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Journal of Philosophy, Inc. 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 Your use of the JSTOR archive indicates your acceptance of the Terms & Conditions of Use, available at . http://www.jstor.org/page/info/about/policies/terms.jsp . JSTOR is a not-for-profit service that helps scholars, researchers, and students discover, use, and build upon a wide range of content in a trusted digital archive. We use information technology and tools to increase productivity and facilitate new forms of scholarship. For more information about JSTOR, please contact [email protected]. Journal of Philosophy, Inc. is collaborating with JSTOR to digitize, preserve and extend access to The Journal of Philosophy. http://www.jstor.org 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 _ ~ ~~~~~~- ~ '- ' 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 ele- ments 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 ap- plies (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 con- ceptionof 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 an- other 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', ab- breviating'is a memberof'. Which sentencesof this language, to- gether 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 mem- bers 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 them- selves.For ifany such set existed,if it containeditself, then, as it con- tains 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 This content downloaded on Sat, 12 Jan 2013 10:53:17 AM All use subject to JSTOR Terms and Conditions 2I8 THE JOURNAL OF PHILOSOPHY 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.
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