M. Schrin - Finitist point of view.pdf - Historia - Logiki - MMarkMMark

HISTORYANDPHILOSOPHYOFLOGIC, 22 (2001),135±161

ExtensionsoftheFinitistPointofView

M ATTHIAS S CHIRN andK ARL -G EORG N IEBERGALL

UniversitaÈtMuÈnchen,SeminarfuÈrPhilosophie,LogikundWissenschaftstheorie,Ludwigstr.31,80539

MuÈnchen,Germany

Revised5April2002

Hilbertdevelopedhisfamous®nitistpointofviewinseveralessaysinthe1920s.Inthispaper,wediscuss

variousextensionsofit,withparticularemphasisonthosesuggestedbyHilbertandBernaysin

GrundlagenderMathematik (vol.I1934,vol.II1939).Thepaperisinthreesections.The®rstdealswith

Hilbert’sintroductionofarestrictedo-ruleinhis1931paper`DieGrundlegungderelementaren

Zahlenlehre’.Themainquestionwediscusshereiswhetherthe®nitist(meta-)mathematicianwouldbe

entitledtoacceptthisruleasa®nitaryruleofinference.Inthesecondsection,weassessthestrengthof

®nitistmetamathematicsinHilbertandBernays1934.Thethirdand®nalsectionisdevotedtothesecond

volumeof GrundlagenderMathematik .Forpreparatoryreasons,we®rstdiscussGentzen’sproposalof

expandingtherangeofwhatcanbeadmittedas®nitaryinhisesssay`DieWiderspruchsfreiheitderreinen

Zahlentheorie’(1936).AstoHilbertandBernays1939,weendona`critical’note:howeverconsiderable

theimpactofthisworkmayhavebeenonsubsequentdevelopmentsinmetamathematics,therecanbeno

doubtthatinittheidealsofHilbert’soriginal®nitismhavefallenvictimtosheerproof-theoreticpragmatism.

Inthe1920s,DavidHilbertdevelopedhis®nitistmetamathematicsquacontentual

theoryofformalizedproofstodefendallofclassicalmathematics.Ontheonehand,

Russell’sandZermelo’sdiscoveryoftheset-theoreticparadoxesshookthelogical

foundationsofnumbertheoryandanalysis.Ontheotherhand,BrouwerandWeyl

hadmountedaseriousattackonclassicalanalysis,especiallyonthe(alleged)validity

ofthelawofexcludedmiddle.Hilbertfeltthathehadtomeetthistwofoldchallenge

byestablishingmetamathematicallyandinapurely®nitistfashiontheconsistencyof

classicalmathematics.Thecentralideaunderlyingmetamathematicalconsistency

proofswastoshowtheconsistencyofformalizedmathematicaltheoriesTbymeans

of`weaker’andmorereliablemethodsthanthosethatcouldbeformalizedinT.

ItwasinthelightofGoÈdel’sIncompletenessTheoremsthat®nitist

metamathematicsturnedouttobetooweaktolaythefoundationsforasigni®cant

portionofclassicalmathematics.Hilbertandhiscollaborators,notablyBernays,

respondedbyextendingtheoriginal®nitistpointofview.Theenvisagedextension

wasguidedbytwocentral,thoughpossiblycon¯ictingideas:®rst,tomakesure

thatitpreservedtheessentialsof®nitistmetamathematics;second,toconduct

indeed,withintheextendedproof-theoreticbounds,aconsistencyproofforalarge

partofmathematics,inparticularforsecond-orderarithmetic(Z 2 ).

WemaycharacterizethenatureofHilbert’s®nitistmetamathematicsofthe1920s

(henceforthreferredtoas`M’)inafewwordsasfollows:

(1)Misanon-formalizedandnon-axiomatized`theory’withanon-mathematical

vocabulary.

*WiththeexceptionofHilbert(1928a)and(1928b)thetranslationsofHilbert’swritingsareourown.For

themostpart,wehaveslightlymodi®edtheexistingtranslations.Astothetranslationoftheword

`inhaltlich’,wehaveadoptedtheneologism`contentual’,coinedbythetranslatorofHilbert1926,Stefan

Bauer-Mengelberg.ThereferencestotheEnglishtranslationsofHilbert(1926)and(1928a)appearin

squarebrackets.

History&PhilosophyofLogic ISSN0144-5340print/ISSN1464-5149online # 2002Taylor&FrancisLtd

http://www/tandf/co/uk/journals/tf/01445340.html

DOI:10.1080/01445340210154295

136 MatthiasSchirnandKarl-GeorgNiebergall

(2)BothManditsformalversionMMcontainonlytruesentences. 1

(3)NeitherMnorMMneedberecursivelyenumerable.

(4)Unboundeduniversalquanti®cationscannotbeformulatedinthelanguageL M of

M,norcangenuineP 0 1 -sentencesorP 0 0 -formulaebeformulatedinthelanguage

L MM ofMM.

(5)MMispossiblymuchweakerthanPrimitiveRecursiveArithmetic(PRA). 2

Toavoidmisunderstandings,afewremarksareinorderhere.Weareawarethat

someifnotmostpeopleworkingonHilbert’sprogrammeandrelatedtopicswill

probablynotagreewithourwayofcharacterizinghispre-GoÈdelianconceptionof

metamathematics.Tobesure,allbutoneofour®vethesesaredenialsofstandard

assumptionsmadebyresearchersinthis®eld.However,ournon-standard

approachisbynomeansduetomaniaforinnovation,costwhatitmay.Onthe

contrary,inourpaper`Hilbert’s®nitismandthenotionofin®nity’(seeNiebergall

andSchirn1998)weattempttoargueinsomedetailfortheplausibilityofthe

theses(1)±(5)bypayingcloseattentiontoHilbert’stexts.Wewishtomakeone

proviso,though.Ontheonehand,Hilbert’sclassicalpapersonprooftheory

between1922and1928arenotfreefromambiguityandvagueness.Ontheother

hand,itgoeswithoutsayingthattheywerenotwrittenfromthepointofviewof

modernmetalogicalconsiderations.Wethereforeconcedethatthetheses(1)±(5)

cannotbeattributedtoHilbertwithabsolutecertainty.

Inanycase,itseemstousthatsomeofourfellowresearchersstilltendtoregard

Hilbert’searlyprooftheoryfromthepointofviewoflaterdevelopmentsin

metamathematics,beginninginthe1930s.However,tothebestofourknowledge,so

farnoonehasfurnishedconclusivetextualevidencethatinthe1920sHilbertheld

thedenialofanyofthetheses(1),(3),(4)and(5),letalonethedenialsofallofthem.

Andwebelievethatsuchevidenceisnotforthcoming.Thequestionwhetherthe

conceptionofmetamathematicscharacterizedaboveisacoherentandanintelligible

one,isanothermatter.To®nditincoherentorunintelligibleiscertainlyno

compellingreasontoclaimthatitcannothavebeeninthespiritofHilbert.

Thecoreofthispaperisacriticaldiscussionoftheextensionsofthe®nitistpoint

ofviewthatHilbertandBernayssuggestin GrundlagenderMathematik (Vol.I1934,

Vol.II1939).Thismonumentalworkhaslargelybeenignoredindiscussionsof

Hilbert’s®nitiststandpoint.TogainacompletepictureofthefacetsofHilbert’s

®nitism,thisgapneedstobe®lled.Weventuretodothisin}2and3ofthispaper.

In } 1,wecanvassHilbert’sintroductionofarestricted o -ruleinhis1931paper

`DieGrundlegungderelementarenZahlenlehre’.Hilbertusesthisruletoachievean

extensionofPAandquali®esitexpresslyas®nitary.Themainquestionwefocus

onhereiswhetherthe®nitist(meta-)mathematicianwouldbejusti®edinaccepting

thisruleasa®nitarymodeofinference.Inthesecondsection,weareespecially

interestedin®guringouttowhatextentHilbert’s®nitistperspectiveof1934tallies

withordeviatesfromhisapproachinthe1920s.Wearguethat®nitist

metamathematicsinHilbertandBernays1934isatleastasstrongasPRA,and

mayincludeevenPRA+Con pra .Thethirdand®nalsectionprovides®rstabrief

accountofGentzen’sconsistencyproofforPeanoArithmetic(PA)inhis1936

1MMresultsfromcouchingMinaformallanguage.

2WeargueforthisclaiminNiebergallandSchirn(1998)andin`Finitism=PRA?OnathesisofW.W.

Tait’(forthcoming).ForadetaileddescriptionofthenatureofMseeNiebergallandSchirn1998.

ExtensionsoftheFinitistPointofView 137

paper`DieWiderspruchsfreiheitderreinenZahlentheorie’,followedbyafew

commentsonhissuggestionofextendingthescopeofpermissible®nitary(or

constructive)methodsinmetamathematics.Gentzen’sapproachprovestobe

importantwhenwecometoconsidermorecloselythepost-GoÈdelianfateof

®nitisminHilbertandBernays1939. 3 Oneessentialpointarguedforisthatin

HilbertandBernays1939thecanonsofM,suchasitsinformalcharacter,its

intuitiveevidence,anditsunquestionablesoundness,arethrownoverboard.

1.Hilbert(1931)andthe o -rule

Restrictedversionsofthe o -rulehavebeensuggestedbothasameansof

explicatingcertainformsof®nitaryargumentsorproofsandasawayofcorrectly

extendingatheoryalreadyaccepted.Inthissection,wewanttodealwiththe

questionastowhetherweakversionsofthe o -rulecanberegardedas®nitary.For

iftheycan,theymayproveusefulfortheconstructionofmetamathematical

theoriesthatclashneitherwithHilbert’sprogrammenorwithGoÈdel’s

IncompletenessTheorems.Inpursuingouraim,wealignourselveswithmost

authorswhodiscussthe o -ruleinconnectionwithHilbert’sprogramme. 4 By

contrast,inhis1931essayHilberthimselfintroducesarestricted o -ruleasameans

ofextendingPA,thoughhedoessoinawaywhichadmitsdifferentinterpretations.

Rule o*:WhenitisshownthattheformulaA( ~ z )isacorrectnumericalformulafor

eachparticularnumeral ~ z ,thentheformula 8 x A( x )canbetakenasapremise. 5

Hilbertquali®esthisruleexpresslyas®nitaryandgoesontoremindusthat

8 x A( x )hasamuchwiderscopethanA( ~ n),where ~ nisanarbitrarilygivennumeral.

Indeed,inthetextthatfollowsheshowsthatonecanproveevery(arithmetically)

trueP 0 1 -sentence 6 inthetheorythatresultsfromadjoiningruleo*toPA.Itis

worthbringingo*intosharpfocus.Weshalldothisbydiscussingtwoquestions

weconsidertobecrucial.LetPA ¡ 2 bethedeductiveclosureofPAand

f8 x j … x †j8 n … j … n †2 P 0 0 ^ PA ` j … n ††g .The®rstquestionisthenthis:does

Hilbertinhis1931paperreallyintendtoextendPAinthesenseofPA ¡ 2 orin

another?Thequestionforcesitselfuponus,becausethede®nitionoftheextension

ofPAsuggestedbyHilbert(letusrefertoitasZ*)isaffectedbyindeterminacy.

Thesecondquestionis:Canrule o *beconstruedas®nitary?Letusturntothe®rst.

(a)Hilbertfailstode®netheterm`numericalformula’.We,forourpart,acceptthe

standardview,accordingtowhichthenumericalformulaearethequanti®er-free

formulae(cf.Feferman1986)orthe P 0 0 -formulae. 7

(b)Wede®nedPA ¡ 2 insuchawaythatitbynomeansemergedastheclosureofPA

underarestricted o -rule,butonlyastheextensionobtainedbyadjoining one

3GoÈdel’s1958proposaltoliberalizeHilbert’soriginal®nitistpointofviewislessimportantforour

concernandwillbeleftforanotheroccasion.

4EspeciallyDetlefsen1979,310comesclosetoourviewwhenhedeniesthatMMneedberecursivelyenumerable.

5Hilbert1931,491.Infact,itisuncleartouswhetherHilbertextendedhisbasesystembyan o -ruleorjust

by oneapplication ofan o -rule.

6Hilbertdoesnot,ofcourse,usetheterminologyofthearithmeticalhierarchy.Weassumefamiliarity

withthearithmeticalhierarchy;cf.Smorynski1977andKaye1991.

7ItisworthnotingthatinAckermann1925,4.Numericalformulaearealreadyde®nedasquanti®er-free

formulae.

138 MatthiasSchirnandKarl-GeorgNiebergall

applicationofthisrule.TheclosureofPAunderruleo*maybede®nedas

PA[o]:= \ {T j PA ‡ T ^ Tisatheory ^8 c[ 8 n (c( n ) 2 P 0 0 ^ T ` c( n )) ) T `

8 x c]}.Itseems,therefore,thatZ*isnotde®nedasPA ¡ 2 ,butratherasPA[o].

Sinceonecanprove`PA+Tr P 0 1 =PA[o]’,thisexplicationaccordswellwiththe

P 0 1 -completenessofZ*whichHilbertclaimstohaveestablished. 8 Itcanalso

beshown,however,thatPA ¡ 2 =PA[o].Forthisreason,wehavetakenthe

libertytopresentZ*asPA ¡ 2 . 9

(c)Onepossiblewayofinterpretingo*isthis.AssoonaswehaveprovedinPAthat

foreach z ,A( z )iscorrect,weareentitledtouse 8 x A( x )asapremise.Sincewe

assumethatA( z )isP 0 0 ,wecande®neacorrectness-predicateinPAasapartial

truth-predicate.Anotherpossibilityistoexplicate`correct’as`Pr t ’fora

representation t ofasuitabletheoryT. o *isthenturnedintothefollowing

re¯ection-rule:IfPA `8 x Pr t (A( x )),then 8 x A( x ).

AleksandarIgnjatovicÂ(1994)rejectedthestandardinterpretationof o *(i.e.

version(b)above) 10 inthesensejustspeci®ed.Fromhispointofview(330),Z*is

renderedpreciselythroughS,whereS:=(QF-IA)+{ 8 x j ( x ) j j 2 P 0 0 ^ (QF-IA) `

8 x Pr qf-ia … j … x †† }. 11 However,Sisbynomeans P 0 1 -complete.SincePAprovesthe

uniformre¯ectionprinciplefor(QF-IA),SisanaxiomatizablesubtheoryofPA.

EvenifwestrengthenSconsiderablyeitherbyreplacing,inthe de®niens ,(QF-IA)

bystrongeraxiomatizabletheoriesorbyabandoningtherestrictionimposedonj,

theresultingtheoriesarestillaxiomatizableand,therefore,farfromP 0 1 -complete.

Now,IgnjatovicÂhimselfobservesthatHilbertclaimstohaveshowntheP 0 1 -

completenessofZ*.Inthelightofthis,wefailtoseewhyIgnjatovicÂconsidersSto

beamorepreciseversionofZ*. 12

Wenowtacklethesecondquestion:Canruleo*beconstruedas®nitary?

Whenversionsoftheo-rulearediscussedinconnectionwithHilbert’s

programme,itisfairlycommontoadjoinoneortheotherofthesetothe®nitary

metamathematicalassumptionsalreadyacceptedratherthantotheformalized

mathematicaltheoryunderconsideration(cf.Detlefsen1979,p.310).Thosewho

castonlyacursoryglanceatHilbert’s1931papermaysuccumbtothetemptation

ofinterpretinghischaracterizationof o *as®nitaryalongtheselines.Theobjective

ofadjoining o *inthesensejustmentionedwouldappearstraightforward:topave

thewayforconsistencyproofsforlargerfragmentsofclassicalmathematicsby

strengtheningthe®nitaryproof-theoreticrepertoire.Closerexaminationof

Hilbert’stext,however,revealsthatheadds o *totheformalizedtheoryPA,that

is,toatheorystatedintheformalizedobject-languageandalreadycontaining

8ByTr P 0 n weunderstandthesetoftrueP 0 n -sentences,by`Tr P 0 n ( x )’aformularepresentingthissetin

a`natural’way.

9Ifweunderstandbya`numericalformula’anarbitraryformulainthelanguageofPA,thenPAadjoined

byoneapplicationoftheresultingo-ruledifferssigni®cantlyfromtheclosureofPAunderthiso-rule.

The®rsttheorycoincideswithPA+Tr S 0 3 +RFN[pa](whereRFN[pa]istheuniformre¯ectionprinciple

forPA),thesecondisequaltoTh( N ),i.e.thesetofallarithmeticallytruesentencesofL PA .

10SeealsoIsaacson1992forthe`standard’view.

11IgnjatovicÂendorsestheidenti®cationof(formalized)®nitistmetamathematicswithPRAor(QF-IA).

(QF-IA)istheconservativeextensionofPRAobtainedbyadding®rst-orderlogictoPRA.Our

de®nitionofSisaslightmodi®cationofIgnjatovicÂ’soriginalde®nition(IgnjatovicÂ1994,330).

12PerhapswemisrepresentIgnjatovicÂbyascribingtohimtheviewthatSisapreciseversionof(QF-IA)

enlargedbyoneapplicationof o *.Wedonot,however,ruleoutthatheregardsSasakindof`rational

reconstruction’ofHilbert’sapproach.Atanyrate,whenIgnjatovicÂcriticizesDetlefsen’sinterpretation

ofHilbert’s o -rule,heseemstotakeversion(c)asan explication ofZ*;cf.especially326.

ExtensionsoftheFinitistPointofView 139

trans®nite statements.Forinstance,hesketchesaconsistencyproof for Z*. 13 Itis

furthernoteworthythataround1930Hilbert’sprogrammedidnotcentre

exclusivelyonconsistencyproofsforformalizedmathematicaltheories.Onthe

contrary,Hilbertlikewiseattachedgreatimportancetoissuesconcerningthe

completenessofsuchtheories.Accordingly,inhis1931paperheshowsthatZ*isa

theoryenjoyingahigherdegreeofcompleteness(namelyatleastP 0 1 -completeness)

thanrecursivelyenumerabletheories. 14 Hewrites(1931,491):

1.Ifastatementcanbeshowntobeconsistent,itisprovable;andfurther

2.IfforasentenceSonecanprovetheconsistencywiththeaxiomsofnumbertheory,

onecannot,atthesametime,proveforStheconsistencywiththoseaxioms.

Iwasabletoprovethesesentencesatleastforcertainsimplecases.Thisprogressis

achievedbyadjoiningtotherulesofinferencealreadyadmitted(substitutionand

inference®gure[i.e. modusponens ])thefollowinglikewise®nitaryruleofinference.

Hilbertgoesontostatewhatwerefertoasrule o *.Theuseoftheword`likewise’

clearlysuggeststhatheintendstoadd o *qua®nitaryruleofinferencetothealready

accepted®nitaryrulesofinference.Yettolayemphasisonthe(allegedly)®nitary

natureof o *inthecontextheisconsideringisscarcelyenlighteningandmaygive

risetoconfusion.WhatcountsforHilbertistheresultthatcanbeachievedby

adjoiningo*toPA;thequestionastothe®nitaryornon-®nitarystatusofo*is,

strictlyspeaking,irrelevant.

ItcomesasnosurprisethatHilbertrefrainsfromemployingo*in

metamathematicalconsistencyproofs.Misvoidofanyrulesofinferenceinthe

ordinarysenseoftheexpression.And,accordingtoHilbert,®nitistinferencesare

eithereffectedintheformofthought-experimentsonintuitivelygivenobjects(cf.

HilbertandBernays1934,32)orarecarriedoutsimplywithconcreteobjects.

Hence,evenifHilbertweretosucceedinjustifyingthepurportedly®nitary

characterofo*,anyattemptatadjoiningo*totheinformal,contentual`modesof

inference’availableinMwouldclearlybeatvariancewiththemethodological

underpinningsofM.WearestressingheretheinformalcharacterofMwiththe

provisothatby1930Hilbertmayhaveundergonesomechangeofmindastothe

natureofM.Thereis,however,noconclusivetextualevidenceforthat. 15 Weshall

thereforeassumethatinhis1931articleheisstillsubscribingtotheessentialideas

concerningMasdevelopedinhispapersbetween1923and1928.

13Bernays1935,216observesthatbyintroducingtheadditionalruleofinferenceo*Hilbertrelinquishes

therequirementofacompleteformalizationoftheinferencesinanarithmeticaltheory.Bernaysspeaks

inthisconnectionofadeviationfromHilbert’soriginalproof-theoreticprogramme.

14SeealsoBernays’s1935,214ff.comments.

15HilbertandBernays1934,43observethatthemethodologicalstandpointofBrouwer’sintuitionism

incorporatesacertainextensionofthe®nitistpointofview.Thereasonisclaimedtobethis:

Brouweradmitsthatanassumptionisintroducedaboutthepresenceofaninferenceoraproof,

withouttheinferenceortheproofbeingdeterminedaccordingtointuitivecharacteristics.Froman

epistemologicalpointofview,suchanextensionofthe®nitistattitudeissaidtoamounttothe

methodofadjoiningtotheintuitiveinsights(gainedin®nitistreasoning)re¯ectionsofagenerally

logicalcharacter.HilbertandBernayspointoutthattheextensionsoconceivedisrequisiteifby

meansof®nitistconsiderationsonewishestogobeyondacertainelementarydomain.Cf.also

Bernays1930,60.Itistruethatinhis1931paper(486)Hilbertseemsalreadytobehintingatthis

kindofextensionwhenhecharacterizestheinsightsaprioriasthoseintuitiveandlogicalinsights

thatwegainwithintheframeworkofthe®nitistattitude.Butagain,wearereluctanttoregardthis

isolatedremarkasfurnishingconclusiveevidenceforthepossibleclaimthatby1931Hilbert’s

originalconceptionof®nitismhadalreadyundergoneasigni®cantchange.

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