Different Systems of Set Theory

7/27/2019 Different Systems of Set Theory

1/10

Different Systems of Set TheoryMichael Potter

Later i n this entry the words 'set' and 'class' will take on precise meanings, so to beginwith we shall use the word 'collection' quite broadly to mean anything whose identityis solely a matter of what it s members are. Which collections exist? Two extremepositions are initially appealing. The first is to say tha t all do. Unfortunately this is in-consistent because of Russell's paradox (see PARADOXES OF SET AND PROPERTY).The second is to say th at none do hu t to talk as if they do when such talk can be shownto he eliminable and therefore harmless. This is consistent but f ar too weak to be ofmuch use. We need a n intermediate theory. Various theories of collections have beenproposed since the 1900s. What they all share is the axiom of extensionality, whichasserts th at if x and y ar e collections then

The fact tha t they share this is just a matter of definition: objects which do not satisfyextensionality a re not collections. Beyond extensionality, however, theories differ. Themost popular among mathematicians is Zermelo-Fraenkel set theory or ZF (see $ 3).A common alternative is von Neumann-Bernays-Gdel class theory or NBG (see $41,which allows for the same sets as ZF but also has proper classes, i.e. collections (suchas the class of all se ts or the class of all ordinals) whose members are sets but whichare not themselves sets. Two general principles have been used to motivate the axiomsof ZF and it s relatives. The f i s t is the iterative conception, according to which setsoccur cumulatively in layers, each layer containing all the members and subsets of allprevious layers. The second is the doctrine of limitation of size, according to which the'paradoxical sets' (i.e. the proper classes of NBG) fail to be sets because they are insome sense too big. Neither principle is altogether satisfactory as a justification forthe whole of ZF: for example, the replacement schema (see $3) s motivated only bylimitation of size; and foundation (see $3) s motivated only by the iterative conception.Among the other systems of set theory to have been proposed the one that h as receivedwidespread attention is Quine's NF (see $ 7), which seeks to avoid paradox by means ofa syntactic restriction but which has not been provided with an intuitive justificationon the basis of any conception of set. It is known that if N F is consistent then ZF isconsistent, but the converse result has still not been proved.

1 Virtual collectionsSometimes when we speak of collections it is just a manner of speaking and no more:if we say that Sarah belongs to the collection of all linguists, we might just as wellsay straight away that she is a linguist; nothing seems to be lost in the translation.This way of regarding collections is quite old: i t is how Peano conceived of them in the1880s. If we already have before us some formal theory which does not have collection-talk in it, then we can add it straightforwardly. If @(z) s a formula of the theoryin question, we introduce a new term {z : a($)} called a collection term) into thelanguage. Collection terms a re suhject to the following two rules:(1) If @(z) s a formula then x E { u : @(y)} s an abbreviation for @(z)


2/10

(2) If @(x) and Q(z) re both formulae of the theory then {x : @(x)} = {x : Q(z)} isan abbreviation for Vx(@(z) t,Q(z)).

We permit the use of collection terms only in contexts from which these two rules allowthem to he eliminated. Collections used harmlessly in this way have been called virtualby Quine and we shall follow this terminology here. Let us introduce the conventionthat lower case Greek letters stand schematically for collection terms. So the secondre-writing rule allows us to assert, for instance, the following extensionality schema:

By this schematic device we generate a considerable amount of the naive theory ofcollections. We can, for instance, introduce the usual Boolean operations as follows:

And so on. The theory thus generated is elegant enough in itself, but i t is difficultto be satisfied with it for long. (We have to remember, for instance, that i n this theoryexpressions such as {x, y} are merely notational devices and do not, despite the appear-ance to the contrary that their 'explicit' definitions might suggest, have any meaningon their own.) Collection terms stand in sentences in some of the places where thegrammar leads us to expect proper names; schematic letters stand where we mightexpect variables. There is a drive to objectify a t work here, to suppose that collectionterms can stand in a ll the places where names can occur. I t is of course this s tep which,if carelessly done, leads straigh t to contradiction. How to do it without contradictionwill be the subject of the rest of this entry.

2 Real collectionsIn the virtual theory just discussed the relation of membership is defined contextually.In the real theory with collections regarded a s objects in their own right membership istaken by almost all authors as primitive. (Lewis(1991) is a rare exception.) Formally,this means tha t the language in which the theory is couched contains, in addition to theusual logical constants, a binary predicate symbol 'E'. I t will be convenient to introducea few notational conventions straight away. We write (Vs E a). . . and (3% E a) asabbreviations for Vx(x E a + . ) and 3x(x E a A . . . respectively. Upper case Greekletters such as @ and Qwill s tand for formulae. The notation @(x) stands for a formulain which the variable z ccurs free: it does not indicate that there are no other variablesoccurring free in the formula. If @ is a formula, we write @(a) for the relatiuization of @to a, .e. the formula obtained from @ by replacing each quantifier Vx or 3x by (Vx E a)or (3xE a) as the case may be. We write zC as a n abbreviation forVz(z E z+ z E y).We introduce ordered pairs hy means of the device

In the virtual theory whether there are any objects which are not collections is scarcelyan issue: since there are not really any collections, if there is nothing else then thereis nothing a t all; the theory therefore collapses into vacuity. In any non-trivial realtheory things a re different: there are plenty of collections to talk ahout whether or notthere a re any non-collections. Since this issue arises uniformly for all the real theories


3/10

we shall be discussing, it is best to address i t now. The objects of a theory which arenot collections are called atoms, individuals or (even in non-German texts) Urelemente.If we are to allow for their presence, we need a primitive unary predicate atom(x) torepresent that z is an atom. We then introduce a s an axiom the following:(2.1) Axiom of exte nsio nalit y (a ppli ed form). VxVy(atom(z) Vatom(y) VVZ(ZE z ttZ E v) + x =y) .This asserts no more than tha t objects which are not atoms are collections, since (as wehave already observed) i t is part of what we mean by a collection that it is completelydetermined by its members. We shall also, if the atoms are to play any role in thetheory a t all, need the following:Collection axiom. 3:/Vz(x E y tt atom(x)).In words: the atoms form a collection. If, on the other hand i t is pure collections thatinterest us, then the beginnings of the theory are a little more concise. We do not needthe predicate atom(%)or the collection axiom, and the axiom of extensionality takes thefollowing simpler form:Axiom of ex tensionality (p ure form). VxVy(Vr(z E x tt z E y)i = y)

Throughout what follows we shall sta te theories in their pure form and adopt theconvention that if T is a pure theory (i.e. one which includes the pure form of the axiomof extensionality) then T U is the corresponding applied theory (i.e. the one obtainedfrom T by deleting the pure form of extensionality and substituting the applied formtogether with the collection axiom). In the virtual theory we had collection terms de-fined contextually, provided that they occurred in certain positions (on the right of the'E' symbol or on either side of the '=' symbol). Now tha t we are regarding collections asreal we cannot be so indiscriminate. Let us say tha t a formula @(z) s collectiuizing inx for a theory T if

Suppose now th at T includes extensionality among its axioms. If @(z) s collectivizingin z for T then by extensionality

it is therefore legitimate to introduce a term {y :@(y)}with the property that

We thereby retrieve the virtual theory of lbut limited to collectivizing formulae. Whatremains is to specie which formulae are collectivizing; it is this that is dealt withdifferently in the various competing theories.

3 Zermelo-Fraenkel set theoryWe start by listing the axioms of a fragment of Zermelo-Fraenkel set theory called Z-:Axiom of extensionality . As before.Axiom sche ma of separation. If @(x) is a formula in which y does not occur free,then this is an axiom:

Empty set axiom. 3xVy 2/ 2/ z.


4/10

Pairing axiom. VxVy3z(x E z A y E z)Union set axiom. Va3xVyVz((z E y A y E a) - E 2).Power set axiom. Vx3yVz(z 2 x + z E y).Axiom of infinity. 3x(0 E z A VyVz((y E x A z E x) + y U {z} E x).These are the axioms of Z-. To get Z we add:Axiom of foundation. Vx(z # 0+ (3y E x)x ny = 0),To get Z F we add:Axiom schema of replacement. If@(x,y) is a formula in which z does not occur freethen this is a n axiom:

Va((Vz E a)3!y@(z,y) - (Vx E a)(Vy E z)@(x, )).To get Z FC we add:Axiom of choice. Va3b(Vx E a)(x # 0 + 3!y(y E x n b)).Many other variants a re possible. For example:

(1) In the presence of the replacement schema the exact form of the axiom of infinityis not critical: any axiom asserting the existence of a set which can be proved tobe infinite will do. Two popular choices ar e the following:Axiom of infinity (weak form). 3x(0 E z A Vy(y E z + y U {y} E x) .Axiom of infinity (Zermelo's form). 3 4 0 E x A Vy(y E x + {y} E x).

(2) In ZF - and stronger theories the pairing axiom is redundant.(3) The separation and replacement schemata and the union se t axiom can be amal-

gamated into the following single schema:Axiom schema of selection and union. If @(x, ) is a formula in which a andbdo not occur free, then this is an axiom:

V@aVz(@(z,y) +x E a) - b3zVa(a E z ti (3y E b)@(x,y)).The history of these axioms for set theory is a s follows. In 1908 Zermelo gave thefirst published axiomatization of set theory. His axioms were those of ZCU - (i.e. theapplied form of Z- plus the axiom of choice) except tha t the axiom of infinity was re-place by Zermelo's form as given above. Zermelo's choice of axioms seems to have beendriven largely by his desire to ground a proof of the well-ordering theorem (see AX-IOM OF CHOICE) which he published in the same year. Although Zermelo's system isstrong enough for many mathematical purposes, it is weaker than ZCU - in one impor-tant respect: it is impossible to develop a satisfactory theory of ordinal numbers (seeSET THEORY) in Zermelo's system. During the 1920s this problem was solved in twoways: ingenious methods were found, principally by Kuratowski, of achieving withoutordinals what is most naturally done with them; and Zermelo's system was strength-ened, principally by Fraenkel and Skolem, so as to permit the theory of ordinals to beembedded in it.


5/10

4 Von Neumann-Bernays-Godelclass theoryZermelo-Fraenkel set theory is adequate for almost all mathematical purposes. How-ever, its restrictions can sometimes be inconvenient. Mathematicians who use Z F andits variants are therefore accustomed to using the device of virtual sets a s described in31, except tha t in order to distinguish virtual sets from the real thing they are calledvirtual classes. If we stop there, we have a good compromise: the technical simplicityof Z F and the convenience of virtual classes when we need them. However, the urge totrea t virtual objects as real is strong, and there is a popular alternative which allows usto say the same things non-schematically, i.e. with classes treated as real. It is calledvon Neumann-Bernays-Godel class theory and denoted NBG. Before we state the ax-ioms we need to express what it is for a relation to be functional: we do this by writingf u n c ( r )as a n abbreviation for the formula V x V y V z ( ( ( z , ) E r A ( z , ) E T ) + y = z ) .Axiom of extensionality. As before.Axiom of classes. 3 x V y ( ( 3 zy E z ) + y E x )Empty set axiom. 3 z ( V y y $ z A 3 z x E z ) .Pairing axiom. (V x E V ) ( V yE V ) ( 3 zE V ) V u ( uE z - = x V u = 9).Axiom schema of class existence. If is any formula with free variables u l , v z , . . . . 1,then this is a n axiom:

Union set axiom. (V x E V ) ( 3 yE V ) V u , ( uE y - u ( u E u A u E x ) ) .Power set axiom. (V x 3: V ) ( 3 V ) V u ( u E y ti u 2 x ) .Axiom of infinity. ( 3 x E V ) ( 0E x A ( V u E x ) ( V u E x ) ( u U { ? I }E z ) )Axiom of replacement. (V x E V ) V r ( f u n c ( r )+ ( 3 y E V ) V u ( uE y tt (3 x ) ( u , u ) E

Axiom of foundation. As before.Most of these axioms are unexciting transcriptions of the axioms of ZF . However,

there is one striking difference: the axiom scheme of replacement has turned into asingle axiom. Even more surprisingly, the axiom schema of class existence can also bereplaced by a finite list of axioms, such as the following:

VaVb3cVx(zE c tt ( x E a A s E b ) )

Va3b(Vx E V ) ( xE b tt ( 3 y E V ) ( ( z ,) E a ) ) .

Va3b(Vz , , z E V ) ( ( Z ,, z ) E b ci ( y , , x ) E a )V a 3 b ( V z , , z E V ) ( ( x , , z ) E b ( x , , I ) E a )


6/10

So NBG is finitely axiomatizable. This is quite striking since by contrast Z F cannot befinitely axiomatized (if it is consistent). And yet NBG and Z F are of equal strength, inthe sense tha t if @ is any formula in the language of set theory then

Z F i- if and only if NBG i- dV ) .Moreover, NBG is consistent if and only if Z F is consistent. Once we have taken thestep of regarding classes as objects, it is puzzling why we should constrain the classexistence schema by insisting that the formula defining a class can mention only sets,not proper classes. So if we are serious about the reification of proper classes, we shouldreplace the class existence schema with the following:Morse-Kelley schema. If@ ( z ) s any formula in which y does not occur free then thisis a n axiom:

The system thus obtained is known as Morse-Kelley class theory and denoted MK. Itis (if consistent) a genuine extension of NBG, even with respect to sets. In other words,there are formulae such that MK k ~ ( ~ 1ut NBG Y dV ) . owever, the diMicultynow is to know why we should stop: if classes are sufficiently rea l to occur legitimatelyin the definitions of sets, there seems no good reason why they should not he membersof still more collections. As long as we refrain from calling the new collections thuscountenanced 'sets' or 'classes', there is no fresh danger of inconsistency. Theoriesarranged in this way are not popular, though, despite their practical convenience.

5 Other axiomatizationsSo far we have said little about the motivation for the axioms. Two principles haveguided traditional accounts of this question: one is the iterative conception; the otheris the doctrine of limitation of size. However, it is not easy to justify all the axioms ofZ F on the basis of either principle on its own. Other axiomatizations have been givenwhich are closer in spirit to these principles. Here we shall describe two. The first isScott's way of axiomatizing the iterative conception of set, according to which se ts aredivided up into levels. The set of all levels belongingto a given level is called its history.The formal details are as follows. First define

acc(a)= { z : (3 a ) ( z E b V z C ) ]if this set exists. ( In words, the accumulation of a set is the set of all the elements andsubsets of its members.) Next define hist,ory(a) o be an abbreviation for

(Vb E a ) b = acc(bn a ) .Then define level(b) to be an abbreviation for

Now adopt the convention that the variables V,V' ,V" are restricted to range over lev-els, so that for instance W . is an abbreviation for W ( l e v e l ( V )t . Then theaxioms of Z' are as follows:Axiom of extensionality. As before.Axiom schema o f separation. As before.Axiom o f creation. V z W x E V .


7/10

Axiom of infinity. 3 V ( W t V)(3Vt' V)(V1 V").The axioms of ZF' are as follows:Axiom of extensionality. As before.Axiom schema of separation. As beforeAxiom schema of reflection. If @(z) s any formula then this is a n axiom:

Then Z' is equivalent to Z and ZF' is equivalent to ZF . The second theory we shalldescribe is due to Ackermann. It takes V as a primitive constant. The axioms of thesystem A ar e a s follows:Axiom of extensionality. As before.Axiom schema of separation. As beforeAxiom of transitivity. VxVy(x E y A y E V i E V).Axiom of subsets. VxVy(x C y A y E V + s E V).Aekermam's schema. If is a formula in which the free variables are 7~1,712,. ,unan d in which V does not occur, then this is an axiom:

The motivation for A is not that of the iterative conception; rather is it based on theidea that anything formed solely from sets is itself a set. The system A* is somethingof a hybrid: it grafts onto A the following axiom, which is derived from the iterativeconception and cannot be justified by the motivating idea of A:Axiom of foundation. (Vx E V)(z 0+ (37~ x)(y n x = 0).If is a formula in which the constant V does not occur then:(1) if Z- t then A k dV);(2) A* a(") f and only if Z F t a.

6 Anti-foundation axiomsWe have already mentioned th at not all the axioms of Z F can be motivated by anyone account of the notion of sethood. In particular, the axiom of foundation cannotbe sustained solely on the basis of the doctrine of limitation of size. But without i twhere do sets get the ir individuation? To tell whether two sets are equal we comparetheir members; if necessary, we compare their members; and so on. In the presence ofthe axiom of foundation, this process cannot go on indefinitely; in its absence, it can.Now if the pattern of membership exhibited by the two sets is the same, then it seemsreasonable to say tha t they are equal: they have done all that can be expected of them,as sets, to be regarded as identical. We are therefore led to the following principle:Axiom of extensionality (strong form). If two sets have isomorphic -graphs, thenthey are equal.


8/10

(The -graph of a set is the membership relation restricted to the set, its members,their members, etc.) As we have already noted, this assertion is provable in ZF, but inZF- (ZF without foundation) we need to add it as an axiom. It ensures, for instance,tha t there is a t most one set il such tha t Cl = {Cl} (since any two such sets would haveto have isomorphic -graphs). However, it does not guarantee t ha t there is such aset Cl. Aczel and others have studied the consequences of adding to ZF- not only thestrong extensionality axiom just mentioned (which is essentially limitative) but also apermissive axiom generating non-well-founded sets (which can be thought of as limitpoints in much the same way tha t irrational numbers are limits of rational numbers).The most natural of the axioms tha t have been studied is the following:Anti-foundation axiom. Every graph is the -graph of some set.

7 Quine's systemsAnother, more radical st rand in attempts to resolve the set-theoretic paradoxes is rep-resented by the various forms of the theory of types (see THEORY OF TYPES). Wecannot, in any of the systems we have been describing so far, collect together all theobjects to form another object, but we can certainly quantify over them; in the theoryof types we cannot. Objects are stratified into types, which are not cumulative but oth-erwise resemble the levels of Scott's theory; variables a re labelled with superscripts toindicate the type they range over. The formula z m E yn is taken to be well-formed onlyif n = m + 1. Now type theory is not, according to the strict definition adopted here,a form of set theory, because it does not have the axiom of extensionality. What i t hasinstead is a schema of axioms, one for each type:

This may seem a pedantic cavil, particularly if the device of typical ambiguity is in-troduced, whereby formulae are written with superscripts omitted and taken to standschematically for any way of adding superscripts which produces a grammatical result.Thus the extensionality schema above is usually represented

Nevertheless, we maintain, this is not the axiom of extensionality. Typical ambiguityis not true generality. To see the difference consider the formula 3yVz z E y : taken astypically ambiguous this s tands schematically for such formulae as 3 y n + l V z n z n E gn+land is therefore true; taken a s a formula in its own right it is false. The restrictionson what can be said in the theory of types have struck many writers as too severe.However, the means by which the theory avoids paradox are as secure as those ofiterative set theories. Quine has proposed a system which uses type theory's method ofparadox avoidance but abandons its grammatical restrictions. Specifically, he regardsformulae as genuine, not typically ambiguous, but calls a formula stratified if it ispossible to decorate it with superscripts so as to make it well-formed according to thetheory of types. The system, which is known as N F because it was proposed in anarticle called Wew foundations for mathematical logic', has the following axioms:Axiom of extensionality. As before.Axiom schema of stratified comprehension. If @ ( z ) is any stratified formula inwhich y does not occur free then this is a n axiom:

N F is very different from ZF. For one thing N F i s finitely axiomatizable. Moreover,since the formula z = z is stratified, it follows a t once that there is a set of all sets: if


9/10

we denote it V then V E V. The axiom of choice is provably false in NF. When we cometo arithmetic, it is possible in a natural way to define zero and the successor operationin NF, and then to define the set of natural numbers to be the intersection of all se tscontaining zero and closed under successors. This works well up to a point, but givesus mathematical induction only for stratified formulae. Unfortunately some usefulformulae are not stratified. For example, we cannot prove in NF, and are thereforeforced to add a s an axiom, the following:Axiom of counting. If n is a natu ral number then it has exactly n predecessors.

In an effort to overcome this inconvenience Quine has proposed a second systemcalled ML which stands to NF in somewhat the same relation that NBG stands toZF. The second system takes V as a primitive constant and has the following axioms:Axiom of extensionality. As before.Axiom schema of class existence. If @(x) is a formula in which y does not occurfreethen this is an axiom:

Axiom schema of set existence. If a($, l , u z ,. ,v,) is a stratified formula in whichno variables other than those listed occurfree, then this is an axiom:

In contrast to NF, ML is not finitely axiomatizable if it is consistent. If is anyformula in which V does not occur then N F t if and only if ML t Thus MLis consistent if and only NF is consistent. However it is still not known whether MLand N F are consistent if ZF i s consistent. If we define the natural numbers in super&cially the same way as in NF, we get mathematical induction even for non-stratifiableformulae, obviating the need for an axiom of counting. However, th is time we cannotprove that the class of natural numbers is a set (if ML is consistent). So now we haveto add this a s an axiom. Thus neither NF nor ML suffices without supplementation asa basis for mathematics.

References and further readingAckermann (1956) 'Zur Axiomatik der Mengenlehre', Math. Ann., 131: 33&45. (Thesource for the system A mentioned in 55.)* Aczel, P. (1988) Non-well-founded sets, Stanford, Ca.: Center for the Study of Lan-guage and Information. (An account of anti-foundation axioms aimed a t the mathe-matically aware.)Boolos, G. (1989) 'Iteration again', Philosophical topics, 17 (2): 5-21. (An interestingbut reasonably technical discussion of the i terative and limitation-of-size conceptionsof set.)Bourbaki, N. (1968) Theory of sets, New York: Addison-Wesley. (A development of settheory in a system equivalent to ZF- together with a strong form of the axiom of choice.The system is notable for its employment of the axiom schema of selection and unionreferred to in 53.)Forster, T.E. (1992) Set Theory with a Universal Set, Oxford: Oxford University Press.(A survey of work on NF, well written but aimed a t logicians.)Fraenkel, A.A. (1922) 'Zu den Grundlagen der Cantor-Zermeloschen Mengenlehre',Math. Ann. 86: 230-7. English translation in van Heijenoort (1967). (This paper is thereason Fraenkel's name is associated with the standard formulation of set theory.)


10/10

Fraenkel, A.A., Bar-Hillel, Y., and Levy, A. (1973) Foundations of set theory, 2nd edi-tion, Amsterdam: North-Holland. (Detailed discussions of most of the systems of settheory mentioned here and a wealth of historical information and references.)G de l , K. (1940) The consistency of the continuum hypothesis, Princeton, N.J.: Prince-ton University Press. (The definitive formulation of NBG.)Hallet, M. (1984) Cantorian set theory and limitation of size, Oxford: Clarendon Press.(A detailed study of the early history of set theory.)van Heijenoort, J. (ed.) (1967) From Frege to Godel: A source book in mathematicallogic, 1879-1931, Cambridge, Mass.: Harvard University Press. (It is possible to tracethe historical development of ZF and NB G through t he papers reprinted here.)Kelley, J.L. (1955) General topology, Princeton, N.J.: Van Nostrand. (The appendixcontains the first published version of MK.)* Kuratowski, K. (1921) Sur la notion d'ordre dans la theorie des ensembles', Fundam.Math. 2: 161-71. (Mentioned in $3.)* Lewis, D. (1991) Pa rts of classes, Oxford: Basil Blackwell.Morse, A. (1965)A theory of sets, New York: Academic Press. (A trea tment of the theoryof classes based on a version of MK.)von Neumann, J. (1925) Eine axiomatisierung de r Mengenlehre', J. reine angew. Math.154: 2194 0. English translation in van Heijenoort (1967). (The first formulation of a naxiomatization similar to what i s now called NBG.)Potter, M.D. (1990) Sets: An introduction, Oxford: Oxford University Press. (An ac-count of set theory based on a Scott-type axiomatization and intended for the mathe-matically minded.)* Quine,W.V.O. (1937) New foundation for mathematical logic', Amer. Math. Monthly44: 70-80. (Principally of historical interest. For up-to-date information about NFconsult Forster.)- 1969) Set theory and i ts logic, revised edition, Cambridge, Mass.: Harvard Uni-versity Press. (Rather idiosyncratic bu t extremely clear and well written.)*- 1951) Mathematical logic, revised edition, Cambridge, Mass.: Harvard Univer-sity Press. (Very technical, but includes a wealth of information.)Scott, D. (1974) 'Axiomatizing set theory' in Axiomatic set theory, Proc. Symp. PureMath. 13, Par t 11, pp.207-14, Providence, R.I.: Am. Math. Soc. (The primary sourcefor Scott's axiomatization of ZF ' described in $5.)Skolem, T. (1922) 'Einige Bemerkungen zur axiomatischen Begrundung de r Mengen-lehre', Wiss. Vortrage gehalten zuf dem 5 Kongress der Skandinau. Mathematiken inHelsingfors, pp.217-32. English translation in van Heijenoort (1967). (The first pre-cise formulation of the axiom schema of replacement and of Zermelo's notion of 'definiteproperty'.)* Zermelo, E. (1908) Untersuchungen iiber die Grundlagen der Mengenlehre 1',Math.Ann., 65: 261-81. English translation in van Heijenoort (1967). (The first publishedaxiomatization of set theory, referred to in $3.)

Documents

Different Systems of Set Theory