupposingtheproposition[R(q)q]wereprovable,itwouldalsobecorrectbutthatmeans,ashasbeensaid,thatqwouldbelongtoK,i.e.accordingto(1),~(Bew[R(q)q])wouldholdgood,incontradictiontoourinitialassumption.If,onthecontrary,thenegationof[R(q)q]wereprovable,then~(nK),i.e.Bew[R(q)q]wouldholdgood.[R(q)q]wouldthusbeprovableatthesametimeasitsnegation,whichagainisimpossible.

TheanalogybetweenthisresultandRichardsantinomyleapstotheeyethereisalsoacloserelationshipwiththe"liar"antinomy, sincetheundecidableproposition[R(q)q]statespreciselythatqbelongstoK,i.e.accordingto(1),that[R(q)q]isnotprovable.Wearethereforeconfrontedwithapropositionwhichassertsitsownunprovability. Themethodofproofjustexhibitedcanclearlybeappliedtoeveryformalsystemhavingthefollowingfeatures:firstly,interpretedastocontent,itdisposesofsufficientmeansofexpressiontodefinetheconceptsoccurringintheaboveargument(inparticulartheconcept"provableformula")secondly,everyprovableformulainitisalsocorrectasregardscontent.Theexactstatementoftheaboveproof,whichnowfollows,willhaveamongothersthetaskofsubstitutingforthesecondoftheseassumptionsapurelyformalandmuchweakerone.

Fromtheremarkthat[R(q)q]assertsitsownunprovability,itfollowsatoncethat[R(q)q]iscorrect,since[R(q)q]iscertainlyunprovable(becauseundecidable).Sothepropositionwhichisundecidableinthesystem PMyetturnsouttobedecidedbymetamathematicalconsiderations.Thecloseanalysisofthisremarkablecircumstanceleadstosurprisingresultsconcerningproofsofconsistencyofformalsystems,whicharedealtwithinmoredetailinSection4(PropositionXI).

Cf.thesummaryoftheresultsofthiswork,publishedinAnzeigerderAkad.d.Wiss.inWien(math.naturw.Kl.)1930,No.19.A.WhiteheadandB.Russell,PrincipiaMathematica,2ndedition,Cambridge1925.Inparticular,wealsoreckonamongtheaxiomsofPMtheaxiomofinfinity(intheform:thereexistdenumerablymanyindividuals),andtheaxiomsofreducibilityandofchoice(foralltypes).Cf.A.Fraenkel,'ZehnVorlesungenberdieGrundlegungderMengenlehre',Wissensch.u.Hyp.,Vol.XXXIJ.v.Neumann,'DieAxiomatisierungderMengenlehre',Math.Zeitschr.27,1928,Journ.f.reineu.angew.Math.154(1925),160(1929).Wemaynotethatinordertocompletetheformalization,theaxiomsandrulesofinferenceofthelogicalcalculusmustbeaddedtotheaxiomsofsettheorygivenintheabovementionedpapers.TheremarksthatfollowalsoapplytotheformalsystemspresentedinrecentyearsbyD.Hilbertandhiscolleagues(sofarasthesehaveyetbeenpublished).Cf.D.Hilbert,Math.Ann.88,Abh.ausd.math.Sem.derUniv.HamburgI(1922),VI(1928)P.Bernays,Math.Ann.90J.v.Neumann,Math.Zeitsehr.26(1927)W.Ackermann,Math.Ann.93.I.e.,moreprecisely,thereareundecidablepropositionsinwhich,besidesthelogicalconstants~(not),(or),(x)(forall)and=(identicalwith),therearenootherconceptsbeyond+(addition)and.(multiplication),bothreferredtonaturalnumbers,andwheretheprefixes(x)canalsoreferonlytonaturalnumbers.Inthisconnection,onlysuchaxiomsinPMarecountedasdistinctasdonotarisefromeachotherpurelybychangeoftype.Hereandinwhatfollows,weshallalwaysunderstandtheterm"formulaofPM"tomeanaformulawrittenwithoutabbreviations(i.e.withoutuseofdefinitions).Definitionsserveonlytoabridgethewrittentextandarethereforeinprinciplesuperfluous.I.e.wemapthebasicsignsinonetoonefashiononthenaturalnumbers(asisactuallydoneon).I.e.acoveringofasectionofthenumberseriesbynaturalnumbers.(Numberscannotinfactbeputintoaspatialorder.)Inotherwords,theabovedescribedprocedureprovidesanisomorphicimageofthesystemPMinthedomainofarithmetic,andallmetamathematicalargumentscanequallywellbeconductedinthisisomorphicimage.Thisoccursinthefollowingoutlineproof,i.e."formula","proposition","variable",etc.arealwaystobeunderstoodasthecorrespondingobjectsoftheisomorphicimage.Itwouldbeverysimple(thoughratherlaborious)actuallytowriteoutthisformula.Perhapsaccordingtotheincreasingsumsoftheirtermsand,forequalsums,inalphabeticalorder.Thebarsignindicatesnegation.[Replacedwith~.]AgainthereisnottheslightestdifficultyinactuallywritingouttheformulaS.Notethat"[R(q)q]"(orwhatcomestothesamething"[Sq]")ismerelyametamathematicaldescriptionoftheundecidableproposition.ButassoonasonehasascertainedtheformulaS,onecannaturallyalsodeterminethenumberq,andtherebyeffectivelywriteouttheundecidablepropositionitself.Everyepistemologicalantinomycanlikewisebeusedforasimilarundecidabilityproof.Inspiteofappearances,thereisnothingcircularaboutsuchaproposition,sinceitbeginsbyassertingtheunprovabilityofawhollydeterminateformula(namelytheq inthealphabeticalarrangementwithadefinitesubstitution),andonlysubsequently(andinsomewaybyaccident)doesitemergethatthisformulaispreciselythatbywhichthepropositionwasitselfexpressed.

2

[DescriptionoftheformalsystemP]

Weproceednowtotherigorousdevelopmentoftheproofsketchedabove,andbeginbygivinganexactdescriptionoftheformalsystemP,forwhichweseektodemonstratetheexistenceofundecidablepropositions.PisessentiallythesystemobtainedbysuperimposingonthePeanoaxiomsthelogicofPM (numbersasindividuals,relationofsuccessorasundefinedbasicconcept).

ThebasicsignsofthesystemParethefollowing:

I. Constants:"~"(not),""(or),""(forall),"0"(nought),"f"(thesuccessorof),"(",")"(brackets).

II. Variablesoffirsttype(forindividuals,i.e.naturalnumbersincluding0):"x ","y ","z ",Variablesofsecondtype(forclassesofindividuals):"x ","y ","z ",Variablesofthirdtype(forclassesofclassesofindividuals):"x ","y ","z ",

andsoonforeverynaturalnumberastype.

Note:Variablesfortwotermedandmanytermedfunctions(relations)aresuperfluousasbasicsigns,sincerelationscanbedefinedasclassesoforderedpairsandorderedpairsagainasclassesofclasses,e.g.theorderedpaira,bby((a),(a,b)),where(x,y)meanstheclasswhoseonlyelementsarexandy,and(x)theclasswhoseonlyelementisx.

Byasignoffirsttypeweunderstandacombinationofsignsoftheform:

a,fa,ffa,fffaetc.

whereaiseither0oravariableoffirsttype.Intheformercasewecallsuchasignanumbersign.Forn>1weunderstandbyasignofn typethesameasvariableofn type.

Combinationsofsignsoftheforma(b),wherebisasignofn andaasignof(n+1) type,wecallelementaryformulae.Theclassofformulaewedefineasthesmallestclass containingallelementaryformulaeand,also,alongwithanyaandbthefollowing:~(a),(a)(b),x(a)(wherexisanygivenvariable). Weterm(a)(b)thedisjunctionofaandb,~(a)thenegationand(a)(b)ageneralizationofa.Aformulainwhichthereisnofreevariableiscalledapropositionalformula(freevariablebeingdefinedintheusualway).Aformulawithjustnfreeindividualvariables(andotherwisenofreevariables)wecallannplacerelationsignandforn=1alsoaclasssign.

BySubsta(v|b)(whereastandsforaformula,vavariableandbasignofthesametypeasv)weunderstandtheformuladerivedfroma,whenwereplacevinit,whereveritisfree,byb. Wesaythataformulaaisatypeliftofanotheroneb,ifaderivesfromb,whenweincreasebythesameamountthetypeofallvariablesappearinginb.

[AxiomsoftheformalsystemP]

Thefollowingformulae(IV)arecalledaxioms(theyaresetoutwiththehelpofthecustomarilydefinedabbreviations:.,,,(x),= andsubjecttotheusualconventionsaboutomissionofbrackets):

I.1. ~(fx =0)2. fx =fy x =y3. x (0).x (x (x )x (fx ))x (x (x ))

II.Everyformuladerivedfromthefollowingschematabysubstitutionofanyformulaeforp,qandr.

1. ppp2. ppq3. pqqp4. (pq)(rprq)

III.Everyformuladerivedfromthetwoschemata

1.v(a)Substa(v|c)2. v(ba)bv(a)

bymakingthefollowingsubstitutionsfora,v,b,c(andcarryingoutinItheoperationdenotedby"Subst"):foraanygivenformula,forvanyvariable,forbanyformulainwhichvdoesnotappearfree,forcasignofthesametypeasv,providedthatccontainsnovariablewhichisboundinaataplacewherevisfree.

IV.Everyformuladerivedfromtheschema

1. (u)(v(u(v)a))

onsubstitutingforvoruanyvariablesoftypesnorn+1respectively,andforaaformulawhichdoesnotcontainufree.Thisaxiomrepresentstheaxiomofreducibility(theaxiomofcomprehensionofsettheory).

V.Everyformuladerivedfromthefollowingbytypelift(andthisformulaitself):

1. x (x (x )y (x ))x =y

Thisaxiomstatesthataclassiscompletelydeterminedbyitselements.

[RulesofinferenceoftheformalsystemP]

Aformulaciscalledanimmediateconsequenceofaandb,ifaistheformula(~(b))(c),andanimmediateconsequenceofa,ifcistheformulav(a),wherevdenotesanygivenvariable.Theclassofprovableformulaeisdefinedasthesmallestclassofformulaewhichcontainstheaxiomsandisclosedwithrespecttotherelation"immediateconsequenceof".

[TheGdelnumberingsystem]

ThebasicsignsofthesystemParenoworderedinonetoonecorrespondencewithnaturalnumbers,asfollows:

0"1"f"3"~"5""7""9"("11")"13

Furthermore,variablesoftypenaregivennumbersoftheformp (wherepisaprimenumber>13).Hence,toeveryfiniteseriesofbasicsigns(andsoalsotoeveryformula)therecorresponds,onetoone,afiniteseriesofnaturalnumbers.Thesefiniteseriesofnaturalnumberswenowmap(againinonetoonecorrespondence)ontonaturalnumbers,bylettingthenumber2 ,3 p correspondtotheseriesn ,n ,n ,wherep denotesthek primenumberinorderofmagnitude.Anaturalnumberistherebyassignedinonetoonecorrespondence,notonlytoeverybasicsign,butalsotoeveryfiniteseriesofsuchsigns.Wedenoteby(a)thenumbercorrespondingtothebasicsignorseriesofbasicsignsa.SupposenowoneisgivenaclassorrelationR(a ,a ,a )ofbasicsignsorseriesofsuch.Weassigntoitthatclass(orrelation)R'(x ,x ,x )ofnaturalnumbers,whichholdsforx ,x ,x whenandonlywhenthereexista ,a ,a suchthatx =(a )(i=1,2,n)andR(a ,a ,a )holds.Werepresentbythesamewordsinitalicsthoseclassesandrelationsofnaturalnumberswhichhavebeenassignedinthisfashiontosuchpreviouslydefinedmetamathematicalconceptsas"variable","formula","propositionalformula","axiom","provableformula",etc.ThepropositionthatthereareundecidableproblemsinthesystemPwouldthereforeread,forexample,asfollows:Thereexistpropositionalformulaeasuchthatneitheranorthenegationofaareprovableformulae.

[Recursion]

WenowintroduceaparentheticconsiderationhavingnoimmediateconnectionwiththeformalsystemP,andfirstputforwardthefollowingdefinition:Anumbertheoreticfunction (x ,x ,x )issaidtoberecursivelydefinedbythenumbertheoreticfunctions(x ,x ,x )and(x ,x ,x ),ifforallx ,x ,k thefollowinghold:

(0,x ,x )=(x ,x )(k+1,x ,x )=(k,(k,x ,x ),x ,x ).

Anumbertheoreticfunctioniscalledrecursive,ifthereexistsafiniteseriesofnumbertheoreticfunctions , , whichendsinandhasthepropertythateveryfunction oftheseriesiseitherrecursivelydefinedbytwooftheearlierones,orisderivedfromanyoftheearlieronesbysubstitution, or,finally,isaconstantorthesuccessorfunctionx+1.Thelengthoftheshortestseriesof ,whichbelongstoarecursivefunction,istermeditsdegree.ArelationR(x ,x ,x )amongnaturalnumbersiscalledrecursive, ifthereexistsarecursivefunction(x ,x ,x )suchthatforallx ,x ,x

R(x ,x ,x )[(x ,x ,x )=0] .

[PropositionsIIV]

Thefollowingpropositionshold:

I.Everyfunction(orrelation)derivedfromrecursivefunctions(orrelations)bythesubstitutionofrecursivefunctionsinplaceofvariablesisrecursivesoalsoiseveryfunctionderivedfromrecursivefunctionsbyrecursivedefinitionaccordingtoschema(2).

II.IfRandSarerecursiverelations,thensoalsoare~R,RS(andthereforealsoR&S).

III.Ifthefunctions()and()arerecursive,soalsoistherelation:()=().

IV.Ifthefunction()andtherelationR(x,)arerecursive,soalsothenaretherelationsS,T

S(,)~(x)[x()&R(x,)]T(,)~(x)[x()R(x,)]

andlikewisethefunction

(,)=x[x()&R(x,)]

wherexF(x)means:thesmallestnumberxforwhichF(x)holdsand0ifthereisnosuchnumber.

PropositionIfollowsimmediatelyfromthedefinitionof"recursive".PropositionsIIandIIIarebasedonthereadilyascertainablefactthatthenumbertheoreticfunctionscorrespondingtothelogicalconcepts~,,=

(x),(x,y),(x,y)namely

(0)=1(x)=0forx0(0,x)=(x,0)=0(x,y)=1,ifx,ybothO

(x,y)=0,ifx=y(x,y)=1,ifxy

arerecursive.TheproofofPropositionIVisbrieflyasfollows:Accordingtotheassumptionthereexistsarecursive(x,)suchthat

R(x,)[(x,)=0].

Wenowdefine,accordingtotherecursionschema(2),afunction(x,)inthefollowingmanner:

(0,)=0(n+1,)=(n+1).a+(n,).(a)

where

a=[((0,))].[(n+1,)].[(n,)].

(n+1,)isthereforeeither=n+1(ifa=1)or=(n,)(ifa=0). Thefirstcaseclearlyarisesifandonlyifalltheconstituentfactorsofaare1,i.e.if

~R(O,)&R(n+1,)&[(n,)=0].

Fromthisitfollowsthatthefunction(n,)(consideredasafunctionofn)remains0uptothesmallestvalueofnforwhichR(n,)holds,andfromthenonisequaltothisvalue(ifR(0,)isalreadythecase,thecorresponding(x,)isconstantand=0).Therefore:

(,)=C((),)S(,)R[(,)),)]

TherelationTcanbereducedbynegationtoacaseanalogoustoS,sothatPropositionIVisproved.

[TheRelations146]

Thefunctionsx+y,x.y,x ,andalsotherelationsx1

xisaprimenumber.

0Prx0(n+1)Prxy[yx&Prim(y)&x/y&y>nPrx]

nPrxisthen (inorderofmagnitude)primenumbercontainedinx.

0!1(n+1)!(n+1).n!

Pr(0)0Pr(n+1)y[y{Pr(n)}!+1&Prim(y)&y>Pr(n)]

Pr(n)isthen primenumber(inorderofmagnitude).

nGlxy[yx&x/(nPrx) &~x/(nPrx) ]

nGlxisthen termoftheseriesofnumbersassignedtothenumberx(forn>0andnnotgreaterthanthelengthofthisseries).

l(x)y[yx&yPrx>0&(y+1)Prx=0]

l(x)isthelengthoftheseriesofnumbersassignedtox.

x*yz[z[Pr{l(x)+l(y)}] &(n)[nl(x)nGlz=nGlx] &(n)[00).

E(x)R(11)*x*R(13)

E(x)correspondstotheoperationof"bracketing"[11and13areassignedtothebasicsigns"("and")"].

nVarx(z)[131 &(v){vx&nVarv&x=R(v)}]

xisasignofn type.

Elf(x)(y,z,n)[y,z,nx &Typ (y)&Typ (z)&x=z*E(y)]

xisanelementaryformula.

Op(x,y,z)x=Neg(y)x=yDisz (v)[vx&Var(v)&x=vGeny]

FR(x)(n){0

(1)

1

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16

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34

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363738

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(17)

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(21)(22)

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GdelsProofofIncompletenessEnglishTranslation

ThisisanEnglishtranslationofGdelsProofofIncompletenessandwhichisbasedonbasedonMeltzersEnglishtranslationoftheoriginalGerman

berformalunentscheidbareStzederPrincipiaMathematicaundverwandterSystemeI.

Note:Headingsinitalicsenclosedinsquarebracketsareadditionaltotheoriginaltext,theseareincludedforconvenience,e.g.,[Recursion]

Contents

Part1

Part2DescriptionoftheformalsystemPTheaxiomsofthesystemPTherulesofinferenceofthesystemPTheGdelnumberingsystemRecursion

PropositionsIIVTheRelations146

1 2 3 4 5 6 7 8 9 1011 12 13 14 15 16 17 18 19 2021 22 23 24 25 26 27 28 29 3031 32 33 34 35 36 37 38 39 4041 42 43 44 45 46

PropositionVPropositionVI

Part3PropositionVIIPropositionVIIIPropositionIXPropositionX

Part4PropositionXI

ONFORMALLYUNDECIDABLEPROPOSITIONSOFPRINCIPIAMATHEMATICAANDRELATEDSYSTEMS1

byKurtGdel,Vienna

1Thedevelopmentofmathematicsinthedirectionofgreaterexactnesshasasiswellknownledtolargetractsofitbecomingformalized,sothatproofscanbecarriedoutaccordingtoafewmechanicalrules.Themostcomprehensiveformalsystemsyetsetupare,ontheonehand,thesystemofPrincipiaMathematica(PM) and,ontheother,theaxiomsystemforsettheoryofZermeloFraenkel(laterextendedbyJ.v.Neumann). Thesetwosystemsaresoextensivethatallmethodsofproofusedinmathematicstodayhavebeenformalizedinthem,i.e.reducedtoafewaxiomsandrulesofinference.Itmaythereforebesurmisedthattheseaxiomsandrulesofinferencearealsosufficienttodecideallmathematicalquestionswhichcaninanywayatallbeexpressedformallyinthesystemsconcerned.Itisshownbelowthatthisisnotthecase,andthatinboththesystemsmentionedthereareinfactrelativelysimpleproblemsinthetheoryofordinarywholenumbers whichcannotbedecidedfromtheaxioms.Thissituationisnotdueinsomewaytothespecialnatureofthesystemssetup,butholdsforaveryextensiveclassofformalsystems,including,inparticular,allthosearisingfromtheadditionofafinitenumberofaxiomstothetwosystemsmentioned, providedthattherebynofalsepropositionsofthekinddescribedinfootnote4becomeprovable.

Beforegoingintodetails,weshallfirstindicatethemainlinesoftheproof,naturallywithoutlayingclaimtoexactness.TheformulaeofaformalsystemwerestrictourselvesheretothesystemPMare,lookedatfromoutside,finiteseriesofbasicsigns(variables,logicalconstantsandbracketsorseparationpoints),anditiseasytostatepreciselyjustwhichseriesofbasicsignsaremeaningfulformulaeandwhicharenot. Proofs,fromtheformalstandpoint,arelikewisenothingbutfiniteseriesofformulae(withcertainspecifiablecharacteristics).Formetamathematicalpurposesitisnaturallyimmaterialwhatobjectsaretakenasbasicsigns,andweproposetousenaturalnumbers forthem.Accordingly,then,aformulaisafiniteseriesofnaturalnumbers, andaparticularproofschemaisafiniteseriesoffiniteseriesofnaturalnumbers.Metamathematicalconceptsandpropositionstherebybecomeconceptsandpropositionsconcerningnaturalnumbers,orseriesofthem, andthereforeatleastpartiallyexpressibleinthesymbolsofthesystemPMitself.Inparticular,itcanbeshownthattheconcepts,"formula","proofschema","provableformula"aredefinableinthesystemPM,i.e.onecangive aformulaF(v)ofPMforexamplewithonefreevariablev(ofthetypeofaseriesofnumbers),suchthatF(v)interpretedastocontentstates:visaprovableformula.WenowobtainanundecidablepropositionofthesystemPM,i.e.apropositionA,forwhichneitherAnornotAareprovable,inthefollowingmanner:

AformulaofPMwithjustonefreevariable,andthatofthetypeofthenaturalnumbers(classofclasses),weshalldesignateaclasssign.Wethinkoftheclasssignsasbeingsomehowarrangedinaseries, anddenotethen onebyR(n)andwenotethattheconcept"classsign"aswellastheorderingrelationRaredefinableinthesystemPM.Letbeanyclasssignby[n]wedesignatethatformulawhichisderivedonreplacingthefreevariableintheclasssignbythesignforthenaturalnumbern.Thethreetermrelationx=[yz]alsoprovestobedefinableinPM.WenowdefineaclassKofnaturalnumbers,asfollows:

nK~(Bew[R(n)n])

(whereBewxmeans:xisaprovableformula).SincetheconceptswhichappearinthedefinitionsarealldefinableinPM,sotooistheconceptKwhichisconstitutedfromthem,i.e.thereisaclasssignS, suchthattheformula[Sn]interpretedastoitscontentstatesthatthenaturalnumbernbelongstoK.S,beingaclasssign,isidenticalwithsomedeterminateR(q),i.e.

S=R(q)

holdsforsomedeterminatenaturalnumberq.Wenowshowthattheproposition[R(q)q] isundecidableinPM.Forsupposingtheproposition[R(q)q]wereprovable,itwouldalsobecorrectbutthatmeans,ashasbeensaid,thatqwouldbelongtoK,i.e.accordingto(1),~(Bew[R(q)q])wouldholdgood,incontradictiontoourinitialassumption.If,onthecontrary,thenegationof[R(q)q]wereprovable,then~(nK),i.e.Bew[R(q)q]wouldholdgood.[R(q)q]wouldthusbeprovableatthesametimeasitsnegation,whichagainisimpossible.

TheanalogybetweenthisresultandRichardsantinomyleapstotheeyethereisalsoacloserelationshipwiththe"liar"antinomy, sincetheundecidableproposition[R(q)q]statespreciselythatqbelongstoK,i.e.accordingto(1),that[R(q)q]isnotprovable.Wearethereforeconfrontedwithapropositionwhichassertsitsownunprovability. Themethodofproofjustexhibitedcanclearlybeappliedtoeveryformalsystemhavingthefollowingfeatures:firstly,interpretedastocontent,itdisposesofsufficientmeansofexpressiontodefinetheconceptsoccurringintheaboveargument(inparticulartheconcept"provableformula")secondly,everyprovableformulainitisalsocorrectasregardscontent.Theexactstatementoftheaboveproof,whichnowfollows,willhaveamongothersthetaskofsubstitutingforthesecondoftheseassumptionsapurelyformalandmuchweakerone.

Fromtheremarkthat[R(q)q]assertsitsownunprovability,itfollowsatoncethat[R(q)q]iscorrect,since[R(q)q]iscertainlyunprovable(becauseundecidable).Sothepropositionwhichisundecidableinthesystem PMyetturnsouttobedecidedbymetamathematicalconsiderations.Thecloseanalysisofthisremarkablecircumstanceleadstosurprisingresultsconcerningproofsofconsistencyofformalsystems,whicharedealtwithinmoredetailinSection4(PropositionXI).

Cf.thesummaryoftheresultsofthiswork,publishedinAnzeigerderAkad.d.Wiss.inWien(math.naturw.Kl.)1930,No.19.A.WhiteheadandB.Russell,PrincipiaMathematica,2ndedition,Cambridge1925.Inparticular,wealsoreckonamongtheaxiomsofPMtheaxiomofinfinity(intheform:thereexistdenumerablymanyindividuals),andtheaxiomsofreducibilityandofchoice(foralltypes).Cf.A.Fraenkel,'ZehnVorlesungenberdieGrundlegungderMengenlehre',Wissensch.u.Hyp.,Vol.XXXIJ.v.Neumann,'DieAxiomatisierungderMengenlehre',Math.Zeitschr.27,1928,Journ.f.reineu.angew.Math.154(1925),160(1929).Wemaynotethatinordertocompletetheformalization,theaxiomsandrulesofinferenceofthelogicalcalculusmustbeaddedtotheaxiomsofsettheorygivenintheabovementionedpapers.TheremarksthatfollowalsoapplytotheformalsystemspresentedinrecentyearsbyD.Hilbertandhiscolleagues(sofarasthesehaveyetbeenpublished).Cf.D.Hilbert,Math.Ann.88,Abh.ausd.math.Sem.derUniv.HamburgI(1922),VI(1928)P.Bernays,Math.Ann.90J.v.Neumann,Math.Zeitsehr.26(1927)W.Ackermann,Math.Ann.93.I.e.,moreprecisely,thereareundecidablepropositionsinwhich,besidesthelogicalconstants~(not),(or),(x)(forall)and=(identicalwith),therearenootherconceptsbeyond+(addition)and.(multiplication),bothreferredtonaturalnumbers,andwheretheprefixes(x)canalsoreferonlytonaturalnumbers.Inthisconnection,onlysuchaxiomsinPMarecountedasdistinctasdonotarisefromeachotherpurelybychangeoftype.Hereandinwhatfollows,weshallalwaysunderstandtheterm"formulaofPM"tomeanaformulawrittenwithoutabbreviations(i.e.withoutuseofdefinitions).Definitionsserveonlytoabridgethewrittentextandarethereforeinprinciplesuperfluous.I.e.wemapthebasicsignsinonetoonefashiononthenaturalnumbers(asisactuallydoneon).I.e.acoveringofasectionofthenumberseriesbynaturalnumbers.(Numberscannotinfactbeputintoaspatialorder.)Inotherwords,theabovedescribedprocedureprovidesanisomorphicimageofthesystemPMinthedomainofarithmetic,andallmetamathematicalargumentscanequallywellbeconductedinthisisomorphicimage.Thisoccursinthefollowingoutlineproof,i.e."formula","proposition","variable",etc.arealwaystobeunderstoodasthecorrespondingobjectsoftheisomorphicimage.Itwouldbeverysimple(thoughratherlaborious)actuallytowriteoutthisformula.Perhapsaccordingtotheincreasingsumsoftheirtermsand,forequalsums,inalphabeticalorder.Thebarsignindicatesnegation.[Replacedwith~.]AgainthereisnottheslightestdifficultyinactuallywritingouttheformulaS.Notethat"[R(q)q]"(orwhatcomestothesamething"[Sq]")ismerelyametamathematicaldescriptionoftheundecidableproposition.ButassoonasonehasascertainedtheformulaS,onecannaturallyalsodeterminethenumberq,andtherebyeffectivelywriteouttheundecidablepropositionitself.Everyepistemologicalantinomycanlikewisebeusedforasimilarundecidabilityproof.Inspiteofappearances,thereisnothingcircularaboutsuchaproposition,sinceitbeginsbyassertingtheunprovabilityofawhollydeterminateformula(namelytheq inthealphabeticalarrangementwithadefinitesubstitution),andonlysubsequently(andinsomewaybyaccident)doesitemergethatthisformulaispreciselythatbywhichthepropositionwasitselfexpressed.

2

[DescriptionoftheformalsystemP]

Weproceednowtotherigorousdevelopmentoftheproofsketchedabove,andbeginbygivinganexactdescriptionoftheformalsystemP,forwhichweseektodemonstratetheexistenceofundecidablepropositions.PisessentiallythesystemobtainedbysuperimposingonthePeanoaxiomsthelogicofPM (numbersasindividuals,relationofsuccessorasundefinedbasicconcept).

ThebasicsignsofthesystemParethefollowing:

I. Constants:"~"(not),""(or),""(forall),"0"(nought),"f"(thesuccessorof),"(",")"(brackets).

II. Variablesoffirsttype(forindividuals,i.e.naturalnumbersincluding0):"x ","y ","z ",Variablesofsecondtype(forclassesofindividuals):"x ","y ","z ",Variablesofthirdtype(forclassesofclassesofindividuals):"x ","y ","z ",

andsoonforeverynaturalnumberastype.

Note:Variablesfortwotermedandmanytermedfunctions(relations)aresuperfluousasbasicsigns,sincerelationscanbedefinedasclassesoforderedpairsandorderedpairsagainasclassesofclasses,e.g.theorderedpaira,bby((a),(a,b)),where(x,y)meanstheclasswhoseonlyelementsarexandy,and(x)theclasswhoseonlyelementisx.

Byasignoffirsttypeweunderstandacombinationofsignsoftheform:

a,fa,ffa,fffaetc.

whereaiseither0oravariableoffirsttype.Intheformercasewecallsuchasignanumbersign.Forn>1weunderstandbyasignofn typethesameasvariableofn type.

Combinationsofsignsoftheforma(b),wherebisasignofn andaasignof(n+1) type,wecallelementaryformulae.Theclassofformulaewedefineasthesmallestclass containingallelementaryformulaeand,also,alongwithanyaandbthefollowing:~(a),(a)(b),x(a)(wherexisanygivenvariable). Weterm(a)(b)thedisjunctionofaandb,~(a)thenegationand(a)(b)ageneralizationofa.Aformulainwhichthereisnofreevariableiscalledapropositionalformula(freevariablebeingdefinedintheusualway).Aformulawithjustnfreeindividualvariables(andotherwisenofreevariables)wecallannplacerelationsignandforn=1alsoaclasssign.

BySubsta(v|b)(whereastandsforaformula,vavariableandbasignofthesametypeasv)weunderstandtheformuladerivedfroma,whenwereplacevinit,whereveritisfree,byb. Wesaythataformulaaisatypeliftofanotheroneb,ifaderivesfromb,whenweincreasebythesameamountthetypeofallvariablesappearinginb.

[AxiomsoftheformalsystemP]

Thefollowingformulae(IV)arecalledaxioms(theyaresetoutwiththehelpofthecustomarilydefinedabbreviations:.,,,(x),= andsubjecttotheusualconventionsaboutomissionofbrackets):

I.1. ~(fx =0)2. fx =fy x =y3. x (0).x (x (x )x (fx ))x (x (x ))

II.Everyformuladerivedfromthefollowingschematabysubstitutionofanyformulaeforp,qandr.

1. ppp2. ppq3. pqqp4. (pq)(rprq)

III.Everyformuladerivedfromthetwoschemata

1.v(a)Substa(v|c)2. v(ba)bv(a)

bymakingthefollowingsubstitutionsfora,v,b,c(andcarryingoutinItheoperationdenotedby"Subst"):foraanygivenformula,forvanyvariable,forbanyformulainwhichvdoesnotappearfree,forcasignofthesametypeasv,providedthatccontainsnovariablewhichisboundinaataplacewherevisfree.

IV.Everyformuladerivedfromtheschema

1. (u)(v(u(v)a))

onsubstitutingforvoruanyvariablesoftypesnorn+1respectively,andforaaformulawhichdoesnotcontainufree.Thisaxiomrepresentstheaxiomofreducibility(theaxiomofcomprehensionofsettheory).

V.Everyformuladerivedfromthefollowingbytypelift(andthisformulaitself):

1. x (x (x )y (x ))x =y

Thisaxiomstatesthataclassiscompletelydeterminedbyitselements.

[RulesofinferenceoftheformalsystemP]

Aformulaciscalledanimmediateconsequenceofaandb,ifaistheformula(~(b))(c),andanimmediateconsequenceofa,ifcistheformulav(a),wherevdenotesanygivenvariable.Theclassofprovableformulaeisdefinedasthesmallestclassofformulaewhichcontainstheaxiomsandisclosedwithrespecttotherelation"immediateconsequenceof".

[TheGdelnumberingsystem]

ThebasicsignsofthesystemParenoworderedinonetoonecorrespondencewithnaturalnumbers,asfollows:

0"1"f"3"~"5""7""9"("11")"13

Furthermore,variablesoftypenaregivennumbersoftheformp (wherepisaprimenumber>13).Hence,toeveryfiniteseriesofbasicsigns(andsoalsotoeveryformula)therecorresponds,onetoone,afiniteseriesofnaturalnumbers.Thesefiniteseriesofnaturalnumberswenowmap(againinonetoonecorrespondence)ontonatura