The Iterative Conception of Set Author(S): George Boolos Reviewed Work(S): Source: the Journal of Philosophy, Vol

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The Iterative Conception of Set Author(s): George Boolos Reviewed work(s): Source: The Journal of Philosophy, Vol. 68, No. 8, Philosophy of Logic and Mathematics (Apr. 22, 1971), pp. 215-231 Published by: Journal of Philosophy, Inc. Stable URL: http://www.jstor.org/stable/2025204 . Accessed: 12/01/2013 10:53

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This content downloaded on Sat, 12 Jan 2013 10:53:17 AM All use subject to JSTOR Terms and Conditions THE JOURNALOF PHILOSOPHY

VOLUME LXVIII, NO. 8, APRIL 22, I97I

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THE ITERATIVE CONCEPTION OF SET A SET, accordingto Cantor,is "any collection.., intoa whole of definite,well-distinguished objects... of our intuitionor thought.'1Cantor alo sdefineda set as a "many, whichcan be thoughtof as one, i.e., a totalityof definiteelements that can be combinedinto a whole by a law.'2 One mightobject to the firstdefi- nitionon the groundsthat it uses the conceptsof collectionand whole, which are notionsno betterunderstood than that of set,that there ought to be sets of objects that are not objects of our thought,that 'intuition'is a termladen with a theoryof knowledgethat no one should believe, that any object is "definite,"that there should be sets of ill-distinguishedobjects, such as waves and trains,etc., etc. And one mightobject to the second on the groundsthat 'a many' is ungrammatical,that if somethingis "a many" it should hardly be thoughtof as one, that totalityis as obscureas set,that it is farfrom clear how laws can combineanything into a whole,that thereought to be othercombinations into a whole than thoseeffected by "laws," etc., etc. But it cannot be denied that Cantor's definitionscould be used by a person to identifyand gain some understandingof the sort of object of which Cantor wished to treat. Moreover,they do suggest-although, it must be conceded,only very faintly-two im- portantcharacteristics of sets: that a set is "determined"by its elements in the sense that sets with exactly the same elements are

I "Unter einer 'Menge' verstehenwir jede ZusammenfassungM von bestimrnten wohlunterschiedenen Objekten m unserer Anschauung oder unseres Denkens (welche die 'Elemente' von M genannt werden) zu einem Ganzen." Georg Cantor, GesammelteAbhandlungen, Ernst Zermelo, ed. (Berlin, 1932), p. 282. 2 .... jedes Viele, welches sich als Eines denken lIsst, d.h. jeden Inbegriff bestimmter Elemente, welcher durch ein Gesetz zu einem Ganzen verbunden werden kann" (ibid., p. 204). 215

This content downloaded on Sat, 12 Jan 2013 10:53:17 AM All use subject to JSTOR Terms and Conditions 2I6 THE JOURNAL OF PHILOSOPHY identical,and that,in a sense, the clarificationof whichis one of the principal objects of the theorywhose rationale we shall give, the elementsof a set are "prior to" it. It is not to be presumedthat the concepts of set and memberof can be explainedor definedby means of notionsthat are simpleror conceptuallymore basic. However, as a theoryabout sets mightit- selfprovide the sort of elucidationabout sets and membershipthat good definitionsmight be hoped to offer,there is no reason forsuch a theoryto begin with,or even contain,a definitionof 'set'. That we are unable to given informativedefinitions of not or for some does not and should not prevent the development of quantificational logic, which provides us with significantinformation about these concepts. I. NAIVE SET THEORY Here is an idea about sets that mightoccur to us quite naturally, and is perhaps suggestedby Cantor's definitionof a set as a totality of definiteelements that can be combinedinto a whole by a law. By the law of excluded middle,any (one-place) predicatein any language eitherapplies to a given object or does not. So, it would seem, to any predicatethere correspond two sorts of thing: the sort of thingto which the predicateapplies (of which it is true) and the sort of thingto which it does not apply. So, it would seem, forany predicate thereis a set of all and only those thingsto which it applies (as well as a set of just those thingsto whichit does not apply). Any set whosemembers are exactlythe thingsto whichthe predicate applies-by the axiom of extensionality,there cannot be twosuch sets-is called the extensionof the predicate. Our thought might thereforebe put: "Any predicate has an extension." We shall call this proposition,together with the argumentfor it, the naive conceptionof set. The argumenthas greatforce. How could therenot be a collection, or set, of just those thingsto which any given predicate applied? Isn't anythingto whicha predicateapplies similarto all otherthings to which it applies in preciselythe respect that it applies to them; and how could therefail to be a set of all thingssimilar to one another in this respect? Wouldn't it be extremelyimplausible to say, of any particularpredicate one mightconsider, that there weren't two kindsof thingit determined,namely, a kind of thingof whichit is true,and a kind of thingof whichit is not true? And why should one not take these kinds of thingsto be sets? Aren't kinds sets? If not, what is the difference? Let us denote by '3C' a certain standardlyformalized first-order language,whose variables range over all sets and individuals (= non-

This content downloaded on Sat, 12 Jan 2013 10:53:17 AM All use subject to JSTOR Terms and Conditions THE ITERATIVE CONCEPTION OF SET 2I7 sets), and whose nonlogicalconstants are a one-placepredicate letter 'S' abbreviating'is a set', and a two-placepredicate letter 'e', abbreviating'is a memberof'. Which sentencesof this language, together with their consequences,do we believe state truths about sets? Otherwiseput, whichformulas of SX should we take as axioms of a set theoryon the strengthof our beliefsabout sets? If the naive conceptionof set is correct,there should (at least) be a set of just those thingsto which q applies, if q5is a formulaof 3C.So (theuniversal closure of) F(3y) (Sy & (x) (xEy 4)))1 shouldexpress a truthabout sets (if no occurrenceof 'y' in k is free). We call the theorywhose axioms are the axiom of extensionality (to whichwe laterrecur), i.e., the sentence

(x) (y) (Sx & Sy & (Z)(zEx +->xy) --->x = y) and all formulas r(3y)(Sy & (x)(xEy <->0))1 (where 'y' does not occur freein 4) naive set theory. Some of the axioms of naive set theoryare the formulas

(3 y) (Sy & (x) (xEy -*xx)) (3y) (Sy & (x) (xEy*- (x=z v x=w))) (3 y) (Sy & (x)(xey -> (3 w) (x & wEZ))) (3y) (Sy & (x) (xEy*-> (Sx & x=x))) The firstof these formulasstates that thereis a set that contains no members.By the axiom of extensionality,there can be at most one such set. The second states that thereis a set whose sole members are z and w; the third,that thereis a set whosemembers are just the membersof membersof z. The last, which states that there is a set that contains all sets whatsoever,is ratheranomalous; forif thereis a set that contains all sets, a universal set, that set contains itself,and perhaps the mind ought to boggle at the idea of something'scontaining itself. Nevertheless,naive set theoryis simple to state, elegant, initially quite credible,and natural in that it articulatesa view about sets that mightoccur to one quite naturally. Alas, it is inconsistent.

Proof of theInconsistency of Naive Set Theory(Russell's paradox) No set can containall and only thosesets whichdo notcontain themselves.For ifany such set existed,if it containeditself, then, as it contains only those sets whichdo not contain themselves,it would not contain itself;but if it did not contain itself,then, as it contains all those sets whichdo not contain themselves,it would contain itself. Thus any such set would have to contain itselfif and only if it did

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not contain itself. Consequently,there is no set that contains all and only those sets whichdo not contain themselves. This argument,which uses no axioms of naive set theory,or any otherset theory,shows that the sentence (3y) (Sy & (x) (xey<-> (Sx & - xex))) is logicallyvalid and, hence, is a theorem of any theory that is expressedin X3.But one of the axioms and, hence, one of the the- orems,of naive set theoryis the sentence (3y) (Sy & (x) (xEy*- (Sx & -x- x X))) Naive set theoryis thereforeinconsistent. II. THE ITERATIVE CONCEPTION OF SET Faced with the inconsistencyof naive set theory,one mightcome to believe that any decision to adopt a system of axioms about sets would be arbitraryin that no explanation could be given why the particularsystem adopted had any greaterclaim to describe what we conceive sets and the membershiprelation to be like than some othersystem, perhaps incompatiblewith the one chosen. One might thinkthat no answercould be given to the question: why adopt this particularsystem rather than that or thisother one? One mightsup- pose that any apparentlyconsistent theory of sets would have to be unnatural in some way or fragmentary,and that, if consistent,its consistencywould be due to certainprovisions that were laid down for the expresspurpose of avoiding the paradoxes that show naive set theoryinconsistent, but that lack any independentmotivation. One mightimagine all this; but thereis anotherview of sets: the iterativeconception of set, as it is sometimescalled, whichoften strikes people as entirelynatural, freefrom artificiality, not at all ad hoc, and one they mightperhaps have formulatedthemselves. It is, perhaps, no more natural a conceptionthan the naive conception,and certainlynot quite so simple to describe.On the other hand, it is, as faras we know,consistent: not onlyare the sets whose existencewould lead to contradictionnot assumed to exist in the axioms of the theoriesthat expressthe iterativeconception, but the morethan fiftyyears of experiencethat practicingset theoristshave had withthis conception have yieldeda good understandingof what can and what cannot be proved in these theories,and at present therejust is no suspicionat all that they are inconsistent.3 a The conceptionis wellknown among logicians; a ratherdifferent version of it is sketchedin chapterIx of JosephR. Shoenfield'sMathematical Logic (Reading, Mass.: Addison-Wesley,1967). I learnedof it principallyfrom Putnam, Kripke, and Donald Martin.Authors of set-theory texts either omit it orrelegate it to back pages; philosophers,in the main,seem to be unawareof it,or of the preeminence ofZF, whichmay be said to embodyit. It is due primarilyto Zermeloand Russell.

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The standard, first-ordertheory that expressesthe iterativecon- ceptionof set as fullyas a first-ordertheory in the language C of set theory4can, is knownas Zermelo-Fraenkelser theory, or 'ZF' forshort. There are other theoriesbesides ZF that embody the iterativeconception:one of them,Zermelo set theory,or "Z", whichwill occupy us shortly,is a subsystemof ZF in the sense that any theoremof Z is also a theorem of ZF; two others, von-Neumann-Bernays-Godelset theoryand Morse-Kelley set theory,are supersystems(or exten- sions) of ZF, but they are most commonlyformulated as second- ordertheories. Othertheories of sets,incompatible with ZF, have been proposed.5 These theoriesappear to lack a motivationthat is independentof the paradoxes in the followingsense: they are not, as Russell has writ- ten, "such as even the cleverestlogician would have thoughtof if he had not knownof the contradictions."6A finaland satisfyingresolu- tion to the set-theoreticalparadoxes cannot be embodiedin a theory that blocks theirderivation by artificialtechnical restrictions on the set of axioms that are imposedonly because paradox would otherwise ensue; these other theoriessurvive only throughsuch artificialde- vices. ZF alone (togetherwith its extensionsand subsystems)is not only a consistent(apparently) but also an independentlymotivated theoryof sets: thereis, so to speak, a "thoughtbehind it" about the nature of sets which mighthave been put fortheven if,impossibly, naive set theoryhad been consistent.The thought,moreover, can be describedin a rough,but informativeway withoutfirst stating the theorythe thoughtis behind. In order to see why a conceptionof set other than the naive conception mightbe desired even if the naive conceptionwere consistent,let us take anotherlook at naive set theoryand the anomalous- ness of its axiom, '(3y)(Sy & (x)(xEy<-- (Sx & x = x)))'. Accordingto this axiom thereis a set that contains all sets, and thereforethere is a set that containsitself. It is importantto realize how odd the idea of something'scontaining itself is. Of course a set can and must includeitself (as a subset). But containitself? What- ever tenuoushold on the conceptsof set and memberwere given one by Cantor's definitionsof 'set' and one's ordinaryunderstanding of 'element','set', 'collection',etc. is altogetherlost if one is to suppose that some sets are membersof themselves.The idea is paradoxical not in the sense that it is contradictoryto suppose that some set is

6? contains (countably many) variables, ranging over (pure) sets, '=', and 'e', which is its sole nonlogical constant. 6 For example, Quine's systems NF and ML. 6 Russell, My Philosophical Development(New York: Simuon& Schuster, 1959), p. 80.

This content downloaded on Sat, 12 Jan 2013 10:53:17 AM All use subject to JSTOR Terms and Conditions 220 THE JOURNAL OF PHILOSOPHY a memberof itself,for, after all, '(3 x) (Sx & xex)' is obviouslycon- sistent,but that if one understands'e' as meaning'is a memberof', it is very,very peculiar to suppose it true. For when one is told that a set is a collectioninto a whole of definiteelements of our thought, one thinks: Here are some things. Now we bind them up into a whole.7 Now we have a set. We don't suppose that what we come up with after combiningsome elementsinto a whole could have been one of the verythings we combined(not, at least, ifwe are combining two or more elements). If (3x)(Sx & xex), then (3x)(3y)(Sx & Sy & xcy &yEx). Thesup- position that there are sets x and y each of which belongs to the otheris almost as strangeas the suppositionthat some set is a self- member. There is of course an infinitesequence of such cyclical pathologies: (3x)(3y)(3z)(Sx & Sy & Sz & xEy & yez & zex), etc. Only slightlyless pathologicalare the suppositionsthat thereis an ungroundedset,8 or that thereis an infinitesequence of sets x0,x1, x2, . , each termof whichbelongs to the previousone. There does not seem to be any argumentthat is guaranteed to persuade someone who really does not see the peculiarityof a set's belongingto itself,or to one of its members,etc., that these states of affairsare peculiar. But it is in part the sense of theiroddity that has led set-theoriststo favorconceptions of set, such as the iterative conception,according to which what they findodd does not occur. We describethis conceptionnow. Our descriptionwill have three parts. The firstis a roughstatement of the idea. It containssuch ex- pressionsas 'stage', 'is formedat', 'earlier than', 'keep on going', which must be exorcisedfrom any formaltheory of sets. From the roughdescription it sounds as if sets were continuallybeing created, which is not the case. In the second part, we presentan axiomatic theory which partially formalizesthe idea roughly stated in the firstpart. For reference,let us call this theorythe stage theory.The third part consists in a derivation from the stage theory of the axioms of a theoryof sets. These axioms are formulasof ?, the language of set theory,and contain none of the metaphoricalex- pressionswhich are employedin the rough statementand of which abbreviationsare foundin the language in which the stage theory is expressed. Here is the idea, roughlystated: A set is any collectionthat is formedat some stage of the following process: Begin with individuals (if thereare any). An individual is 7 We put a "lasso" aroundthem, in a figureof Kripke's. 8 x is ungroundedif x belongsto someset z, each ofwhose members has a meim- berin commonwith z.

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an object that is not a set; individualsdo not contain members.At stage zero (we count fromzero instead of one) formall possible collectionsof individuals. If thereare no individuals,only one collection, the null set, whichcontains no members,is formedat this0th stage. If thereis only one individual,two sets are formed:the null set and the set containingjust that one individual.If thereare two individuals, foursets are formed;and in general,if thereare n individuals, 2n sets are formed.Perhaps there are infinitelymany individuals. Still, we assume that one of the collectionsformed at stage zero is the collectionof all individuals,however many of themthere may be. At stage one, formall possible collectionsof individualsand sets formedat stage zero. If thereare any individuals,at stage one some sets are formedthat contain both individuals and sets formedat stage zero. Of course some sets are formedthat contain only sets formedat stage zero. At stage two, formall possible collectionsof individuals,sets formedat stage zero, and sets formedat stage one. At stage three,form all possible collectionsof individuals and sets formedat stages zero, one, and two. At stage four,form all possible collectionsof individuals and sets formedat stages zero, one, two, and three. Keep on going in this way, at each stage formingall possiblecollections of individualsand sets formedat earlierstages. Immediatelyafter all of stages zero,one, two,three, . . ., thereis a stage; call it stage omega. At stage omega, formall possible collectionsof individualsformed at stages zero, one, two,.... One of these collectionswill be the set of all sets formedat stages zero, one, two,

Afterstage omega thereis a stage omega plus one. At stage omega plus one formall possible collectionsof individualsand sets formed at stages zero, one, two..., and omega. At stage omega plus two formall possible collectionsof individualsand sets formedat stages zero, one, two,. . ., omega, and omega plus one. At stage omega plus threeform all possible collectionsof individualsand sets formedat earlierstages. Keep on goingin this way. Immediatelyafter all of stages zero, one, two,..., omega, omega plus one, omega plus two,. . ., thereis a stage,call it stage omega plus omega (or omega times two). At stage omega plus omega formall possible collectionsof individualsand sets formedat earlierstages. At stage omega plus omega plus one...... omega plus omega plus omega (or omega times three)...... (omega timesfour) ...... omega timesomega ...... Keep on going in this way.... Accordingto thisdescription, sets are formedover and over again:

This content downloaded on Sat, 12 Jan 2013 10:53:17 AM All use subject to JSTOR Terms and Conditions 222 THE JOURNAL OF PHILOSOPHY in fact,according to it, a set is formedat everystage later than that at whichit is firstformed. We could continueto say thisif we liked; instead we shall say that a set is formedonly once, namely,at the earliest stage at which,on our old way of speaking,it would have been said to be formed. That is a rough statementof the iterativeconception of set. Ac- cordingto thisconception, no set belongsto itself,and hence thereis no set of all sets; forevery set is formedat some earlieststage, and has as membersonly individualsor sets formedat stillearlier stages. Moreover,there are not two sets x and y, each of which belongs to the other.For ify belongedto x, y would have had to be formedat an earlierstage than the earlieststage at whichx was formed,and if x belonged to y, x would have had to be formedat an earlier stage than the earlieststage at whichy was formed.So x would have had to be formedat an earlierstage than the earlieststage at which it was formed,which is impossible.Similarly, there are no sets x, y, and z such that x belongsto y, y to z, and z to x. And in general,there are no sets xo, x1,X2,. . ., xn such that xo belongs to x1,x1 to x2,. . ., Xn-1 to xn, and x,,to xo. Furthermoreit would appear that thereis no sequenceof sets xo,X1, X2, X3,. . . such thatx1 belongsto XO, x2be- longs to x1, x3 belongs to X2, and so forth.Thus, if sets are as the iterativeconception has them,the anomalous situationsdo not arise in which sets belong to themselvesor to othersthat in turn belong to them. The sets of whichZF in its usual formulationspeaks ("quantifies over") are not all the sets thereare, ifwe assume that thereare some individuals,but only thosewhich are formedat some stage underthe assumptionthat thereare no individuals.These sets are called pure sets. All membersof a pure set are pure sets,and any set, all of whose membersare pure,is itselfpure. It may not be obvious that any pure sets are ever formed,but the set A, whichcontains no membersat all, is pure, and is formedat stage 0. {A } and { {A} } are also both pure and are formedat stages 1 and 2, respectively.There are many others.From now on, we shall use the word 'set' to mean 'pure set'. Let us now try to state a theory,the stage theory,that precisely expressesmuch, but not all, of the contentof the iterativeconception. We shall use a language, J,in whichthere are two sortsof variables: variables 'x', 'y', 'z', 'w',...., whichrange over sets, and variables 'r', 's', 't', whichrange over stages. In additionto the predicate letters'E' and '=' of ?, Jalso contains two new two-placepredicate letters'E', read 'is earlier than', and 'F', read 'is formedat'. The rules of formationof Jare perfectlystandard.

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Here are some axioms governingthe sequence of stages:

(I) (s) ' sEs (No stage is earlierthan itself.) (II) (r) (s) (t)((rEs & sEt) -- rEt) (Earlierthan is transitive.) (III) (s) (t)(sEtv s=tv tEs) (Earlierthan is connected.) (IV) (3 s) (t)(t#s -* sEt) (Thereis an earlieststage.) (V) (s) (3 t)(sEt & (r) (rEt-- (rEs v r= s))) (Immediatelyafter any stagethere is another.) Here are some axioms describingwhen sets and theirmembers are formed: (VI) (3 s) ((3 t)tEs& (t) (tEs-- (3 r)(tEr & rEs))) (Thereis a stage,not the earliestone, which is not immediatelyafter any one stage. In therough description, stage omega was sucha stage.) (VII) (x)(3s)(xFs & (t)(xFt-- t=s)) (E very set is formedat some uniquestage.) (VIII) (x) (y)(s) (t)((yex & xFs & yFt)-- tEs) (Everymember of a set is formedbefore, i.e., at an earlierstage than, the set.) (I X) (x) (s) (t)(xFs & tEs-- (3 y)(3 r)(yex & yFr & (t= rv tEr))) (If a set is formedat a stage,then, at or afterany earlierstage, at least one ofits membersis formed.So it neverhappens that all membersof a set are formedbefore some stage, but theset is notformed at thatstage, if it has notbeen formed already.) We may capture part of the contentof the idea that at any stage every possiblecollection (or set) of sets formedat earlier stages is formed (if it has not yet been formed) by taking as axioms all formulasr(s)(3y)(x)(xey *-+ (x & (3t)(tEs & xFt)))l, where x is a formulaof the language Jin which no occurrenceof 'y' is free.Any such axiom will say that forany stage thereis a set of just thosesets to whichx applies that are formedbefore that stage. Let us call these axioms specificationaxioms. There is still one importantfeature contained in our rough de- scriptionthat has not yet been expressedin the stage theory: the analogy between the way sets are inductivelygenerated by the pro- cedure described in the rough statementand the way the natural numbers0, 1, 2 ... are inductivelygenerated from 0 by the repeated application of the successoroperation. One way to characterizethis featureis to assert a suitable inductionprinciple concerning sets and stages; for,as Frege, Dedekind, Peano, and othershave enabled us to see, the contentof the idea that objects of a certainkind are inductivelygenerated in a certainway is just the propositionthan an appropriateinduction principle holds of those objects. The principleof mathematicalinduction, the inductionprinciple governingthe natural numbers,has two forms,which are interderiv-

This content downloaded on Sat, 12 Jan 2013 10:53:17 AM All use subject to JSTOR Terms and Conditions 224 THE JOURNAL OF PHILOSOPHY able on certain assumptions about the natural numbers.The first versionof the principleis the statement

(P) [ (PO & (n) [Pn -+ PSn]J) -* (n)Pn] which may be read, 'If 0 has a propertyand if whenevera natural numberhas the propertyits successordoes, thenevery natural number has the property'.The second versionis the statement

(P)[ (n) ((m)[m < n ->Pm] -Pn) -* (n)Pn] It may be read, 'If each naturalnumber has a propertyprovided that all smallernatural numbers have it, thenevery natural number has the property'. The inductionprinciple about sets and stages that we should like to assertis modeledafter the second formof the principleof mathematical induction.Let us say that a stage s is coveredby a predicate if the predicate applies to every set formedat s. Our analogue for sets and stages of the second formof mathematicalinduction says that if each stage is coveredby a predicateprovided that all earlier stagesare coveredby it, thenevery stage is coveredby thepredicate. The fullforce of this assertioncan be expressedonly witha second-order quantifier.However, we can capture some of its contentby taking as axioms all formulas

F(s) ((t)[tEs -* (x) (xFt -+ 0) -* (x) (xFs x)) 2 (s) (x) (xFs x) wherex is a formulaof 9 that containsno occurrencesof 't' and 0 is just like x except forcontaining a freeoccurrence of 't' whereverx contains a freeoccurrence of 's'. [Observe that '(x) (xFs -* X)' says that X applies to every set formedat stage s and, hence, that s is covered by x.] We call these axioms inductionaxioms. III. ZERMELO SET THEORY We complete the descriptionof the iterative conceptionof set by showinghow to derive the axioms of a theoryof sets fromthe stage theory.The axiomswe derivespeak onlyabout sets and membership: they are formulasof ?. The axiomof the nullset: (3 y)(x) - xEy. (Thereis a set withno members.) Derivation.Let X = 'x = x'. Then (s) (3y) (x) (xEy -+ (x = x & (t) (tEs & xFt))) is a specificationaxiom, according to which,for any stage,there is a set of all setsformed at earlierstages. As thereis an earlieststage, stage 0, beforewhich no setsare formed,there is a set thatcontains no members.

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Note that,by axiom(IX) ofthe stage theory, any set withno membersis formedat stage 0; forif it wereformed later, it wouldhave to have a member(that was formedat or afterstage 0). The axiom of pairs: (z)(w)(3y)(x)(xEy-+ (x=zv x=w)). (For any sets z and w, not necessarilydistinct, there is a set whosesole membersare z and w.) Derivation.Let X = '(x= z v x= w)'. Then (s) (3 y)(x) (xEy- ( (x= z v x= w) & (3 t)(tEs & xFt))) is a specificationaxiom, according to which,for any stage, there is a set of all sets formedat earlierstages that are identicalwith either z orw. Any set is formedat somestage. Let r be thestage at whichz is formed;s, the stageat whichw is formed.Let t be a stagelater than both r and s. Then thereis a set of all sets formedat stagesearlier than t thatare identical withz or w. So thereis a set containingjust z and w. The axiom of unions: (z) (3 y)(x) (xEy*-+ (3 w) (xEw& wEz)). (For any set z, thereis a set whosemembers are just the membersof membersof z.) Derivation.'(s) (3 y) (x) (xEy(-+ ((3 w) (xEw & wEZ) & (3 t) (tEs & xFt)))' is a specificationaxiom, according to which,for any stage,there is a set of all membersof membersof z formedat earlierstages. Let s be the stage at whichz is formed.Every memberof z is formedbefore s, and hence everymember of a memberof z is also formedbefore s. Thus thereis a set of all membersof membersof z. The power-setaxiom: (z) (3 y) (x) (xEy+ (w)(wEx -- wez)). (For any set z, thereis a set whosemembers are just thesubsets of z.) Derivation. '(s) (3 y) (x) (xEy(-+ () (wEx-4 wEZ) & (3 t) (tEs & xFt)))' is a specificationaxiom, according to which,for any stage,there is a set ofall subsetsof z formedat earlierstages. Let t be thestage at whichz is formed and let s be thenext later stage. If x is a subsetof z, thenx is formedbe- fores. For otherwise,by axiom (IX), therewould be a memberof x that was formedat or aftert and, hence,that was nota memberof z. So there is a set ofall subsetsof z formedbefore s, and hencea set ofall subsetsof z. The axiomof infinity:

(3 y) (( x) (xEy & (z) - ZEx) & (x) (xEy -> (3 z) (zEy & (w) (wEz -+ (wEx v w= x))))) (Call a set nullif it has no members.Call z a successorof x ifthe members ofz are just thoseof x and x itself.Then there is a setwhich contains a null set and whichcontains a successorof any set it contains.) Derivation.Let us firstobserve that every set x has a successor.For let y be a set containingjust x and x (axiomof pairs),and let w be a set containingjust x and y (axiom of pairs again), and let z containjust the membersof membersof w (axiomof unions).Then z is a successorof x,

This content downloaded on Sat, 12 Jan 2013 10:53:17 AM All use subject to JSTOR Terms and Conditions 226 THE JOURNAL OF PHILOSOPHY forits membersare just x and x's members.Next, note that if z is a successor of x, x is formedat r, and t is the next stage afterr, then z is formed at t. For every memberof z is formedbefore t. So z is formedat or beforet, by axiom (IX). But x, which is in z, is formedat r. So z cannot be formed at or beforer. So z cannot be formedbefore t. Now, by axiom (VI), there is a stage s, not the earliest one, which is not immediatelyafter any stage. '(s) (3 y) (x) (xEy*-+ (x = x & (3 t) (tEs & xFt)))' is a specificationaxiom, accordingto which,for any stage, thereis a set of all sets formedat earlier stages. So there is a set y of all sets formedbefore s. y thus contains all sets formed at stage 0, and hence contains a null set. And if y contains x, y contains all successors of x (and there are some), for all these are formedat stages immediatelyafter stages befores and, hence, at stages themselvesbefore s. Axioms of separation (Aussonderungsaxioms):All formulas r (z)(o y)(xw (x) noy (xez & y)) i where is aI formulaof in which? no occurrenceof 'y' is free. Derivation.If 0 is a formulaof 2 in whichno occurrenceof 'y' is free,then F (s) (3 y) (x) (xEy*-+ ((xEz & 4) & (3 t) (tEs & xFt))) 1 is a specificationaxiom, which we may read, 'For any stage s, there is a set of all sets formedat earlier stages, which belong to z and to which4 applies. Let s be the stage at which z is formed.All membersof z are formedbefore s. So, for any z, thereis a set of just those membersof z to which4 applies, whichwe would write,F(z) (3 y) (x) (xiy *-+ (xEz & 0)))1. A formalderivation of an Aussond- erungsaxiom would use the specificationaxiom described and axioms (VII) and (VIII) of the stage theory. Axioms of regularity:All formulas r(3 x)) -* (3 x) () & (y) (yEx- ))1, where4 does not contain 'y' at all and t'is just like 4 except forcontaining an occurrence of 'y' wherever 4 contains a free occurrenceof 'x'. Derivation.The idea: Suppose 4 applies to some set x'. x' is formedat some stage. That stage is thereforenot covered by F41. By an induction axiom, there is then a stage s not covered by r 4), although all stages earlier than s are covered by rF,-4. Since s is not covered by r-4)1, there is an x, formedat s, to which r,0)i does not apply, i.e., to which 4 applies. If y is in x, however,y is formedbefore s, and hence the stage at which is is formedis covered by rF_)1. So rF 4)1 applies to y (which is what r ,t" says). For a formalderivation, contrapose, reletter, and simplifythe induction axiom F(s) ((t) (tEs -- (x) (xFt - 4))) (x) (xFs - 4))) (s) (x) (xFs ) 4))1 so as to obtain r (3 s) (3 x) (xFs &4)) (3 s) (3 x) (xFs & 4 & (y) (t) (tEs & yFt-r-- t))l Assume r(3 x)Wl. Use axiom (VII) and modus ponens to obtain F(3 s) (3 x) (xFx & 4 & (y) (t) (tEs & yFt---ti) 0)

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Use axioms (VII) and (VIII) to obtain r(3 x) (4 & (y) (yex--->- i))1 from this. The axioms of regularity(partially) expressthe analogue forsets of the version of mathematical induction called the least-number principle:if there is a numberthat has a property,then thereis a least numberwith that property.The analogue itselfhas been called the principleof set-theoreticalinduction.9 Here is an application of set-theoreticalinduction. Theorem:No set belongsto itself. Proof.Suppose that some set belongsto itself,i.e., that (3 x)xex. (3 x)xex -> (3 x) (xEx & (y) (yex - yey)) is an axiomof regularity.By modusponens, then, some set x belongsto itselfthough no memberof x (not even x) belongsto itself.This is a contradiction.

The axioms whose derivationswe have given are thosestatements whichare oftentaken as axioms of ZF and whichare deduciblefrom all (sufficientlystrong10) theories that can fairlybe called formaliza- tions of the iterativeconception, as roughlydescribed. (The axiom of extensionalityhas a special status, which we discuss below.) Other axioms than those we have given could have been taken as axioms of the stage theory.For example,we could have fairlytaken as an axiom a statementasserting the existenceof a stage, not immediatelylater than any stage, but later than some stage that is itselfneither the earlieststage nor immediatelylater than any stage. Such an axiom would have enabled us to deduce a strongeraxiom of infinitythan the one whose derivation we have given, but this strongerstatement is not commonlytaken as an axiom of ZF. We could also have derivedother statements from the stage theory,such as the statementthat no set belongsto any of its members,but this statementis nevertaken as an axiom of ZF. We do not believe that the axioms of replacementor choice can be inferredfrom the iterative conception. One of the axioms of regularity,

(3 x)xEz-+ (3 x) (xEz & (y)&yEx yez) is sometimescalled theaxiom of regularity;in the presenceof other axioms of ZF, all the otheraxioms of regularityfollow from it. The name 'Zermelo set theory'is perhaps most commonlygiven to the 9 By Tarski,among others. 10'Sufficiently strong' may herebe taken to mean "at least as strongas the stagetheory."

This content downloaded on Sat, 12 Jan 2013 10:53:17 AM All use subject to JSTOR Terms and Conditions 228 THE JOURNAL OF PHILOSOPHY theory whose axioms are '(x) (y) ((z) (zEx * xEy) -> x=y)', i.e., the axiom of extensionality,and the axioms of the null set, pairs, and unions,the power-setaxiom, the axiom of infinity,all the Aussonder- ungsaxioms,and the axiom ofregularity." With the exceptionof the axiom of extensionality,then, all the axioms of Zermelo set theory followfrom the stage theory. IV. ZERMELO-FRAENKEL SET THEORY The axioms of replacement.ZF is the theorywhose axioms are those of Zermeloset theoryand all axioms of replacement.'2A formulaof ? is an axiom of replacementif it is the translationinto ? ofthe result "substituting"a formulaof ? for 'F' in

F is a function-> (z) (3y) (x) (xEy<- (3w) (wEz & F(w)=x)) There is an extensionof the stage theoryfrom which the axioms of replacement could have been derived. We could have taken as axioms all instances (that can be expressedin 9) of a principlewhich may be put, 'If each set is correlatedwith at least one stage (no matterhow), then forany set z thereis a stage s such that foreach memberw of z, s is later than some stage withwhich w is correlated'. This boundingor cofinalityprinciple is an attractivefurther thought about the interrelationof sets and stages, but it does seem to us to be a furtherthought, and not one that can be said to have been meant in the roughdescription of the iterativeconception. For that thereare exactlyw, stages does not seem to be excluded by anything said in the roughdescription; it would seem that Rco (see below) is a model forany statementof ? that can (fairly)be said to have been implied by the rough description,and not all of the axioms of replacement hold in Rw1.'3Thus the axioms of replacementdo not seem to us to followfrom the iterativeconception. Adding the axioms of replacementto those of Zermeloset theory enables us to definea sequence of sets, {Ra }, with which the stages of the stage theorymay be identified.Suppose we put Ro = the null set; Ra?il= R,a, the powerset of Ra, and R), = U0<;R#(X a limitordi-

11In his 1908 paper, "Investigations in the Foundations of Set Theory I" ["Untersuchungen uber die Grundlagen der Mergenlehre," MathematischeAn- naler, LX (1908): 261-281; reprintedin Jean van Heijenoort, ed., From Frege to Godel (Cambridge, Mass.: Harvard, 1967), pp. 119-215], Zermelo took as axioms versionsof the axioms of extensionality,the null set, pairs (and unit set), unions, the power-set axiom, the axiom of infinity,the Aussonderungsaxioms,and the axiom of choice. 12Sometimes the axiom of choice is also consideredone of the axioms of ZF. 13 Worse yet, Raj would also seem to be such a model. (a, is the firstnonrecursive ordinal.)

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nal)-axioms of replacementensure that the operation R is well- defined-and say that s is a stage if (3 a)s = R,a, that x is formedat s if x is subset but not a memberof s, and that s is earlierthan t if, forsome a, ,B,s = Ra, t = Ro, and a < A. Then we can prove as the- oremsof ZF not only the translationsinto the language of set theory of the axioms of the stage theory,but also those of all those stronger axioms assertingthe existenceof stages furtherand further"out" that mighthave been suggestedby the roughdescription (and those of the instancesof the boundingprinciple which are expressiblein 9 as well). ZF thus enables us to describeand assert the fullfirst-order contentof the iterativeconception within the language of set theory. Althoughthey are not derived fromthe iterativeconception, the reason foradopting the axioms of replacementis quite simple: they have many desirable consequences and (apparently)no undesirable ones. In addition to theoremsabout the iterativeconception, the consequences of replacement include a satisfactoryif not ideal'4 theoryof infinitenumbers, and a highlydesirable result that justi- fiesinductive definitions on well-foundedrelations. The axiom of extensionality.The axiom of extensionalityenjoys a special epistemologicalstatus shared by none of the otheraxioms of ZF. Were someoneto deny anotherof the axiomsof ZF, we would be rathermore inclined to suppose, on the basis of his denial alone, that he believed that axiom false than we would if he denied the axiom of extensionality.Although 'there are unmarriedbachelors' and 'there are no bachelors' are equally preposterousthings to say, if someone were to say the former,he would far more invite the suspicionthat he did not mean what he said than someone who said the latter. Similarly,if someone were to say, "there are distinctsets with the same members,"he would therebyjustify us in thinkinghis usage nonstandardfar more than someonewho assertedthe denial of some other axiom. Because of this difference,one might be tempted to call the axiom of extensionality"analytic," true by virtue of the meaningsof the words containedin it, but not to considerthe other axioms analytic. It has been persuasivelyargued, by Quine and others,however, that until we have an acceptable explanationof how a sentence (or what it says) can be true in virtue of meanings,we should refrain fromcalling anything analytic. It seems probable, nevertheless,that whatever justificationfor accepting the axiom of extensionality theremay be, it is more likely to resemblethe justificationfor ac- 14An ideal theorywould decide the continuum hypothesis, at least.

This content downloaded on Sat, 12 Jan 2013 10:53:17 AM All use subject to JSTOR Terms and Conditions 230 THE JOURNAL OF PHILOSOPHY ceptingmost of the classical examples of analyticsentences, such as 'all bachelors are unmarried'or 'siblingshave siblings' than is the justificationfor accepting the other axioms of set theory.That the conceptsof set and beinga memiberof obey the axiom of extensionality is a far morecentral feature of our use of themthan is the fact that they obey any otheraxiom. A theorythat denied, or even failed to affirm,some of the other axioms of ZF mightstill be called a set theory,albeit a deviant or fragmentaryone. But a theorythat did not affirmthat the objects withwhich it dealt were identicalif they had the same memberswould only by charitybe called a theoryof setsalone. The axiomof choice. One formof theaxiom of choice, sometimes called the "multiplicativeaxiom," is the statement,'For any x, ifx is a set of nonemptydisjoint sets (two sets are disjointif nothingis a member of both), thenthere is a set, called a choiceset for x, that contains exactly one memberof each of the membersof x'. It seems that, unfortunately,the iterativeconception is neutral with respectto the axiom of choice. It is easy to show that,since, as is now known,neither the axiom of choice nor its negationis a theorem of ZF, neitherthe axiom nor its negationcan be derivedfrom the stage theory.Of course the stage theory,which is supposed to formalizethe rough description,could be extended so as to decide the axiom. But it seems that no additional axiom, which would decide choice, can be inferredfrom the roughdescription without the assumptionof the axiom of choice itself,or some equally uncertain principle,in the inference.The difficultywith the axiom of choice is that the decisionwhether to regardthe roughdescription as imply- ing a principleabout sets and stages fromwhich the axiom could be derived is as difficulta decision, because essentiallythe same decision,as the decisionwhether to accept the axiom. Suppose that we tried to derive the axiom by arguing in this manner:Let x be a set of nonemptydisjoint sets. x is formedat some stage s. The membersof membersof x are formedat earlierstages than s. Hence, at s, if not earlier,there is a set formedthat contains exactlyone memberof each memberof x. But to assert thisis to beg the question. How do we know that such a choice set is formed?If a choice set is formed,it is indeed formedat or befores. But how do we know that one is formedat all? To argue that at s we can choose one memberfrom each memberof x and so forma choice set forx is also to beg the question: "we can't choose" one memberfrom each memberof x if thereis no choice set forx.

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To say this is not to say that the axiom of choice is not both obvious and indispensable.It is only to say that the justificationfor its acceptance is not to be foundin the iterativeconception of set. GEORGE BOOLOS Massachusetts Instituteof Technology

A PLEA FOR SUBSTITUTIONAL QUANTIFICATION * IN thisnote I shall discussthe relevanceto ontologyof whatis called the substitutionalinterpretation of quantifiers.According to this interpretation,a sentenceof the form'(3x) Fx' is true if and only if there is some closed term 't' of the language such that 'Ft' is true. This is opposed to the objectualinterpretation according to which '(3x)Fx' is true if and only if thereis some object x in the universeof discoursesuch that 'F' is trueof that object.' Ontology is not the only connection in which substitutional quantificationhas been discussed in recent years. It has been ad- vocated as a justificationfor restrictionson the substitutivityof identityin intensionallanguages2 or has been found necessary to * I owe to SidneyMorgenbesser and W. V. Quine the stimulusto writethis paper.I am also indebtedto Hao Wangfor a valuablediscussion, and to Quinefor illuminatingcomments on an earlierversion. 1 It is noteworthythat the substitutionalinterpretation allows truthof, say, first-orderquantified sentences to be givena directinductive definition, while in theobjectual interpretation the fundamental notion is truthof (satisfaction),and truthis definedin termsof it. Davidson'sversion of the correspondence theory of truthwould not be applicableto a substitutionallanguage. See "True to the Facts," thisJOURNAL, Lxvi, 21 (Nov. 6, 1969): 748-764. Quine has pointedout to me that in a languagewith infinitely many singular terms,a problemarises about defining truth for atomic formulas. Objectual quanti- ficationcan be nontrivialfor a languagewith no singularterms at all, but,if there are such,the problem can be resolvedby an auxiliarydefinition which assigns them denotations,relative to a sequencethat codes an assignmentto thefree variables. In the substitutionalcase, we need to definethe truthof an atomic formula Pt, . . . tmdirectly for closed termsti ... t,,,,for example, inductivelyby reducing the case formore complex terms to that forless complexterms. In the usual languageof elementarynumber theory, closed terms are constructedfrom 'O' by applicationsof variousfunction symbols which (except for the successorsymbol, say 'S') haveassociated with them defining equations that can be regardedas rules forreducing closed terms to canonicalform, namely, as numerals(a numeralis 'S' appliedfinitely many times to '0'), so thatan atomicformula Pt, ... tmis trueif and onlyif Pn1 .. . nmis true,where ni is the numeralcorresponding to ti. The truthof predicatesapplied to numeralsis definedeither trivially (as in the case of '-') or in a similarrecursive manner. Quine discusses this issue in ?6 of an unpublishedpaper, "Truth and Disquotation." 2 RuthBarcan Marcus, "Modalities and IntensionalLanguages," Synthese, XIII,

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