potter iterative set theory

7/28/2019 Potter Iterative Set Theory

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Iterative Set TheoryAuthor(s): M. D. PotterSource: The Philosophical Quarterly, Vol. 43, No. 171 (Apr., 1993), pp. 178-193Published by: Blackwell Publishing for The Philosophical QuarterlyStable URL: http://www.jstor.org/stable/2220368 .

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ThePhilosophicaluarterlyol. 3,No.171ISSN 0031-80942.00

ITERATIVE SET THEORYBY M. D. POTTER

This article' s intendedforphilosopherswho are interestedn the roleset theoryhas played in the foundations f mathematics ince 1900.Much ofwhat is said here is elaborated at greater ength,withthetechnicaldetailsspeltout, inmy set-theoryextbook.2 shouldat theoutsetwarn thoseof a sensitivedispositionthat thisarticle containsplatonist anguage whichtheymayfind ffensive.I. THE ITERATIVE CONCEPTION

What I shall bediscussingsan axiomatization, f sortwhichhas beendeveloped in the last twentyyearsor so, of the traditional terativeconceptionof ets.That is tosay,sets re collections-as-one,ackswithobjects n them, bjectswitha lasso around them,whatevermetaphoryou please. So to some extent, t least, thechangeswhichwould bewrought fwe adopted these axioms would be cosmetic:the surfacedescriptionwould be different, ut the underlyingreality beingdescribed would be the same. Thus this s not at all thesortofprojectQuine, for xample,was engagedinwhen he wroteMathematicalogicand New Foundations: he notionof set which Quine was trying oaxiomatize, f herewas one,must, think, ave beenentirely asedonan analysisofthesyntactic xplanationfor heknownparadoxesintheso-called naive conceptionofwhat a set is, and thereforemusthavebeen essentially egative n character.Quine's was, in otherwords,aone-step-back-from-disasteriew.Axiomatizations fthe terative onception, n the otherhand, are,it is surely now generally recognized, positive in character. No

' This paper isderivedfrom alks gave to the ?-club at Cambridge n 1990and theLogic ndLanguageonference t St Andrews t Easter 1991. I amgrateful o themembersofthose udiences whomade helpful emarks,nd toPeterSullivan.2 M. D. Potter, ets:an ntroductionOxfordUP, 1990).? Theeditorsf The hilosophicaluartrly,993.Published yBlackwell ublishers,08CowleyRoad,Oxford X4 IJF,UKand 238 Main Street, ambridge,MA 02142,USA.


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ITERATIVE SET THEORY 179mathematician eriously oubts that terative ettheory s consistent.And thereasonfor his act snot ustthatthe xiomshave beenaroundforeightyyears (mostof themanyway) and no one has yetfound acontradiction.Afterall, mathematiciansroutinelyuse only a tinyfragmentof the generality permitted them by the theory, andpresumably only by pushing the theory to its limits could acontradiction e obtained.Notonlythat,buthardly nyone as far s Iknow) has seriouslytried to find one.3 Nor is there any meansavailable to us ofdecidinghow long is an appropriatetime to waitbefore eclaringa theory ecure.So our confidencen theconsistency f ZF is notdue simplyto thefactthatwe have notfounda contradictionyet;rather s it due to arecognition fthe ntuitive asisfor heaxioms.This view isgenerallyacceptedbymathematicians ow,but t sworthnotinghowrarelyt sto befound xpressednthe iterature efore he1950s.What wasmuchmore common until then was to view theparadoxes as exhibitinggenuine contradictionin our intuitiveconception. For instance,HermannWeyl in 1949: The attitude sfrankly ragmatic;one curesthe visible symptoms of the paradoxes] but neitherdiagnoses norattacks the underlyingdisease.'4 Or, even more starkly,Quine in1941: 'Common sense is bankrupt,for t wound up in contradiction.Deprived of his tradition,the logician has had to resort to myth-making.'5Accordingto thisview, the object of a good axiomatizationis toretain as manyas possibleof thenaive set-theoreticrgumentswhichwerememberwithnostalgiafrom urdays nCantor'sparadise,but tostop ust shortofpermitting hoseargumentswhich lead toparadox.This certainly ppears to have been themotiveofZermelo's axiom-atization: There isat thispoint nothingeft orus todo buttoproceedintheoppositedirection nd,starting rom ettheory s it shistoricallygiven, to seek out the principles required for establishing thefoundations fthismathematicaldiscipline. n solving hisproblemwemust, n theonehand,restrictheseprinciples ufficientlyoexclude allcontradictionsnd,ontheother, akethem ufficientlyide toretain llthat is valuable in this theory.'6 In other words, absence of

3 The position is differentfor Quine's systemswhich I mentioned earlier:mathematicians ave set out tofind ontradictionsnthem, nd did at first chieve somesuccess n thatenterprise eforeQuine adjustedhisaxiomsappropriately.C. H, H. Weyl,PhilosophyfMathematicsndNatural ciencePrincetonUP), p. 231.5 W. V.O. Quine, Mathematicalogic Cambridge,Mass.: Harvard UP), p. 153.6 E. Zermelo, UntersuchungeniberdieGrundlagenderMengenlehre ', translatedinJ. vanHeijenoort ed.), From rege oGodel: Source ook nMathematicalogic, 879-1931(Cambridge,Mass.: Harvard UP, 1967), p. 200.? The editors fThePhilosophicaluarterly,993.


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180 M.D. POTTERcontradictions to be regarded, nBourbaki'swords, as an empiricalfactrather han as a metaphysicalprinciple'.7 here was a timewhenit was popular to regard thisgung-ho pragmatism s an inevitableconcomitant fplatonism: The platonist an stomachanything hortofcontradiction; nd whencontradiction oes appear,he iscontent oremove it with an ad hoc restriction.'8 his characterization ofplatonism s a positionwithoutprinciplesnowseemsa trifle nfair.

II. HISTORICAL REMARKS ABOUT ZERMELO'SAXIOMATIZATIONI quoted just now from ermelo's 1908paper, Untersuchungen iberdie Grundlagen der Mengenlehre', in which he gave his axiomat-ization. I think t would be helpfulto make a few remarks aboutZermelo'spaper and theway inwhich his deas weredevelopedlater.The first hingto observe is that althoughZermelo 1908 is oftenquoted as thebirthplaceofthe cumulative terativehierarchy fsets,there s infactno mention f t there: fZermelo knewabout itthen,hewas keepingvery uiet about it. In particular, lthoughhedid assumethe axiom of choice, he did not assume foundation, theFundierungsaxiom,hich is reallythe keyto the iterative onception.That was not ntroduceduntil the ate 1920s.One ofZermelo'spre-publication rafts fhissystem oes have it asan axiomthatnosetcan belongto tself, hich s a consequenceof butweaker than) foundation.To understandwhyhe dropped even thisweak versionfromhissystemwe need to rememberwhyhewrotethispaper. His first roof ftheexistence fa well-orderingn anysethadappeared in 1904.9 t had been criticizednotonlybecause it used theaxiomof choice (whichhe statedexplicitly or hefirst ime)but alsobecause it made use of transfiniterdinal numbers.Remember thatthefactthat the class of all ordinals is not a set,which is now gen-erally called the Burali-Fortiparadox despite never having beenknown oBurali-Fortiimself,asbythenwellknown: thadbeenknown oCantor in at least 1899 and was explicit in Russell's PrinciplesfMathematics, hich appeared in 1903. So therewas a widespreadsuspicionthat therewas somethingfishy bout ordinals,a suspicion

7 N. Bourbaki, Foundations of Mathematicsfor theWorkingMathematician',J.Symb. ogic,14 (1949), p. 3.8 W. V. O. Quine, From LogicalPointof View Cambridge,Mass.: Harvard UP,1953), p. 127.9 E. Zermelo, Beweis,dass ede Menge wohlgeordnetwerdenkann',Math.Ann., 9(1904).? The editors f ThePhilosophicalyarltrly,993.


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ITERATIVE SET THEORY 181whichZermelo tried o cater tobyproviding secondproof fthewell-orderingprinciple, ogetherwithan axiomatizationofa system fsettheoryn which thisproof ould be formalized.Now Zermelo was at this tage (and perhapslatertoo) verymuchmorea workingmathematicianthan a philosopher.So his approachwas pragmatic.His axiomatizationconsists implyofthoseprincipleshe needed in order to make his second proofof the well-orderingprinciplework. o I suspect hatthereasonhe eft utthe xiomthatnosetbelongsto itselfwas simply hathe did not use it nhisproof.This pragmatic pproach seems tohave beenrepeatedseveraltimesin thedevelopment fthe standardform fZF which tookplace overthenexttwenty-fiveears.Take theordinals s an example. Zermelo'ssystem id notinclude ordinals.So proofswhich make use ofordinalsdid not seem to be formalizable n it. For this reason Kuratowski n192210 eneralizedthemeansbywhich Zermelohad gotfromhisfirstproof f thewell-orderingheorem whichusedordinals)tohissecond(whichdidnot).Kuratowskiprovided generalmethodfor onvertinganyproofwhich uses ordinals nto one whichdoes notand which, s aconsequence, s formalizablenZermelo'ssystem.Now it was thisworkwhich ed Kuratowski todiscoverZorn's lemma tenyearsbefore orndid. But the fact that it is not called Kuratowski's lemma tells ussomethingbout theworking ractices fmathematicians: hey re notgoingto giveup a useful ool ust because it cannot be formalized nZermelo'ssystem.What happened insteadwas that when Mirimanoffand thenvon Neumann developed an explicit representation f theordinals n settheoryone which, ncidentally, ermelo seems to havetoyedwith n 1915 or so"), Zermelo's axioms werenotstrong noughtogive representationsfenoughoftheordinalsto be useful.The keydevelopment here was when von Neumann saw that he couldguarantee the existenceof representations f enough ordinals byaddingwhat s now called the xiom schemeofreplacement.Ofcourse,I should make it clear that it was not the need fora satisfactoryembedding of the theoryof ordinals in set theorywhich led to theinventionfthereplacement xiom: Cantorhad stated t nformallynaletterto Dedekind'2 in 1899 as an expressionof his (to some extenttheologicallymotivated) limitation-of-sizeoctrine,and there s no

10K. Kuratowski, 'Une methode d'elimination des nombres transfinis desraisonnementsmathematiques', und.Math.,5 (1922)." See M. Hallett,CantorianetTheoryndLimitationf ize (OxfordUP, 1985),p. 276ff.12 An Englishtranslation f this etter ppears in van Heijenoort ed.), From rege oGodel,pp. 113-17. See also I. Grattan-Guinness,The Rediscoveryof the Cantor -Dedekind Correspondence', ahresber.tsch.Math.-Ver.,3 (1974).? The editorsfThePhilosophicaluarterly,993.


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182 M.D. POTTERevidence that he saw the connection von Neumann drew with thetheory fordinals because there s no evidence that t had occurredtoCantor to model theordinals nsettheory ymeansoftheMirimanoff/von Neumann trick). But it was that connection which led to itswidespreadadoption.ven von Neumann himself dmitted that theaxiomofreplacement,ust regardedas an axiom without egardto itsuseful onsequences, goes a bit too far'.13Ifyouwant another xampleofthe role ofpragmatic onsiderationsin thedevelopment fZF, consider hewaythat ndividuals sometimescalled atoms) were eliminatedfrom t. Zermelo's originalsystemhadcertainly nvisagedthe existenceof ndividuals,that is to say objectswhichare in themselves et-theoreticallypaque, so thatnothing anbe said in the anguage of ettheory bout theirnternal tructure,utwhich can serve as buildingblocks fromwhich sets can be formed.What happened in the1920swas that et-theoreticeductionism theprogrammeof showing that other mathematical theories can bemodelled n settheory was an outstanding uccess. To within verysmallmarginof error ll mathematicaltheories an be so modelled.Moreover the presence or absence of individuals has no effectwhatsoeveron this claim. So postulatingtheexistenceofindividualsincreases heontological ommitmentsf he ystemwithoutncreasingits xplanatorypower.What issurely urprisingsthemoral whichwasdrawnfrom hisfact,namelythat we shouldoutlaw individualsfromthetheory. his simple idestephas,itseemstome, ed toa greatmanymisunderstandingsnthe ubsequent iterature.What started ut as nomore than a relativeconsistency esult- if Zermelo's set theory sconsistent,hen eano arithmetic s consistent oo- has been distortedintoa bizarreheresywhichholds thatthenumber2 isthe et (, {(}}. Itissurely nexceptionable hatwhatever lse thenumber2 isor snot, tiscertainly et-theoreticallypaque: it s, na word, n individual.Theeconomyofontological ommitment hichwegainbypretending hatthenumber2 is {(, ))}} s spurious.It is not that the advantages ofpostulatingwhat we need are thoseof theft ver honesttoil', to useRussell's famousphrase.14t is rather that having stolen (when weaxiomatizedset theory)enough to last us a lifetimewe now have nofurther eed ofcrime.

13J. von Neumann, Eine Axiomatisierung er Mengenlehre',J. reinengew.Math.,154 (1925), p. 227.14B. Russell, ntroductionoMathematicalhilosophyLondon: GeorgeAllenand Unwin,1919),p. 71.? The editorsf ThePhilosophicaluarlerly,993.


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ITERATIVE SET THEORY 183III. THE SCOTT/DERRICK AXIOMATIZATION

The discussion o farhas beenan attempt o focus n the ystem nowninthe et theorists'rgot s ZA' ('Z' for ermelo, A' for toms).This isessentially ermelo's original systemwithout he axiom ofchoice butwith heaxiom offoundationnd a slight echnical trengtheningfhisaxiom of infinity o ensure the existence,unfortunately ot quiteprovablein Zermelo'ssystem, f the set ofall hereditarilyinite ets.The axiomatizationof this ystemwhich shall state sbased,moredirectlyhan s thetraditional ne,on the terative onception f et, nintuitive escription f whichhas beengiven ndetailbyBoolos.15ZA,then, s a first-orderheory xpressedna languagewithustonebinarypredicatesymbol e'. We shall assume for onveniencethatwe have aRussellian definite escription perator i!', so that t!x0(x)' means theunique x such that b(x).

Definition {x: #x)} = Iy(VZ) Z y > 0 z)).Definition Aisa setfA = {x: x e A};otherwise isan individual.Axiomscheme ofseparation {x: x e Aand#((x)}xists orny etA.Accordingto the terative onception, ets are createdstage-by-stage,usingas their lements nlythosewhichhave been created at earlierstages. n order to stress hetemporalmetaphor shall call thestagesdays; nd insteadof ayingthatoneday is a member fanother, shallsay itis earlier. he set of all days earlier thana given day is called itshistory.he accumulationf historys the etwhichhas as its lements llthe ndividuals ogetherwith ll theelements nd subsets f ll thedaysbelonging o thehistory.With thisterminology,heguiding principleof the iterative onception s thateach day is the accumulationof tshistory.This way ofdescribing he iterative onceptionwas formalizedbyDana Scott.16 he disadvantage of his account is that it treatsthenotionofa 'day' as primitive. t wasJohn Derrick (in unpublishedwork)who saw that fwe proceed in theoppositedirection,definingfirstaccumulation',then history' nd 'day', we can achievewhatwewant without he need for n extraprimitive.Definition acc(A) = {x:x s an individualor 3B A)(x Borx _ B)}.Definition ! is a historyf V_ e ) (D = acc _ D)).

15G. Boolos, 'The IterativeConception of Set', J. Phil., 68 (1971); reprinted nP. Benacetraf nd H. Putnam (eds), PhilosophyfMathematics:elected eadings,nd edn(CambridgeUP, 1983).16 D. Scott, Axiomatising et Theory', in Axiomaticet Theory,art II (Providence,R.I.: AmericanMathematicalSociety,1974).? The editors fThe hilosophicalyjarlerly,993.


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184 M.D. POTTERDefinition If. is a history,cc(.) is called a day.Theorem (Scott/Derrick) The days are well orderedbymembership.Axiom of creation Everysetbelongstosomeday.Definition Adayis called a limitf t s not theearliest ay andhas noyesterdayi.e.,noimmediatepredecessorwithrespect omembership).Axiom of infinity There is a limitday.These axiomssuffice o deriveas theorems ll thesentenceswhicharetaken as axioms in themore traditionalaxiomatizationof ZA. Thedetails are in mybook. (The system xiomatized there s actually aslightly tronger ne called GA, but the differences notimportantnthepresent ontext nd theproofs re easytoadjust.)

IV. HOW FAR CAN YOU GO?The only doubt which remains about the adequacy of ZA formathematicalpurposesconcernswhethertheprocessofset creationgoes on long enough. There is a limitday, we have asserted,but ouraxiomsstop ust short frequiring here o be a secondlimitday. As Iexplainedinpart I, mathematicianshave longbeen used toassumingtheaxiom schemeofreplacement n orderto ensurethe existenceofenough ordinals fortheirpurposes (to prove Hartogs' lemma, forexample). However, t turns ut thatwe do not needreplacementfwestartfrom differentefinitionf'ordinal'.Fregeand Russelldefined hecardinalof setA tobe the etof ll setsequinumerouswith t;by analogywemaydefine he ordinalof a well-ordered et A,


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ITERATIVESET THEORY 185equinumerouswithA. In the sameway,wedefine heordinalfa well-ordered et A,< ) to be the etof ll well-orderedets f arliest ossiblebirthday somorphic o (A,


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186 M.D. POTTERthe title set' is cheating: fsomeonereplies When I said "all sets", Imeantall', there snothingwe can sayin response.

The fact hatwe cannot form setof ll sets, venthoughwe appearto be able toquantify ver all sets,has beenusedbysome as a stickwithwhich to beat theplatonist onceptionof settheory, ince fwe are toimagine the process of set creation to be already finished as theplatonist emandsthatweshould),then t sdifficulto understandwhywe shouldnot now collect all the setstogether o form newone, orindeed why they are not already so collected. The alternative(canvassed,for xample,by Lear20) s to imaginethe extension f theconcept et s notyetfixed, o ndex sets ccordingtowhenthey ome tofall under the concept, and consequently to adopt intuitionisticsemantics or ettheory.The platonistneed not concede thisground. t ispossibleto treat hismatter nsteadas an example- Skolem's paradox is another oftheimportanceofkeepingtrackof the linguistic tandpointfromwhichassertionsre made. When we talk about setswithin fixed anguage,we cannot talk of a setof all sets:graspingthis fact s an immediateconsequenceofgrasping he terative onceptof et t all. When westepoutside such a language into an identical meta-language,we maycoherently alkof a set (in themeta-languagesense) of all sets in theobject-languagesense). This is neithercontradictory or surprising.What ismore, t s not evidence thattherereally' s a setof ll sets:whenwe talk about all sets,we succeedin talking bout all sets ust becausewe understand heuniversal uantifier.Moving intoa meta-languageto demonstrate relativityn the meaning of the quantifierwouldsucceed in provingwhat it claimsonly four graspofmeaning n theobjectlanguage came tous via themeta-language nsomeway.But itdoes not.

V. IS SET THEORY SOUND FOR ARITHMETIC?So (pacedescriptiveettheorists) hetheory A which have describedis adequate formathematics.Butis it sound formathematics? ecauseof the success of set-theoretic eductionismwhich I have alreadyalluded to,this s a questionmathematicians, t anyrate,tend not toaddress: they earn in thecradle that mathematicsust is part of settheory; nd ifthat s right henthequestionsimplydoes not arise.But set-theoretic eductionisms,when correctly iewed,no morethana series frelative onsistencyesults: made thatpointearlier. o

20 J. Lear, 'Sets and Semantics',J. Phil.,74 (1977).? The editors fThePhilosophicalyarterly,993.


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ITERATIVESETTHEORY 187we have to address,forexample, the questionwhether et theory ssoundfornumbertheory. o fix deas letus considerthepositionof anumber-theoristwhom I shall forconvenience riskassumingto bemale) whoisfamiliarwith thecountingnumbers nd thearithmeticaloperations upon them,and believes thatPeano's axioms are all truewhen nterpreteds assertions bout them,but has no viewseitherwayabout the existenceof the infinite etswhich a set theory uch as ZAposits. Can we persuade him to believe that all the arithmeticalassertionswhichwecan proveset-theoreticallyboutco i.e., theobjectinZA which we have pickedout to model Peano Arithmetic) re trueabout thecountingnumbershe knows nd loves?Notice straightaway hat if the axioms of Peano Arithmetic recomplete i.e., strong noughtodetermine hetruth rfalsityfeveryarithmetical roposition)and set theory s consistent,hen he has nochoice in the matter.To see this,suppose that 0 is an arithmeticalproposition nd O(/) s thecorrespondinget-theoretic laim about co.Certainlyfq sprovablearithmeticallyi.e., nPeanoArithmetic),hen0(o) is provable in set theory.And converselyfO() is provable in settheory,hen-i (/)snotprovable since et heorysformallyonsistent).Hence-q 0 is notprovable nPeanoArithmetic. o 0 isprovable nPeanoArithmeticsincewe are assuming hat t scomplete).Conclusion:0 isprovable nPeano Arithmeticf nd only f () isprovable nsettheory.All thatremains,then, n order to legitimize he use ofset-theoreticarguments in number theoryis the task of showing that PeanoArithmetics completeand settheorys consistent in otherwords,aversion fHilbert'sprogramme.But Peano Arithmetic s not complete. Godel's incompletenesstheorem tells us that there are explicit arithmetical (finitisticallycomprehensible) ssertionswhich are true nsomemodels and false nothers. But more than that: there are mathematically nterestingexamples of such sentenceswhichare provable in set theory.This ismuch morerecent nformation: he firstxamplesweregivenbyParisand Harrington n 1978. The easiest to understand is Goodstein'stheorem; nordertoexplainwhatitsays,we need someterminology.If n is a natural number > 1, thenany natural number mmay beexpresseduniquely n the formm= blnc + b2n2 ... + bknk,whereb, ..., bk nd c, ..., ck re natural numbers,m> Cl > c2 > ... > Ckand 0 < b;< n 1 < i < k).This expressions called thenormalformfmto the base n.Now express ach oftheexponents l, . , ck n normalform o the base n,and then do the same to all theexponents n these? The editors f'The hilosophicalyarterly,993.


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188 M.D. POTTERexpressions,tc. In a finite umberofstepsthisprocesswillterminateand we shall be leftwith an expression orm nwhich no number> nappears.This expressions called thecompleteormalformfm othebasen.Now we are in a positionto define heGoodsteinequencef a naturalnumberm:the firsterm f thesequenceism;the n + 1)th sobtainedby expressing he nth n completenormal formto the base n + 1,changingall the occurrences fn + 1 in thatexpression on + 2, andthensubtracting .

Example TheGoodsteinequence f51 starts s follows21:51 = 222 + 222 2 + 133 + 1 + 333+ 3 101344 + + 44' + 3 1015555 +1 + 555+ 2 102185666+ + 666 1 103630677+ + 77'7 10695975

888 + 888- 1 015151337Thus thenumbers n thisGoodsteinsequence increasewithverygreatrapidity.t isplainthat51 isnot pecial n this espect; he amepatternemerges or nynumber> 3. The followingheorem s thereforet firstglance quite surprising.Goodstein's theorem22 The Goodstein equenceof any positiveintegereaches ina finite umber f teps.

In ZA, or ndeed nany theorytrong noughtoprovethattheordinals


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ITERATIVESETTHEORY 189that thesamearithmetical entences re true n all Dedekindalgebras.But as we all know,there s no contradiction: t simply s not thecasethateverymodeloffirst-ordereano Arithmeticsa Dedekindalgebra(althoughthe converse ertainly olds).The ambiguitywhich leads to theinitialmisunderstandingies inwhat we are prepared to count as 'properties'forthepurposeof theinduction ule.The definitionfco s a Dedekindalgebra nterpretshisrulebyrequiring hat coshould have no propersubsetscontaining0and closed underthe uccessor peration.Buttheonlypropertieswhichthe number-theorist eems obliged to count are those which areexpressiblenhisspecificationanguage,i.e., arithmetical nes. In set-theoretic erms his mountsroughly o requiring nlythatco shouldhave no proper recursive subsets containing0 and closed undersuccessors. t is the mpredicativityn theseparationaxiom acting nconcertwiththeaxiom of nfinity hichmakesthis mbiguity ossible.Ofcourse,from hepointofview ofthe et-theoristt would be perverseto restrict he nductionprinciple o arithmetical roperties, ut thereisnoreasonfor henumber-theoristo sharethisviewunlesshe can bepersuadedofthereality f the nfinite ets referredo in theproofsnquestion.So should the number-theoristccept our set-theoretic roofofGoodstein'stheorem? upposefor hemoment hathe does believethatsettheorys formallyonsistent.At firstightone might uppose thatthis s notenough.And strictlypeakingthat s correct.However,hemustaccept everynumerical nstanceofGoodstein'stheorem.This isbecause thequantifier-freeartofarithmetic that s to say, explicitnumericalcalculations is complete, o that fyoudo a calculation nPeano Arithmetic nd thendo itagain in settheory, ougetthe sameanswer.Of course, cceptingeverynumerical nstance snotthe sameas acceptingthegeneralization youneed theco-rule omakethat tep- but I do notregardthat as an importantdistinction ere. I wouldregarda number-theorist ho accepted everynumerical nstanceofGoodstein's theorem ut notthetheoremtselfs a harmless ccentric,not a dangerousschismatic.What I have just said seems uite significant:t appears that thenumber-theoristan use set theory at least in the specificcase ofGoodstein'stheorem)withoutbelieving t.All hehas to believe n is itsformal onsistency.fthis s tobe a genuineadvance, however, t mustbe possibleto come to a belief n theconsistencyf settheoryn somewayother hanvia a belief n tstruth. his is not mpossible o magine.It is,for nstance,conceivable thatwe mightprovetheconsistencyf(co, ) where is the uccessor perationon the et co of naturalnumbers.? The editors fThePhilosophicaluarterly,993.


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190 M.D. POTTERZA by cut-eliminationusing transfinite nduction as far as someordinal. (It is a matter of conjecturewhat that ordinal would be,although twouldcertainly e considerably reater hanE0.)Once onewas inpossession f uch a proof,t would be possibletocome to a beliefin theconsistency f ZA (and thereforen thevalidityof certainset-theoreticmethods nnumbertheory) ia a beliefnot nsets,butmerelyin ordinals.Of course, for this possibilityto be interestingwe need to bepersuadedthat omeonemightbelieve nordinalswithoutbelievingnsets.There is nodoubtthatthis s nprinciple ossible,but t seemsmostimplausible. Certainly anyone worried by the incompletenessofiterative et theory s to heightought to be equally worried aboutordinals.What is moreplausible at first ight s that someonemightneverthelessbelieve in ordinals who claimed not to believe in theimpredicative ower-set perationwhich s the fundamentalnotionoftheclassicalconception f ettheory.However, t s difficultosee whatreason such a person mighthave forbelievingin the existenceofordinals of the size which would presumablybe required for theconsistency roofwe are hereenvisaging.

In short, the possibilitythat one mightbe justifiedin using areasonably trongmathematical heory ya beliefn tsconsistency otderivedfrom beliefn thetheorytselfs,as a philosophical dvance,largely llusory.Anyotherrouteto thebelief n consistencys almostcertainto involveessentially hesame philosophical problemsas thedirect route via truth. I cannot, therefore,ee any reason why anominalist bout setsshould believe in theformal onsistency fanysuch reasonably strongtheory. This is, I think, fataldifficultyorField.)It may be that the number-theoristwill be able to refinehisspecificationothepointwhen t can decide Goodstein'stheorem. Hecould, for example, convince himself on intuitivegrounds thatGoodstein's theorem itself hould be added to the list of axioms.However, t s not at all clear whatform his ntuitionmight akeor howhe should go about obtainingit: Dan Isaacson has suggested25hatthere s no way ofperceivingnpurely rithmetical erms hetruth fanyarithmetical roposition,uchas Goodstein'stheorem,which s notprovable fromPeano's axioms.) In any case this - obtaining anintuition bout the truthor falsity fGoodstein's theorem wouldmerelypostpone the problem, since any specificationfor numbertheory tronger han Peano Arithmeticwhichsatisfies he obviously

25 D. Isaacson, 'ArithmeticalTruth and Hidden Higher-orderConcepts', in LogicColloquium85 (Amsterdam: lsevier,1987).? The editors fThePhilosophicaluarterly,993.


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ITERATIVESET THEORY 191essentialrequirementthat its proofsbe mechanicallycheckable isincompletebyGodel's incompleteness heorem.

Nor is the matter immediatelysettledfor a liberated number-theoristwho acceptsthereality f nfiniteetsof numbers.For there sstill hen choice tobe made about whatsort f nfiniteetsofnumbersexist. In particular, t is conceptually possible that there could be a(consistent)variant of set theory n which Goodstein's theorem sprovably false. (Of course, the conception of infinite ets whichmotivated such a theorywould have to be verydifferent rom theiterative ne whichmotivates he axiomsystem have describedhere:it would have to be a theory n which the ordinals could not bedeveloped, and it would therefore resumablybe highlynon-well-founded.)Thisphenomenon- alternative iews bout thebehaviourofinfinite ets (what we might n Hilbert's terminologyall 'the idealpart') resultingndifferenthings eingprovableabout numbers 'thereal part') - arises evenmorepressinglyn thecase of the continuumhypothesis, ince in thiscase there is not even a consensusamongmathematiciansboutwhether oacceptit (incontrast othe ituationwithordinals).Let uscomparehere theattitude faffineeometers owards lines atinfinity': rojectivemethodscan provideus with moreelegant,moreintuitive rmoreconciseproofs ftheorems faffineeometry, uttheydo notpermit stoprove thingswhichwereotherwise nprovable.Theaffine eometers re therefore ree oregard ines at infinitys no morethan a technicaldeviceand need take no view about their xistence rreality, nce theformal onsistencyfthemethodhas been established.(Projective geometers will of course see the matter somewhatdifferently.)imilar remarks pplyto certainusesofcomplexnumbersin the theory f real polynomialsand to the use ofproperclasses inZermelo-styleet theory what Quine calls 'virtual' classes). But thenumber-theoristmust take a different iew about infinite ets: thebeliefshe has about sets affectwhat he must accept as true aboutnumbers.

VI. UNDECIDABLE SENTENCES ANDSECOND-ORDER AXIOMSI drew in the last part an analogy betweenGoodstein'stheorem ndotherknown ncompletenessesuchas the continuumhypothesis. hepattern unning hrough hese xamples s thatthey reprovable nthestronger heorybecause it ensuresthe existence ffresh bjectswhichcan serve in the place of free variables in the predicate used to

? The editorsf ThePhilosophicalyarterly,993.


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192 M.D. POTTERinstantiate the first-order xiom scheme of the weaker theory.Goodstein'stheorem s not decidable in first-ordereano Arithmetic.The continuumhypothesiss notdecidableinfirst-orderettheorywiththeaxiomofchoice. Both areprovable ncertain tronger ystemse.g.,set theory n the case of Goodstein's theorem, et theorywith theconstructibilityypothesisnthecase ofthe continuumhypothesis).nbothcases, too, t sdifficultbutperhapsnot mpossible)to come to aviewabout the truth f theresultndependent fviewsabout thetruthof the axioms ofthestronger ystemnwhich the result s knowntobeprovable. So ifmost mathematicians elieve Goodstein'stheorembutnot the continuumhypothesis the fact hatwe call one a theorem ndtheother n hypothesisndicatesthis lready- it can onlybe becausemostmathematicians elieve the xioms of ettheory utdo not believetheconstructibilityypothesis.But there s another imilarity: eingshown theproofs hatthey reundecidable in the weakersystemn questiondoes notweakenone'sconviction hatthey re decidable questions.This is in contrast o theposition.forhequestionofhowmany daysthere re,which discussedearlier.There was neveranysuggestion fthisbeinga questionwhichhad an answer fwe could onlysee it. The reason for hisdifferencesthatour failure odecide Goodstein's theoremnPeano Arithmetic,rthe continuumhypothesisn basic set theory, an be traced to ourfailure oexpress he (platonistic) maginedcontent f a second-orderaxiombymeans of first-orderxiom scheme the nduction xiom inthe case ofPeano Arithmetic,heseparationaxiom in the case of settheory.When we firstwrite down these axiom schemes we see them assecond-order ssertions.It is worthnoting hateventhisdistinctionsby no means a naive one.) The set-theoretic latonist ntends thepower-set f a set to containall its subsets n as strong sense of'all' aspossible;and in the same way the number-theoreticlatonist ntendsinduction oapplytoall subsets f o.But thefirst-orderxiom schemescan onlymake theseclaims for ets which can be definedbymeansofformulae n thelanguage ofthefirst-orderheory n question.Thereseemstobe nowayavailable tous ofexpressing hestrong enseof'all'which we need,otherthanbysaying t louder, n what I described nmybook as the mannerofCzech borderguards' (whichustshowshowquicklya book can date).This point s notnew; it has beenmade byKreisel26: urfeeling hatthesequestionsoughtto be decidable stemsfrom he factthatthey re

26 G. Kreisel, InformalRigourand Completeness roofs',n I. Lakatos (ed.), Problemsinthe hilosophyfMathematicsAmsterdam:North-Holland,1967).? The editors fThePhilosophicalyarterly,993.


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ITERATIVESET THEORY 193indeed decidable in thesecond-order ystems. o much theworse forfirst-orderogic, you might think. But Kreisel's remark is anexplanationfor n observedfact bout ourpsychology, ota proposedprogrammefor ettling hecontinuumproblem. t would be better osay: thecontinuum uestion sdecidable in second-order ettheory othe extent hat econd-orderogic sdecidable. The problems nvolvedhereare exactlythe same as before: hereformulationn second-orderterms ransfershedifficultiesrom ettheory ologic; itdoes not solvethem.Fitzwilliam ollege, ambridge

? The editors f ThePhilosophicaluarterly,993.

potter iterative set theory

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