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CITAPTER 2 GENERAL DEVELOPMENTS $ 21 Preliminaries: Formulas and Definitions. Let us now (a) exphieitly define the notion of formula required in the axiom schema of separation (and later in the axiom sehema of replacement) and (b) state the approach to definitions we adopt in introducing the many defined symbols needed. In Chapter 1, theimportant, distinction was made between tho object language, that is, the language in which wc talk about sets, and the meta- language, that is, the language in which we discuss tho object language itself. We use the metalanguage, which for us is ordinary English aug- mented by a certain amount of intuitivo mathematical language, to de- seribe exactly the object language. Tt may be helpful to look upon this description as analogous to an exact characterization ot a game like chess or bridge. But this analogy is not to be carried too far, for most of the expressions of our object language have a definite meaning in terms of intuitive mathematical ideas, whieh the positions or moves in a game like chess do not. We begin with a fivefold classification of the symbols of the object language into constants, variables, sentential connectives, quantifiers or operators, and punctuation or grouping symbols.* The two primitive con- stants of the language arc the membership relation symbol “€', introduced informally in Chapter 1, and the constant “0”, which denotes the empty set. In addition, wo take from logic the predicate constant “=”, which is the identity symbol. “The general variables ranging over all objects are the letters “r', 9, - with and without subscripts or superscripts. The sentential connectives are the five mentioned in 8L2— +, &, V, >, &; the three quantificrs or logieal operators wc uso— Y, 3, E!— were also mentioned in $1.2. Finally, left- and right-band parentheses gre our only punctuation symbols. *This classification originates with von Neumann [1927], For detailed discussion of these matters ave the first chapter of Church [1956]. 14 GENERAL DEVELOPMENTS 15 Did apressions of the object language are finite sequences of the five classes É symbols of the language. Certain of these expressions, simply because &néir structure, are called primitivo formulas of the object language. inôw define such formulas so that merdly by looking at the form of an piéssion we cam automatically decide in a finite mimber of steps trhether op not it is a primitivo formula. Although this definition is purely syntac- tical:ór structural, it is just the oxpressions satisfying it which have a clear “atuitive meaning. An expression like ( — € q” is not a ptimitive formula id has no intuitive meaning. We first define primitive atomic formulas. A primitivo atomic Jormula is an expression of lhe form (v E w), or of the form (y = w), where v and w are eilher general variables or the con- stant A Thus ze gy and 2 = O ate primitive atomic formulas. “We may now givo what is usually called a recursive definition of primitive urmulas: = (a) Every primitive atomic Jormula is a primilive formula; (b) If Pisa primitive formula, then =P is a primitive formula; (o) If P and Q are primitivo formulas, then P&Q), Pv O), (P> O), and (PO O) are primitivo formulas; (d) IfPisa primitive formula and v is any general variable then (VwP, (Av)P and (Tl)P are primitive Jormulas; (e) No expression of the object language is a primitive formula unless its being so follows from rules (n) - (d). The following arc examples of primitive formulas of the object language which are not atomie: (M)(Vy)-(yCa), ccy=>ve?, CEl2) (0 = 2). In terms of this definition, an exact formulation of the axiom sthema of separalion is then: Any primitive formula of the object language of the form (Mw n)lm vv e=D&(Vwluevoweukç) is an axiom, provided the variable v às distinei from u and w, and às not free in the primitivo formula q. *In this definition, as elsewhere, wo uso tho boldiace letters “w?, 7, w', “uy, 'vy, Sw, as motamathematical variables which take as values variublos “2, 4, &, é. or the constant (0º of the objcet language. And we use boldfuce lettera, 'P', Q', ., as well as Greek letters “9 and “Y, as metamathemaical variables which talte as values lormulas of the object language. The conventions about use and mention fol- lowed here, shich are probably obvious, are lhat ()) the constants E! and *=", the sentential connectives, the quantifier syrabols, and the lefl and tight parentheses aro “used as names aí themselves, and (3) justapasition of nan ionss denoles hinsry operation on expressions which yields now expressions (for example, “rey “ycê = “cy &e ye). Fora more detailed discussion of these conventions, see Chapter 6 of Suppes [1957]. 16 GENERAL DEVELOPMENTS Cuar, 2 Justifying reasons for the restrictions on tho variable y arc given in the next scetion, In principle all of the axioms aud theorems of set theory that we stato in the folowing pages can be written as primitive formulas oÍ the objcet language — indeed, our official objcet language shall consist of these prim- itive formulas. For working purposes it will be useful and convenient to introduce by definition considerable additional notation. We shall in practice apply the axiom schema of separation to formulas which are not written solely in primitive notation; but since at any point in our develop- ment only a finite number of definitions will have preceded, such a formula can be replaced by a primitive formula by a finite number of substitutions. Regarding definitions, then, our viewpoint is that they ave informaliy admitted if clear recipes are given for eliminating new symbols from any context. We thus require that a formula of the object language which introduces » new symbol must satisfy the following: Crirenrox or ELiminasiITr. À formula P introducing a new symbol satisfics the oriterion: of eliminability if and onty if: whenever Qy is a formula in which the new symbol occurs, then there is à primitive formula Qs such that P> (O,0 Qs) is derivable from the axioms. Notice that we have stated this eriterion without giving an exact definition of formula (as opposed to primitive formula). Such a definition is straight- forward if we list all the defined symbols introduced in this book, and then proceed in terms of this list as wc did before with the primitivo notation. This tedious task we shall not perform, but we do want'to mention a second eriterion wo expect our definitions to satisfy,'namely, our definitions must not be creative. CrrrerioN or Non-Creativiry. A formula P iniroducing a new symbol satisfies the eriterion of non-creativity if and only if: there is no primitive formula Q such that P—> Q is derivable from the axioms but Q is not. Tn other words, a definition should not function as a crentivo axiom per- mitting dcrivation of some previously unprovable formula in which only primitive notation oceurs, The classical problem of the theory of definition for any exactly stated mathematical theory is to provide rules of definition whose satisfaction entails satisfaction of Lhe two criteria just stated. We may restrict our- selves here to rules for defining operation symbols. Sligbt modifications yield appropriate rules for defining relation symbols and individual con- stants.* In these rules we refor to preceding defiwmitions, which implies that the definitions are given in & fixed sequence and not simultancously; *Individual constants may in fact bo treated as operation symbols of degree zero. Sec. 2.1 GENERAL DEVELOPMENTS 17 £his approach permits use of defined symbols in the definitions of new sjmbols.* “ Proper definitions of operation symbols may be either equivalences or identities. We begin with the former. An equivalence P introducing a new n-place operation symbol O is a proper definition if and onky if Pis of the form Ovi. and the follmving restrictions are satisfied: (1) vi... Va w are distinct variables, (ii) Q has no free variables other than vs... 3 Ya w, CD) Q is q formula in which the onty non-logical constants are the primitive or previousty defined symbols of set theory, and (iv) the formula (El)Q às derivable from the axioms and preceding defimitions. a) =voQ Regarding the phrasc “non-logical constants” in (iii), the only logical con- Stants ave those introduced in $1.2; all other constants are non-logical. Justification of the various restrictions is easily given. Jlere we shall emphasize only the importance of (iv). Consider the following definition in elementary arithmetic of the pseudo-operation *, (1) ságy=ete
A=5. and the aziom schema of seporation: Bv re Be zcA E o(2)). “Tg'is understood in the axiom sehema of separation that the variable “B” is not free in g(x). An exact metamathematical formulation cf this schema, “was given in Lhe preceding section.* For intuitivo working purposes we «hall hold to the form just now used, shich is a mixture of the object sjanguage and metalanguage; but the reader should keep clearly in mind hat this is an axiom schema, not a single axiom, The restriction that “B” tiot be free in q(x) is essential, for without it we can derive a contradiction whenever A is a non-cmpty set. To see this, let q(x) be (ze B)' and let :À. be the set consisting of the empty set (the existence of À follows from ólher axioms in this chapter). Then we have: (ABOCBONCA &-(0€ B), Which, since 0€ 4, imphes: GB)(OCBeS-(0€B), an absurdity. On the other hand, it is legitimate and sometimes necessary “for e(x:) to contain free variables other than simply “”; only 'B' is excluded. We turn now to systematic developments. We first define the standard “«notation 'g'' for something not being a member of something clse. DEFINITION 2. yo rey). Correspondingly, we use from logic the notation “v =< 2º for “=(x = 2). As the first theorem, we have: Timorem il. vg0. rroor. Taking (x) as “: * 2” we have from tho axiom schema, of ““Soparation: (45) Aves rede 2). ““Suppose now that some 2 is in A; then by (1), 2 * x, which is absurd. Hence we conclude: (2) (Valve 4), *[tis worth noting that strictly speaking the formulation with set variables is wenker “than the metamathematieal ane with only one kind of variable, for the former reads: zisa set (Jy)(y is a set & (Vo) (rey o sex & p(2)), syheroas the carlior metamathematical version docs not have this conditional form. On the other hand, it is of no real interest to have the axiom apply whem 2 is an individual, 22 CENERAL DEVETLOPMENTS Cuar: 2 and therctore by Definition 1 (3) A=0. The theorem follows from (2) and (3). QED. We next prove a simple theorem concerning the uniqueness of the cmpty sob, Tarornm 2. (ValzgA) SA = 0. rroor. If 4 =0, then by Theorem 1, zg 4. On tho other band, if forevery 2,2 4, then Lhere is no clement in the set A and by Definition 1, 4=0. QED. The remainder of Lhis section is concerned with the notions of inclusion aud proper inclusion of sets. If A and B are sets such that cvcry member of À is a member of B, then we say that A ís included in B, or that A isa subsct of B, which we symbolize: 4 € B. “Thus we may write: The set of Irishmen is included in the set of men or: “The set of Irishmen is a subset of the set of men or simply: “The set of Irishmen Ç the set of men. Formally, we have: Derinimon 3. ACBO(ValcAsze BB). From & formal standpoimt Definition 3 is a conditional definition of a two- place relation symbol. The fact that it is à conditional'definition is masiced by the use of capital letters in the manner agreed upon. This same remark applies to all the definitions introduced which use set variables. Treorem 3. ACA. rroor. Sinco it is a truth of logic that (Vol ASze 4), it follows immediately ftom Definition 3 that ATA. QED. Theorem 3 asserts simply that inclusion is reflexive; tho next theorem asserts that it has the property of antisymmetry, as it is usually called. Turogem 4. ACB&BCASAÃO=B. HAC BandBCA, then it follows from Definition 3 that (Volrctozes). Henee by the axiom of extensionality, 4 = B. PROOF, QED. 506. 2.2 GENERAL DEVELOPMENTS 23 Tyronem ô. 4CO>4=0. zroor. By virtuc of Definition 3 and the hypothesis of the thcorem, fz cçAthenzco0. But by Theorem 1zg0, Hence for every x, vg 4, “and thus by Theorem 2, 4 = 0, QE.D, Yransitivity of inclusion is asserted in the following theorem. Trronem 6. ACBEBCOCSAÇOE. rroor. Consider an arbitrary cloment x. Sincc AC BifzCA then “see B, but BC C;henccif rc B then se C. Thus by the transitivity of implication É zC À then ze C. QED* A typical procedure used in informal proofs is exemplified here. We need to prove something about all elements 2. To do this it suffices to give the argument for an arbitrary 2. The use of the phrase “arbitrary element” corresponds in the language of Logic to introducing a Íree variable in a premise (in this case the premise: z€ 4). We now definc proper inclusion. DrrIsITION É ACBOACB&A AB. “Thus using informally the braces notation not yet formally defined, we have: t1,2) C (1,23) but it is not the case that (1,2) C t1,2). (Here we doseribe a sct by writing down names of its members separated by commas and enclose the result with curly braces.) The next four theorems assert expected properties of proper inclusion; the prooís are left às excreisos, THEeoREM 7. (AC A) Trzonem 8. ACEBSBCA) Tueorem 9. ACBEBCC>ACO. Terorem 10, ACBSACB. EXERCISES 1. Prove Theorems 7-10. ; 2Z Formulate Definition 4 as a conditional definition without using set variables A, 'Bº, ete. E Nolransitivity of implication” just mcans that from P > Q and Q > R we may infer P>R. 26 GENERAL DEVELOPMENTS CHar, 2 Tarorsm 13. ANB=BNA, Tmeonzu 14. (ANB)NC = ANMBNO. A binary operation is idempotent if the operation, when performed on any element with itself, results in just that element. “The next theorem asserta the idempotence of intersection. Taronzm 15. ANA = À. rroor. By Thcorem 12 LE ANAGECÁÉICA, but scA&rcÃSrEA, and thus by Definition 5 ANA =A, Q.E.D. Three intuitively obviõus theorems are next stateã; only the first is proved. Turonem 16, ANO =0, proor. By virtue of Theorem 12 se ANDOzeÃA&rEO, but by Thcorem 1 ago. Tence tg ANO, and since the argument holds for every x, by Theorem 2 ANO = 0. QED. Urzorsm 17. ANBCA. Tarongm 18 ACBoAÁNB = We now tum to the theorem justifying the operation of union of sets. The proof of this thcorem involves the first uso of the union axiom. Tmsoruy 19. (MiO)(vValzeCoszcAV ae B). »Roor. Similar to the proof of Theorem IL, but using the union axiom - in place of the axiom schema of separation. Derinrrion 6. AUB=yo(VelzeyezeAVieD) &y às q set, Sea, 2.3 GENERAT DEVETLOPMENTS 27 For working purposes we immediately need a theotem for the operation of union analogous to Theorem 12. Trrormm 20. vrCAUBOSCAÃVvICB. rroor. Similar to the proof of Theorem 12, Since many of the proofs of theorems about union of sets parallel those about intersection, wc may often dismiss them with reference to the vorresponding theorem about intersection, as we have done in the proofs of Theorems 19 and 20. The next threc theorems assert the commutativity, associativity, and = idempotenee of union. Texonem 21. AUB = BUA, Trrorem 22. (AUBJUC = A U(BUC). Trrorem 23. AUA=A. Further facts are asserted in the next four thcorems. Taroram MM. AUO=A, Turorem 25. 4C AUB, Tmrorey 26. 4ACBoAUB=B. Tutorum 27. ACOCEBCOSAUBCCO. We now stato two fundamental distributive laws for intersection and union, and prove the first one. THEroREM 28. (AUB)NC = (ANC)U(BNO). »roor. Let a be an arbitrary clement. By virtue of Theorem 12 TE (AUVBNCe ze AUB&LCC, and by Theorem 20 vc AUB&scCo(reAVICB)&ZEC, and by the distributive laws of sentential logic” WeAvVIEcB) EreCo(RcA&reO)vVrcB&rxe O. “Using again Theorem 12 tresk&recO)VigcB&ze O) ezcANtv re Bne, *The law in question is that ftom (Pv O) & R we may infer (P&R) v(Q ER) ênd convorely, where P, Q, and R are any formulas. 28 GENHRAL DEVELOPMENTS Cmap. 2 Sec. 2.3 GENERAL DEVELOPMENTS 29 “98, As with that theorem, the proofs depend upon exploiting formal proper- tjes of the sentential connectives analogous to the formal properties asserted “fi the theorems. The proof of only the first of thesc theorems is given hore, TuroREM 33. A-(ANB) = AB. and now using Theorem 20 again ve ANCV Ze BnCoze (ANQU(BNO). Hence from the transitivity of equivalence we infor: zE (AUBNCo ze (ANC)U(BNC), proor. Let a be au arbitrary element. Then LE A-(ANB) o rc A &-ze ANB) by Theorem 31 ore &- re A&zeB) by Thcorem 12 and thus by the axiom of extensionality (AUB)NC = (ANC)U(BNC). Q.ED. In the proof of Theorem 28 a device which is used over and over again has been employed: in order to prove to sets identical, we begin by con- sidering an arbitrary element of one of the sets and show that it belongs to this sct if and only if it belongs to the other. Using the axiom of exten- sionality wc immediately obtain the identity of the two sets. Tweorem 29. (ANBJUC = (AUC)N(BUC). We next state the justifying theorem and definition of the operation of set difference. Tazorem 30. (ECXVo)lre Core A & og B). proor. Similar to Theorem 11, but here taking q(x) as “rg Bº. DErINIMNON 7. AmB=y(VilzeyorcA rg DB) &y is a set. orcAb(zeAV es) by sentential logic o(reA&rgAa)vigcÃA&argB) by sentential logic orcA&argB by sentential logic. QED. “A staccato style, wbieh consists of displaying a series of equivalences and is similar to that often used for a chain of identities, has been adopted here. Some readers may find this method of presentation clcarer than the More prolix one used in the proof of Theorem 28. Terorem 34. AMAB)=A-B. “VHgoREM 35. (A B)UB = AUB. Turorem 386, (AUB)-B= 48 Taeonem 37. (ANB) = B = 0. Tanonem 38. (4 B)nB = 0, THsorkM 39. 4 (BUC) = (A BIN(A O). THrorex 40. A-(BNC) = (As BJU(A =), Trzorex 3], CA vBozcA&rgB. zroor. Similar to the proof of Theorem 12. The next theorem obviousty entails the fact that difference of sets is not idempotent, granted the existence of non-cmpty sets, Tizorkm 32. 4-4 =0. rroor. By Theorem 1 In von Neumann set theory the universe V, which is the class of all sets, exists. Tho complement A of a set À can then be defincd as mà = VA. são. Ilence by sentential logic 2E0msEAbsçA, “Bug this is not possible in Zermelo-Fraenkel set thcory, and it may be of “interest to see why not in some detail. Analogous to the justifying thcorems for the definitions of the three operations already considered, we would need to prove: (e) BB(VoecBozxçA) nd they we would define complementation by: and thus by Definition 7 A-A=0, Q.E.D. The remaining thcorems of this section state facts relating the set opera- tions of intersection, union and difference. The proofs of Lhe theorems are easily given by employing an approach similar to that used for Theorem 32 GENERAL DEVELOPMENTS Cear. 2 Case 1. x =. Then by virtue of (1), x = Case 2. vg. In view of (1), either e = uory =“ Suppose t =* 4. Then y = wand by (8)z=e On the other hand, supposc y = 4 Then e=uand by (4) y= Q.E.D. It is convenient for subsequent use to define unit sets, triplet sets and quadruplet sets. The naturalness of the definitions is at once apparent. te) = (ua) tona) = (oybu fem) = fey)u u, and by virtuc of (2) y = v. Derinrtox 9. te few). As an immediate corollary of Theorem 44 we have an intuitively obvious theorem about unit sets. The proof is leit as an exercise. tel We ute now in à position to define ordered pairs in terms of unit sets and unordered pair sets. “This definition, which was historically important in vedueing the thcory of relations to the theory ol sets, is due to Ruratowski [1921], but the earliest definition permitting; tbis reduction is to be found in Wiener [1914]. Tenor 45. tozr=g Derrxrrron 10. (2) = (e), ley). Within set theory, as we shall see in the next chaptex, relations are defined as sets of ordered paírs, Without something like the present definition at hand it is impossible to develop the theory of relations unless the votior ofor der ed pairs is taken as primitive. Lessentially our onky intuition about pes in a giver ! The following theorem establi with respeet to this idea; namely, two ordered pairs are identical only when the first member of one is identical with Lhe first member of the other, and similarly for the two second members, Tmeonem 46. (uy)=(up)>oz=ubkgy=o. rroor. By virtue of Definition 10 and the bypotbesis of the theorem Vel, tro) = tio), fue), and thus by Theorem d4 (D Co) = fu) & ley) = [nov do) = tuo) & boy) — tub). Suppose the first alternative of (1) holds. Then since toy = fu) on 10 is adequate CENERAL DEVELOPMENTS 33 y; Theorem 45 “and hence by Thcorem 44 and the supposition that (2,yh = [w,) =», “which establishes the desired result. Suppose now that the second alternative of (1) holds. Then since g) = (2,2), by Theorem 44 and similarly Henee z=u&y=». QED. Ordered pairs play a prominent role in tho section on Cartesian products Ja this chapter and throughout the next chapter, which is concerned with Folations and functions. EXERCISES 1. Prove Theorem 45. “2 Prove that z=g=(ey)= tell. 3. Isitalways true that if (09,2) = feto) then 2 = 1? 4, Show, by proving a theorem like Theorer 46, that the following definition «ofórdered pers is adeguato: Com) = flmo0), ty, fo). $ 2.5 Definition by Abstraction. In many branches of modem mathematics it is customary to uso the notation: tz: q(x)) to designato the set of all objects having the property e. For example, le:a > +42) às tho set of all real numbers greater than 4/2; 28 another example, tr:l 42) 1) VE the notation ich is absurd. QED. f=:— y “Similar to Theorem 41, we also have: binds the variable “»”. Turorwm 50. 0 = [z:z =). Derinrrion Scaema 11, tr ela) = 0 [(Vo)(rey — (a) &y is a set] V ly = 0 &- “(ae) (Va)(acB O ele). Tt is immediately clear from the definition that fa: point of the second member of the dis] Q fe: (x) equal to the empty set if there is no non-crapty set having as members just those entities with property q. The mammer of translating formulas in which set variables arc bound by abstraction is straightforward. “Thus the schematic formula: We may prove as theorems simple formulas which could bc used to define “ntérsection, union, and difference ol sets. Tarorzu 5l. AnB=[recA&reB) ela)j isa s “proor. Use Thcorem 11 and Definition 11. Turonem 52. AUB= (rzcAVrEBL Trrorem 53. A-B=izrcA&agB). A point of methodological in is that if these last three theorems are sed as definitions of the three operations, no justifying thcorem is needed tior to the definition. (Note that the requirement of such a theorem is not needed in the rule for defining operation symbols by identities in $2.1.). “From Definition 11 we know that if the intuitively appropriate set of “elements does not exist, then the result of performing the operation is the “empty set. However, in order to do any serious work with the operations wê.need the existence ol the intuitively appropriate sot, which comes down faying that when we definc operations by abstraction the justifyimg itheorems may come ajler rather than before the definition. This point is ilhistrated below in 82.7. Definitions which do not need à justifying theorem arc often called axiom-free. In tho seque] it is convenient to have a more flexible form of definition ty abstraction than that provided by Definition 11. In particular we want “to be able to put complicated terms before the colon rather than simply ingle v: For example, in $2.8 wc define the Cartesian product tA: ta) às transtated: s a sot & g(x)). There are a number of intuitively obvious theorem schemata about the abstraction operation, some of which we now state and prove. te: = a Twronzm Scaema 47. ve tu: gla)) — el). proor. Hyc fz: q(x)), then teze(o)) = 0, and so by Definition Schema 11 ve te: de) er o(y); of our theorem: el). QED:S we may conclude from the hypoth Tuzonny 48. 4A=irecA) e D, AXB=legzcá&ycB), but on the basis of Dofinition 11 we necd to replace (1) by the more awk- Ward expression: AXB=la(IDINpcA&eEc BE = Go). Iithe style of Definition 11, we have: »roor. Ibtisa truth of logic that (Vale A ore A). Thus taking o(x) in Definition 11 as 2 € A', we obtain the thcorem at once. Q.ED. 38 GENFRAL DEVELOPMENTS Crar, 2 and then sgB. Henec by Theorem 55, for every 2 seg Uto). it were postulated that there are no individuals, that is, that every object is a set, then we could prove that if UA = 0, then either A=00r A = (0). Asitis, the sums of many different sets may be empty, in fact the sum of any set whose only members are individuals and the empty set. QED. UIAJ=A. U(4,B) = AUB. Tarorzm 58. TrgoREM 59. zroor. By virtue of the fundamental property of unordered pair sets if Ce [4,B) then C=AVC=B. Thus by Thcorem 55 (eb) ze UIAB|oOsEAVZEB, and the infcrence of the desired conclusion from (1) is obvious. Q.E.D. Treorgy 60. U(AUB) = (UA) U (UB). PROOF. ze U(AUB) O (JO)lve C& Ce AUB) by Theorem 55 (Ive C&ECE AV (reC&CEB) by Theorem 20 and sentential logie ES (IMRCCECE AV (IOKEC&CEB) by quantifier logic* orve UAVvZsC UB ze UAUUB by Theorem 55 by Theorem 20. QED. *Clearly, from (gv) (P V Q) we may infer (av)P V (ay)Q, and conversely. CENFRA4L DEVELOPMENTS Terorsx 61, 4CB> UACUB. PROOF. se USAS (ICE C&CECA) >(Ogec&ceB) by Theorem 55 by hypothesis of the thcorem by Theovem 55. QED, >2€ UB ACB>ACUB. (VAXA E BS ACO) = UBCO. (VAXACBESANC=0>(UBNC=O. THEoREM 62, THrorEM 68. “Terorex 64. Turorem 65, Utay)= [ey]. PROOF. Uta) = Ulte), toy) by definition of ordered pairs = (ejufzy) by Theorem 59 = tg) QLD. Turoruu 66. UU(4,B) = AUB. We now turn to the definition and properties of the intersection of a amily of sets. Tho intuitive content of this notion should be clear from “the discussions of union. If, as before, 4 = (f1,2), [2,33, (4), Janc Austen) ben n4 =0, ince there is no number common to all the sets which are members of A. Às a second example, if B = (1,2), t23)) “then NB = (1,2) nt28) = f2). The formal deyelopments which follow require Jittle comment. NA = fe (VBBEA >zE BI. There is no theorem related to Definition 14 in the way that Theorem 55 “às. related to Definition 13, that is, we cannot prove: (1) 2ENAS(VBABCAS2CB), Deriximon 14. 40 GENERAL DEVELOPMENES Crarp. 2 and the reason is obvious. If 4 has no sets as members, then the right member of (1) is always truc and every 2 must be a member of NA. But there is no set having every entity x as a member, a fact which was estabe lished by Theorem 41. What we are able to prove is the more restricted result: Tauonzu 67. 2CNAS(VBBEA>ZTEB) & (IB)BE 4). »Roor [Necessity]. By hypothesis ze NA. Hence NA = 0. Thus by virtue of Definition 14 and the goneral properties of definition by abstraction, we infer that (1 «CNAS (NBBEASTEB). Now let us make the supposition that (2) “ABX(BEA). “Then vacuously it is true that é (WBBCA EB), from which we may infer: (3) (VBBCA-rEB) O r=4. Equivalences (1) and (3) yield tbat for every 2 açNt£os=a, »hence by Theorem 54 fr:ze NA = ( a, but by virtue of Thcorem 48 the left-hand side is NA and by Theorem 50 the right-hand side is the empty set, and so we infer nA=0 «which contradicts the hypothesis that » € NA and proves our suppesition. é <2) e. [Sufficiency]. By hypothesis there is a set, say B*, which is a member: of 4, Hence we may apply the axiom schema of separation to obtain: (4) JOE Cesc E(VBBECA ze B). Since the fact that 2 € Bº follows from BºC A, and the other part of our hypothesis, namely that (VBBCA >2€ B), we infer from (4) that (5) JOW unrece(VBIBEA >2€B). nó: 2.6 GENERAL DEVELOPMENTS 4 “Fhat 2 € NA follows from (5), the definition of NA and tho defining rônditions for definitions by abstraction. QED. - Tr this proof a square-bracket notation, which is often convenient in “proving an equivalence, has been used. We consider the formula which is “the right member of the equivalence as asserting a necessary and sufficient, gondition for the formula which is the left member to hold. Thus if we Want to prove a theorem of the form Pé Q, we establish that Q is a “fgcessary condition for P by assuming P and deriving Q. We establish that Q is a suficient condition for P by deriving P from Q, The order of development of this section is somewhat deceptive. The eproof of Theorem 67 does not depend on the sum axiom, and thus the elementary theory of the N operation could have preceded consideration 6f this axiom. THroREM 68. N0-=0. troor. Suppose that NO =0. Then there is an v in NO, and by Theorem 67 there is à set BC 0, vhich is absurd. QULD, , Tt is worth remarking that in von Neumann set theory, which admits “Sets which are not members of any other set, that is, proper cla; the “operation symbol “MN is defined im such a way that Theorem 68 is false. Ta fact, the theorem is then that (é) No =Y, There V is the universe, that is, the proper class which has as members verything which is a member of something. The radical difference betwcen D) and Thcorem 68 emphasizes the slightly artificial character of any form dt axiomatic set theory. Intuitively (1) may scem preferabte to Thcorem 68; but (1) entails the admission of proper classes, which appear rather bizarre from the standpoint of naive, intuitivo set theory. nioj=a. Proor, Suppose thero is an element x io N(OJ. cfinition 14 2 € O, which is absurd. Q.F.D, Tiko Lhe previous two theorems the next four theorems are concerned it the intersection of extremely simple families of sets. Two of the broofs are omitted. “Tirmorem 69, Then by virtue of THeoREM 70. NÍAJ=A. .PRoor. If ve NtA), hen since 4 € (4), by Definition 14 SEA, GENERAL DEVELOPMENTS Char. 2 Terorzm 79, NAG UA. rRoor. Ifzc NA, then by Thcorem 67 a) (VBXBCA>2E B) & (IBXBE 4). Tt follows from (1) that GB)(BEA&zE B), rhence by Theorem 55, ze UA. QE.D. Tunonem 80. UNC(A,B)= A. NU (A,B) = ANB. In many mathematical contexts in placo of UA and NA the notetion TueorEM 81. [45] UR BEA and (2) nB BEA is often seen. In fact, notation more flexible than (1) and (2) is convenient in later chapters for the development of ordinal number theory. In. introducing at this point the appropriate definitional schema, wo use the ion “r(x)' for a term schema just as we have used “pla” for a for- mula schema, Derinitsos ScHeva 15. (a) Dirt) = Uty: (Boy = (0) &re 4) (5) 049 = Ny (gy =) se A), Thus, E A = (1,23) and r(x) = fz) U (4), then Url) = UCS), (2,8), (3,4]) z€A = [123,4] and Drs Suill other notational devices for union and intersection of families of sets are often useful, but will not be formally introduced. For instance, Uz = Uleig(a)). 6x) GENERAL DEVELOPMENTS 45 Trrom a logical standpoint there is a sharp difference between Definitions 13 and 14 on the one hand and Definition 15 on the other. The first two atroduce operation symbols, whercas Definition 15 introduces an operator Which provides a new way of binding variables, and in this stands dgether with Definitions 11 and 12. We state without proof some theorems concerning the notions introduced by Definition 15. Note that these theorems, like Theorem 78, assert important general distributive luws, Some additional results are given im isto exereises, Trsonam 82. Uz= UA, s€A Tesorea 88. Nar=NA, A Trxonrem 84. 4NUB=U(ANO). cer Trorex 85. (ID(DEB) > AUNB=N(ÁUC), ces Finally, we conclude this section by showing that the union axiom is tedundant. Since this thcorem is not about sets but about our particular siioms for set thcory we list it as 2 metathcorem, that is, as a metamathe- iatical theorem. MeTArHsOREM 1. The union axviom is derivable from the axiom of extenstonality, the pairing aviom, and the sum axiom. rmoor. Given any two sets 4 and B, by the pairing asiom, we have the set t4,B). Now zc U[AB o (IDADE AB] &zE D) by Theorem 55 S(D(D=AVD=BjEgzeD) by Theorem 43 SrcÁV2EB by quantifier togie. From the above cquivalences il is a simple matter of quantifier logic to :derive that : (JO(VereCerc Avec), tthich is precisely the union axiom. QD, In conuection with the above provf, it is easy to check that Thcorems 43 and 55 depend on no more than the three axioms men! Precise identification of the points where the axiom of extensionality is needed ÀS left as am exercise, 46 GENERAL DEYELOPMENTS Crap. 2 EXERCISES 1. Given that 4= ((12), 2,03, (1,8)) find UA, NA, NUA. 2. Given that A = (f(L2), (U)), ttLON) find UA, NA, UUA, NNA4, UNA, NUA, a Give a specific set A which vill servo as a comterexample to the general assertion that n4=0>4=0V 4 = (0). 4. Give specifie sets 4 and B which will yield a counterexample to the general assettion that NA nNB= MANB. 5. Givea definition of UA by means of an equivalenoe and without usc of the abstraction notation. 8. Tn Zermelo's original paper [1908], he defined “Nº in such a way that if A has an individual as an element, N4 = 0, Reformulate Definition 14 to conform to Zormelo's, and show by means of counterexamplos which of the ticorenas in this section do not hold when this revisod definition is used. 7. Prove 'Lhcorem 58. 8. Prove Theorems 62, 68, and 64. 9. Prove Theorem 66. 10. Prove Lhcorems 71 and 72. 11. With respeet to the existential part of the hypothesis of Theorem 74, show. :: by an example that if à is omittod the resulting statement is not a theorem, 12. Prove Theorems 75, 76, and 77, 13. Prove Theorems 80 and 81, 14. Prove that 0€45N4=0. 15. Prove that (VOCECA(IDDEBECCD)=>UACUB. 16. Prove Theorems 82 and 83. 17. Prove Theorems 84 and 85. 18. Prove that ND (e) B)=N fi-B. s€A s€4 19. Explain at what points the axiorm of extensionality is needod in tho prof of Metatheorem 1. $27 Power Set Axiom. In this section we are concemed with the notion of tho set of all subscts cf a given set, 'This set is called the powér set of à given set. The name “power set has its origin in the fact that Ei e. 2.7 GENERAL DEVELOPMENTS 47 g'set A has n elements, then its power set (in symbols: PÁ) has 2º elements. As an illustration of the notion, if A= 1,2] hen SA = (0, (13, (2), 43. “Tt should be intuitively clear that, as in this example, the empty set is a member of the power set oi any set; moreover, any set is a member of its ym power set. 'The appropriate formal definition should be obvious. Derisirion 16. PA =([BLIBCÇA). This definition is axiom-free in the same sense that Definitions 13 and 14 are, but in order to prove the desited theorem concerning PÁ, the power set axiom guaranteeing the existence of the intuitively appropriate set is “nceded: GABVOCEBSCC A). Tt is worth noting that we could have taken the weaker formulation (IBVOMC CASC EB) and then used the axiom schema of -Séparation to get the present axiom. We may immediately prove: Tarorey 86. BEGÁSGBCA, proor. Use Definition 16, power set axiom and properties of definition “-by abstraction. THrorEM 87. AC GA. rroor. By Theorem 5 ACA, “xhence by Theorem 86 we obtain the desired result. QE.D. Trronex 88. OC OA, THzorEMm 89. 40 = (0). pRoOor. Since0CO, 0€ 90. ““Moreover, if A € €0, then by Theorem 86 ACO, ut then by Theorem 4 QED, 50 GENERAL DEVELOPMENTS CHar. 2 To establish the converse implication it will suífice to show that (3) implies (4) 2E GOA U B), since by virtue of (1) it will then be obvious that (3) implies (2). Thus we need show only that (3) implies (4). Now by (3) and the definition of ordered pairs Ely, tunel), and since by hypotbesis y € 4 and z€ B, we have: bICAUB, and tn) CAUB, whence by virtue of Theorem 86 je ota o B) and fee HA U B). Thus ty, tn) SA UB, that is, 2 MAUB), but by virtue of Theorem 86 again, wc then have: ve CPA U B), whieh is what we desired to prove. Q.E.D. Wc have then almost immediately the following two useful thcorems. 2EAXBO(INA)geA&reB&o = (va). Trrorem 97. (L)CAXBOZrEA &yEB. THEOREM 96. Wo next turn to a number of thcorems whosc intuitive content is obvious. Several of the proofs arc omilted and left as exercises. Turonnu 98. AXB=04=0VB=0, proor [Necessity]. We use an Indirect argument. By hypothesis 4 X B = O, Suppose now that AZO&BA0. GENERAT DEVELOPMENTS 51 'hen by Theorem 2 ye A) & (Joe B), nd thus by Theorem 96 (mjCAXB, hich contradicts the hypothesis and proves our supposition false. [Sufliciency]. From the condition that 4 = 0 or B = 0 and Theorem “2, we infer Lg») =(Iy(ye A)v -(do)(ze B), and it follows from (1) that, (II )ye A &zcB&s = (yo), ànd thus by Theorem 96, for every « agAaAXB. “Hence by Theorem 2 4x8B=0, QED. Taroram 99). AXB=BXÃAS(A=0VB=0VA-B). rroor [Necessityl. Suppose that 40, BAZO0 and AB, ie, suppose the condition does not hold. Since 4 = B, there is an & such that either cc A &rgBorzgA&arcB. For definiteness, let us àssume the first alternative holds, and let 3 be an element of B (there are Such clements, since B 0). Then by Theorem 97 ten)cAXB, and thus from the hypothesis that 4 x B= B X A, we have: CMCBXA but by virtue of Theorem 97 again 2€ B, Which contradicts our assumption that 2 g B. [Sufficiency]. Of the three possibilities we may use Theorem 98 to Combine two; namely, from 4 =0V B=0 we infer that AXB=0=BXA. Assume now the third possibility: 4 = B. Then since it is a truth Of logic that AXA=AXA 52 GENERAL DEVELOPMENTS CraP, 2 28 GENERAL DEVELOPMENTS s3 we have at once that EXBROISES AXB=BXA. QED. Trrorey 100. AZ0&AXBGAXC>BCC. Prove Thcorems 96 and 97. Prove "Vheorem 101. Prove Theorems 103 and 104. Give a simple countcrexample to show that in general it is not the case that AU(BXO =(AXBJU(AXO. so oca rroor. If B = 0, the proof is trivial, so assume B »* 0. Sine by”: hypothesis 4 z< O, let rc A&gEB. . cdday 5. Ts the Cartesiau product operation associutive? IF so, prove it. Tf not, givo a counterexample, “8. Prove that Then by Theorem 97 (CncAXB, and thus by hypothesis AxNB = (AXO. ENEAXO. : a . . tes XE $ 2.9 Axiom of Regularity. Tt is difficult to think of a set which Hence by use of Lheorem 97 again “night reasonably be xegarded as a member of itself. Certainly the set of (8) vet. “ali men, for cxample, is not a man and is therefore not a member of itself. “Perhaps it might be argued that in intuitive set theory the set of all abstract bjeuts or the set of all scts should provide an example of a set which is a pember of £, but as wo saw in the first chapter, the set of all sets is itself a paradoxical concept. These remarks suggest we take as an axiom: (1) AgA. owever, the assumption of (1) would not prohibit the counterintuitive situation of there being distinct sets 4 and B such that (2) ACR&BEA. £ you do not believe (2) is counterintuitive, try to give a simple example £. sets A and B satisíying (2).) Wurthermore, if we took (2) as am axiom, longer counterintuitive cycles of membership would not be vuled out — like ihe existence of distinct sets À, B, and O such that 6) ACB&BECEACEA. Since 3 is an arbitrary clement of 8, (1) establishes that BCC QED; “Fhe proof of the néxt theorem is left as an excreise. Trmorem 101. BCC=>AXBCAXE. “The next three thcorems state three distributive laws for the operatio of forming the Cartesian product of two sets. Only the first one is proved. here. Turorem 102. AX(BNO=(AXB) NAXC). PROOF. GPCAXBNODOLECA&yEBNC by Theorem 97 orcA&ycB&gveC by Theorem 12 oscA&yeB&IçA&gEl by sentential logic é StrncAXBELN)CAXO by Theotem 97 etrpe(áxXB) n(AXO) by Theorem 12. QED:: “We prevent such cyeles of any length x by adopting am axiom which is, om the assumption of our other axioms, including the axiom of choice, Squivalent to the non-existence of infinite descending sequences of sets ie, Arm C A). The form of the axiom which we adopt, the axiom of togularity, is due to Zermelo [1930], although an essentialy equivalent; but more complicated axiom was given carlicr in von Neumann [1929, é os1J* AOS (Arc A &(Vn(yes —>yg A). *The essential ide tor to that in Trmoney 103. AX(BUC)=(AXB)U(AXO). Tenonux 104 AX(B-C=(AXB)-(AX O). xvas formulated even earlier in von Neumann [1925, p. 239] and iximanoff [1917]. 
