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Both follow from D2 and '1'4. aUb = b U a and a U (b U c) = (aUb)Uc. The first follows from B2, and the second follows from Ba and T4. aUa'=1. This follows from D2, T4, B1, and aUO=a. This follows from D2, D1, and T4. 6 3 I Another Formulation of the Theory 257 Tll. afl (aUb) =a. Pr. 1. b' fl (a fl a') = 0, 2. a (l (a' fl b') = 0, 3. a fl (a' fl b')" = 0, 4. a f l (a' f l b')' = a, 5. a f l (a U b) = a, T12. an (anb)'=aflb'. Pr. 1. aflb"fl(aflb)'=0, 2. a fl (a 0 b)' fl b" = 0, 3. a fl (a (1 b)' (1 b' =afl(aflb)', 4. a fl b' fl (a fl b)' =afl(afl1')', 5. a flb'fl(aflb)' by T3 and Dl. from 1, by B, and B2from 2, by T4. from 3, by B3. from 4, by D2. by Dl and T4. from 1, by B1. from 2, by B3. from 3, by B1. T13. T14. = a fl b' fl (b' U a'), 6. a fl b'f (b'Ua') =aflb', 7. a fl(a(lb)'=aflb', by T7 and B1. byTll. from 4, 5, and 6. a fl c = a, a fl c' =0 and a U c = c are equivalent properties. Pr. Left as an exercise. aflc=aandbflc=bimply (aUb)lc=aUb. Pr. Assume that a fl c = a and b fl c = b. Then a U c = c and b U c = c, by T13. By TI 1, T15. Pr. 1. (aUb |
)fl [(aUb)Uc]=aUb. Two substitutions within the brackets give the desired result. an (bUc) _ (a fl b)U(aflc)and a U (b fl c) _ (a U b) fl (a U c). (a fl b) fl [a fl (b U c) ] =aflbfl (bUc) =aflb, by B2, Ti, andT11. 2. (a fl c) n [a fl (b U c) ] = a fl c, similarly. [(a fl b) U (a fl c) ] fl [a fl (bUc) ] 3. = [(a fl b) U (a fl c) ), from 1, 2, and T14. 4. [a fl (bUc) ] fl [(a fl b) U (a U c) ]' =afl (bUc)fl(aflb)'fl(aflc)', = a ll b' ( c, r) (b U c), =afl(bUc)'fl(bUc), = 0. by T7; by B, and T12; 258 Boolean Algebras I C It A P. 6 5. [an(bUc)Jn [(a n b) U (a n c) I =an(bUc), 6. a n (1 U c) = (a n b) U (an c), from 4byT13. from 3 and 5 by B,. The proof of the other distributive law is left as an exercise. The set of axioms in the new formulation of the theory of Boolean algebras is independent. A proof of this requires the determination of a system (13, n, ');, which satisfies all the axioms except B i = 1, 2, 3, 4. Below arc defined four systems which demonstrate the independence of the axiom with the corresponding label. c a b c (B,) B = (a, bB2) B = (a, bB,) B = (aB4) B = (A C v'(Z+) I7+ - A is a finite set). n is set intersection.'is defined as follows. We note that for each A in B there exists a least positive integer a such that [a, the set of all integers x |
> a, is included in A. Then A is the disjoint union of [a and A0, a subset of, a - 2} (unless A = Z+, in which case A = [1). Now we (1, 2, define A' to be A U [(a + 1), where A is the complement of AD in, a - 11 (unless A = Z+, in which case A' = [2). (1, 2, Some hints for the analysis of this example, which establishes the independence of B4, appear in Exercise 3.2. Possible substitutes for B4 are described in Exercise 3.3. EXERCISES 3.1. Prove T13 and the remaining distributive law in the proof of Theo- rem 3.1. 3.2. Regarding the system (B, n, '), which, it is asserted, establishes the independence of B4, it is clear that B, and B2 hold. Prove that the system 6.4 I Congruence Relations for a Boolean Algebra 259 satisfies B3 but not B4. hint: for B3, show that if C = Co U [c, then C n C' _ [(c + 1), and, if A = Ao U [a and B = Bo U [b, then AnB'= {(AflB)U[b+l) (Ao n B') U [a ifa<b if a> b 3.3. Show that each of B6, B6, - - -, B,o defined below implies B4 in the presence - - -, B9 together with B1, B2, and B, of B,, B2, and B,. Infer that each of B6, B6, yields a formulation of the theory of Boolean algebras. For some calculations it is convenient to use the fact that if (B, n, ') satisfies B1, B2, and B3, then (B, <), where a < b means a n b = a, is a partially ordered set. So prove this first. B6. For all a and b, ana'=bnb'. Bs. For all a, a" = a. B. There exists in B an element m such that whenever x n m = x, x = M. Bs. There exists an integer n > 1 such that for all a, the nth iteration of |
a under'is equal to a. B9. For all a and b, a < b implies b' < a'. B,o. B is finite. 4. Congruence Relations for a Boolean Algebra We turn to an examination of an aspect of the two given sets of axioms for a Boolean algebra that has not been touched on. It is sufficient to consider the second set of axioms, since the reader will readily see what alterations are required for our remarks to apply to the first set. When the statements labeled B1, B,, B,, and 134 were introduced, no mention was made of the precise meaning to be assigned the relation symbolized by " ="; rather, it was intended that the reader supply his own version of equality. Suppressing any preconceived notions that we might have in this connection, let us determine a set of conditions which are adequate for our purposes. An analysis of the proofs of TI-T15 in the proof of Theorem 3.1 reveals that the following is a sufficient set of conditions. (E) " =" is an equivalence relation. (S) Let F be an element of the Boolean algebra (B, n, ') resulting - of B using the operations in B, and let fi-oin elements a, b, a = a,, b = b,, - - -. Then, if P; is an element which results from F by the replacement of some or all occurrences of a by al, b by b,, - - -, then F = F,. - - 260 Boolean Algebras I CHAP. 6 Now (S) can be derived from the following two simple instances of this substitution principle. If a = b, then a fl c = b fl c for all c. If a = b, then a' = V. C) ( The proof, which we forego, is by induction on the number of symbols in the element F. Thus (E) and (C) insure (E) and (S), and, hence (E) together with (C), which are clearly necessary properties of equality, are also sufficient for our purposes. As such, equality is an instance of a congruence relation for a Boolean algebra, a notion which we discuss next. Before focusing our attentions on congruence relations for Boolean algebras we make several remarks about this concept in a general setting. When one is presented with, or constructs, some specific mathematical system, there is among its |
ingredients a "natural" congruence relation either explicitly or implicitly defined. This means that there is present an equivalence relation which is preserved under the operations at hand in the sense suggested by (C) above. Normally one symbolizes this relation by "_," calls it equality, and uses it without comment. For example, in the case of sets, the relation is that of set equality; it is a congruence relation on any collection of sets. If one is attempting to demonstrate that a particular system (5 has properties BI-B4, he will interpret the occurrences of the equality sign in these as the natural equality for 0-L For example, in the verification that ((P(X), f1, ') is a Boolean algebra, " _" will be taken to denote set equality. In summary, the equality symbol, as used in B1-B4 need have no absolute nature, but merely a relative one. It suffices that it stand for some congruence relation. We return to the general discussion with the remark that when one -), there are often is studying any specific mathematical system (X, compelling reasons for identifying elements of X which are distinct relative to the natural congruence relation. This amounts to the introduction of an equivalence relation p other than the natural one. One then directs his attention to X/p, whose elements are the p-equivalence classes, and regards it as the basic set. If p is not merely an equivalence but a congruence relation, then it is possible to introduce into X/p faithful analogues of whatever operations and relations are defined for X. We proceed to discuss this matter in detail for the case of Boolean algebras. 6.4 1 Congruence Relations for a Boolean Algebra 261 Let (B, (l, ') be a Boolean algebra, and let B be a congruence relation on it; that is, let 0 be an equivalence relation on B such that the following hold. (C,) Ifa0b, then a()c0b flcfor all c. (C2) Ifa8b, then a'8b'. We shall be concerned solely with proper congruence relations, that is, those congruence relations different from the universal relation on B. We now derive from (CI) an instance of the earlier substitutivity property (S). (C3) If a O c and bbd, then a ll b U c f d. |
For proof, assume that a 0 c and b 0 d. Then a fl b 8 c (l b and b fl c 0 d f1 c, by (Ci). Since the meet operation is commutative and 8 is transitive, the result follows. The derivation of the dual of (C3) is left as an exercise. If B/8 is the set of B-equivalence classes if, then in B/8 the foregoing result (C3) becomes the following. Ifa=candb=rl,then aflb=cfd. This means that the relation (((a,b),a flb)jaCB/8andbCB/8; is a function on (B/8) X (BIB) into B/8, that is, an operation in B/8. We shall denote this operation in B/8 by fl and its value at (a, 6) by a () 6. So, by definition, afb=a(lb. Next, it follows directly from (C) that if 'a = b, then a = b. Hence, the relation {(a, a )Ja C B/8} is a function on B/e into B/8. We denote this function by'and its value at a by a'. So, by definition, a = W. It is a straightforward exercise to verify that (B/8, (l, ') is a Boolean algebra. For example, to verify B3, assume that a fl b' = c f) c'. Then, in turn, anb cflc, a fT b' = c (1 c, afb'8cflc', (a n b')' 0 (c (1 c')', by definition of x'; by definition of To fl y; xeyiffx=y, by (Ca) ; 262 Boolean Algebras CHAP. 6 W U b 0 1, (a'Ub) fl a01 (l a, an boa, anb=a, anb=a, by property of (B, by (C1) ; by property of (B, fl, '); x =yiffxey; by definition of x fly. In summary, we have shown that from a Boolean algebra (B, f), ') and a proper congruence relation 0 on it one may derive a Boolean algebra (B/0, f |
), ') whose elements are 9-equivalence classes and whose operations are defined in terms of those of the original algebra using representatives of equivalence classes. If 0 is different from the equality relation in B, then the derived algebra may be essentially different from the parent algebra. This is true in the first of the following examples. EXAMPLES 4.1. Consider the Boolean algebra (D'(Z), fl, ') whose elements are the subsets of Z, the set of integers. J We recall the definition of the symmetric difference, A + B, of two sets as the set of all objects which are in one of A and B but not both. For A and B in 61(Z) let us define A 0 B to mean that A + B has a finite number of elements. It is easily verified that 0 is an equivalence relation on 6'(Z). Further, if A 0 B, then A f l CO B (-l C, since, for all A, B, and C, (AfC)+(BfC)=(A+B)() C, and, hence, if A + B is finite, then so is (A fl C) + (B fl C). Finally, if A 0 B, then A' 0 B', since A + B = A' + B'. Thus, 0 is a proper congruence relation on the given algebra, and a new Boolean algebra whose elements are B-equivalence classes results on defining 3o 3=AflB and 4'=A. That a substantial collapse of elements has taken place on transition from the first to the second algebra is indicated by the fact that, in the first the zero element is 0, whereas in the second the zero element, Qf, is the collection of all finite subsets of Z. 4.2. The symmetric difference operation used in the preceding example can be defined in any Boolean algebra. By the symmetric difference of elements x and y of a Boolean algebra, symbolized x + y, we understand the element (xflY')U(x'fly). It is an easy exercise to prove that this operation is commutative, associative, and nilpotent (x + x = 0). Other properties which we shall need later are f We prefer to use prime symbols to denote the operation of complementation relative to Z in this example, so the bar symbol will be available to denote equivalence classes. 6.4 I Cong |
ruence Relations for a Boolean Algebra 263 x+0=x, (x+y)nz= (xnz)+(ynz), x'+y'=x+y. Further, since the symmetric difference is defined in terms of union, intersection, and complementation, if' O is a congruence relation on a Boolean algebra, then x 0 y implies that x + z 0 y+ z. At this point it becomes desirable to simplify our notation by identifying an algebra simply by its basic set. Thus, we shall use the phrase "the Boolean algebra B" in place of "the Boolean algebra (B, n, ')." Let us consider now the relationship of a Boolean algebra B/0 to the algebra B from which B/0 is derived using a proper congruence relation. Let p be the natural mapping (see Section 1.9) on the set B onto the set B/0, that is, the mapping p: 13 -*- B/0, where p(b) = b. Since anb =a-7)_b and 27' = a-7, 1)(a n b) = p(a) n p(b) and p(a') _ (p(a))'. That is, p is a "many-to-one" mapping (unless 0 is the equality relation on B) which preserves operations. A mapping g on one Boolean algebra, B, onto another, G, which takes meets into meets and complements into complements, that is, g(a n b) = g(a) n g(b), g(a') = (g(a))', is called a homomorphism of I3 onto C, and C is called a homomorphic image of B. If, in addition, g is one-to-one, then g is called an isomorphism of B onto C. If g is an isomorphisin of B onto C, then g-' (which exists) is easily proved to be an isomorphism of C. onto B, and each algebra is called an isomorphic image of the other and each is said to be isomorphic to the other. Returning to the case at hand, we may say that p is a hornonrorphisrn and 13/0 is a homomorphic image of B. That is, each proper congruence relation on a Boolean algebra determines a homomorphic image. Conversely, each homomorphic image C of a |
Boolean algebra B determines a proper congruence relation on B. Indeed, if f: B -} C is a homomorphism, then the relation 0 defined by a 0 b iff f(a) = f(b) is a proper congruence relation on B. The proof is left as an exercise. We continue by showing that BIB, the algebra of 0-equivalence classes, is isomorphic to C. For this we introduce the relation g, which is defined to be 264 Boolean Algebras I CRAP. 6 ((x,f(x))Iz C B/0}. It is easily seen that g is a function which maps B/0 onto C in a one-toone fashion and that g(x n y) = g(x -n Y) = f(x n y) = f(x) n f(y) = g(Y) n g(y), g(x') = g(x`) = f(x') = U(x))' _ (g(X))', that is, g is an isomorphism. Moreover, if p is the natural mapping on B onto B/0, then we observe that for the given homomorphism f : B -- C we have f = g o p. The next theorem summarizes our results. THEOREM 4.1. Let B be a Boolean algebra and 0 be a proper congruence relation on B. Then the algebra B/0 of 0-equivalence classes is a homomorphic image of B under the natural mapping on B onto B/0. Conversely, if the algebra C is a homomorphic image of B, then C is isomorphic to some B/0. Moreover, if f : B -->- C is the homomorphism at hand, then f = g e p, where p is the natural mapping on B onto B/0 and g is an isomorphism of B/0 onto C. It should be clear from the foregoing results that the homomorphisms (onto) of a Boolean algebra are in one-to-one correspondence with the proper congruence relations on the algebra. The importance of the role which proper congruence relations play suggests the problem of practical ways to generate them. One way is provided by a distinguished type of subset of a Boolean algebra, which we define next. A nonempty subset I of a Boolean algebra B is called an ideal iff (i) x C I |
and y Cl imply x U y C 1, and (ii) x C I and y C Bimply xnyC I. For example, if a C B, then {x C Bl x < a } is an ideal; this is the principal ideal generated by a, symbolized (a). To show that (a) i§, an ideal, we note that if x C (a) and y C (a), then a is an upper bound of {x, y } and, consequently, is greater than or equal to x U y, the least upper bound of x and y (see Theorem 2.2). Thus, x U y C (a). Finally, if x C (a) and y C B, then x n y < a, since x < a. Two trivial ideals of B, namely, 101 and B, are both principal; indeed, 101 = (0), and B = (1). The ideal (0) is the zero ideal, and the ideal (1) is the unit ideal of B. An ideal of B which is different from B is called a proper ideal. The relationship between, proper ideals of B and proper congruences on B is given in the following theorem. 6.4 I Congruence Relations for a Boolean Algebra 265 THEOREM 4.2. If 0 is a proper congruence relation on a Boolean algebra B, then I = { x C BIx 0 0 } is a proper ideal of B and x O y if x + y C I. Conversely, if I is a proper ideal of B, then the relation 0 defined by x O y if x + y C I is a proper congruence relation on B such that I = { x C Bjx 0 0). Thus, the proper congruence relations on B are in one-to-one correspondence with the proper ideals of B; each 0 corresponds to the ideal I of elements 0-related to 0. Proof. Let 0 be a proper congruence relation on B and let I = { x C Bjx 0 0 1. Then I C B and, if x, y C I, then, in turn, x'ny'01 ny', x'ny'Oy', xUy0y. x'01, x00, The last fact, when combined with y 0 0, implies that x U y 0 0, which proves that I satisfies the first of the defining conditions for an ideal |
. Next, let x C I and y C B. Since x 0 0 implies x n y 0 0, the second condition is also satisfied, and I is an ideal. We prove next that x 0 y if x + y C I. Let x + y C I; that is, x + y 0 0. Then (x +,y) + y 0 0 + y, and hence x 6 y (where we have used properties of the symmetric difference stated in Example 4.2). Conversely, x 0 y implies that x+ y 0 y+ y; that is, x + y 0 0. Turning to the converse of the foregoing, let I be an ideal of B and define 0 as stated in the theorem. Then 0 is reflexive (since x + x = 0 C I), symmetric (since x + y = y + x), and transitive (since the symmetric difference of two elements of I is in I). Further, x O y implies that x fl z o y fl z, since if x 0y, then, in turn, x +y C I, (x + y) n z C I, and (x n z) + (y n z) C I. Finally, x 0 y implies that x' 0 y', since x + y = x' + y'. To complete the proof of the converse we must show that x 0 0 if x E I. This follows from the identity x + 0 = X. From the two preceding theorems there follows the existence of a one-to-one correspondence between the homomorphisms of a Boolean algebra and its proper ideals. If f is a homomorphism of an algebra B onto an algebra C, the associated ideal I, which is called the kernel off, is the set of all elements of B which f maps onto the zero element of C. If 0 is the congruence relation on B that corresponds to 1, then we will often write B/I instead of "B/O," and call the algebra so designated (an isomorphic 266 Boolean Algebras I CH A P. 6 image of C.) the quotient algebra of B modulo I. If f is an isomorphism, then 0 is the equality relation on B and I is the zero ideal. Conversely, it is clear that if the kernel of a homomorphismf can be shown to be the zero ideal, then f is an isomorphism. Therefore, a hom |
omorphism is an isomorphism iff its kernel is the zero ideal. We conclude this section with several general remarks about homomorphisms. Since the operations of union and symmetric difference and the ordering relation are expressible in terms of intersection and complementation, it follows that a homomorphism of a Boolean algebra preserves each of the former. Further, the fact that if f is a homomorphism, then f(a () a') = f(a) l (f(a))', implies that f(O) is the zero element of the image algebra. By a dual argument, f(1) is the unit element of the image algebra. EXERCISES 4.1. Prove the dual of property (Ca) for a congruence relation 0, namely, (Ca)' IfaOcandb8d,then aUbBcUd. 4.2. Complete the proof of the assertion in the text that (B/B, is a Boolean algebra if (B, 1, ') is a Boolean algebra and 0 is a proper congruence relation on B. 4.3. Prove that the symmetric difference operation has the properties stated in Example 4.2. 4.4. Prove that if g is an isomorphism of the Boolean algebra B onto the Boolean algebra C, then g-' is an isomorphism of C onto B. 4.5. Prove the assertion prior to Theorem 4.1 that if f : B -- C is a homomorphism, then the relation 0 defined in B by a B b if f (a) = f (b) is a proper congruence relation on B. Further, prove that f = g o p, where g and p are the mappings defined in the text. 4.6. Prove the assertion following Theorem 4.1 that the homomorphisms (onto) of a Boolean algebra B are in one-to-one correspondence with the proper congruence relations on B. 4.7. Draw the diagram of the algebra (t; of all subsets of {a, b, c, d}. Locate the members of the ideal ({a}) on the diagram. Then use the diagram to determine the B-equivalence classes of the relation B corresponding to ({a}) in accordance with Theorem 4.2. Finally, draw the diagram of the algebra a/B. 4.8. In the next section an atom of a |
Boolean algebra is defined to be nonzero element a such that if b < a, then either b = 0 or b = a. Show that there are no atoms in the Boolean algebra of equivalence classes defined in Example 4.1. 4.9. Referring again to Example 4.1, let A 01 B mean that A 0 B and that 3 is not a member of A + B. Prove that Bt is a congruence relation on P(Z). Determine the atoms of'P(Z)/Bi. 6.5 1 Representations of Boolean Algebras 267 5. Representations of Boolean Algebras The set-theoretical analogue of our second formulation of the theory of Boolean algebras is that of an algebra of sets. Since it was essentially the structure of such a system that motivated the creation of the axiomatic theory under discussion, an obvious representation problem arises : Is every Boolean algebra isomorphic to an algebra of sets? This we can answer in the affirmative. We shall begin with the case where the set B has a finite number of elements, although our first definition is applicable to any Boolean algebra. An element a of a Boolean algebra is an atom if a 0 and b < a implies that either b = 0 or b = a. For x in B let A(x) denote the set of all atoms such that a < x. We next derive several properties of atoms and of the sets A(x) for the case of an algebra (B, n,') such that B is finite. A,. If x 0 0, there exists an atom a with a < x. Proof. This is a direct consequence of the finiteness assumption. The details are left as an exercise. A2. If a is an atom and x C B, then exactly one of a < x and an x= 0 holds. Alternatively, exactly one of a < x and a < x' holds. Proof. Since a n x< a, either an x= a or a n x= 0. Moreover, both cannot hold, since a 0. A3. A(x n y) = A(x) n A(y). P r o o f. First we note that x n y is the meet of two elements in B, and A(x) n A(y) is the set of those elements common to A(x) and A(y). Now, assume that a C A(x n |
y). Then a< x n y, and hence a< x and a< y. Thus a E A(x) n A(y). hence A(x n y) 9A(x) n A(y). Reversing the foregoing steps establishes the reverse inequality, and hence equality. A4. A(x') = A(1) - A(x). Proof. First we note that A(1) is the set of all atoms of B. Now let a C A(x'). Then, by A2, it is false that a C A(x). Hence, a E A(1) A(x). Conversely, if a E A(1) - A(x), then a V A(x). Hence, by A2, aEA(x'). 268 Boolean Algebras I CHAP. 6 A5. A(x) = A(y) iff x = y. Proof. Assume x 0 y. Then at least one of x < y and y < x is false. Suppose that x < y is false. Then x n y' F6 0, so that by Al there exists an atom a< x n y'. By A3, a C A(x) and a C A(y'). Thus, a C A(x) and, by A4, a V_ A(y). Hence, A(x) 0- A(y). The same conclusion follows similarly if it is assumed that y < x is false. A6. If al, 612, {ai, a2,..., a&}., ak are distinct atoms, A(al U a2 U... U ak) _, ak} C A(ai U a2 U Proof. Clearly, jai, all, verse, assume that a CA (a, U a2 U 2, af(aIUall U...Uak)= (anal)U(a(l which is impossible., k. Then, by A2, a n ai = 0, i = 1, 2, U ak) and a 5-6 a;, U ak). For the coni = 1,, k., and hence a = a2)U...U(af ak) =0, THEOREM 5.1. Let B be a Boolean algebra of n elements. Then B is isomorphic to the algebra of all subsets of the set of atoms |
of B. If m is the number of atoms of B, then n = 2"`. Proof. Let T be the set of m atoms of B. Then the mapping A : B -N 6'(T) is one-to-one by A5 and onto 6'(T) by A5. According to A3, the image of a meet in B is the meet of the corresponding images in 6'(7). According to A4, the image A (x') of x' is the complement of the image of x, that is, the relative complement of A(x) in 7'. Thus, A is an isomorphism. Then n = 2"' follows from the fact established earlier that the power set of a set of m elements has 2" members. COROLLARY. Two Boolean algebras with the same finite number of elements are isomorphic. The proof is left as an exercise. EXAMPLE 5.1. For B we choose {1, 2, 3, 5, 6, 10, 15, 30}, the set of divisors of 30. For a and b in B define a n b as the least common multiple of a and b and a' as 30/a. It is an easy matter to verify that (B, n,') is a Boolean algebra. The partial ordering relation introduced for the elements of a Boolean algebra takes the following form for this algebra: a <.b ifFa is a multiple of b. Thus, 30 is the least (and zero) element, and 1 is the greatest (and unit) element of the algebra. The atoms are 6, 10, and 15, and, consequently, the algebra is isomorphic to 6.5 I Representations of Boolean Algebras 269 that determined by all subsets of {6, 10, 15} with the usual operations. The mapping which establishes this isomorphism matches 2 with {6, 10} and 30 with 0, for example. It is left as an exercise to verify that a U b, which in our second formulation of a Boolean algebra is defined as (a' (1 b')', is the greatest common divisor of a and b. Thus, if at the outset we had introduced in B, along with the operation (l, a second binary operation U by defining a U b as the greatest common divisor of a and b, the outcome would have been the same. However, in the process we would have |
had to verify the distributive laws, which, in this case, is not a particularly simple matter. Before continuing with the representation theory we urge the reader to pause and reflect on the extent to which Theorem 5.1 clarifies the structure of finite Boolean algebras (that is, algebras having a finite number of elements). Indeed, it leaves nothing to be desired in the way of a representation theorem. Possibly its definiteness, both with respect to its arithmetical aspect and the inclusion of an explicit recipe for constructing the asserted isomorphism, will be more fully appreciated when the corresponding result for the infinite case is obtained. For this, a different approach must be supplied, since there exist Boolean algebras without atoms (see Exercise 4.8). In the infinite case the substitute for an atom is a distinguished type of ideal, which we describe next. Let S be the set of all proper ideals in the Boolean algebra B. Since 10 } C S, it is nonempty. Further, the members of S may be characterized as the ideals of B which do not contain 1. As is true of any collection of sets, S is partially ordered by the inclusion relation, and the concept of a maximal element of S is defined. A maximal element of S is a maximal ideal of B. The existence of maximal ideals in an infinite Boolean algebra is secured by an application of Zorn's lemma. THEOREM 5.2. Maximal ideals of a Boolean algebra exist. Indeed, there exists a maximal ideal which includes any preassigned proper ideal. Proof. We consider the partially ordered set (S, C) defined above. If a is a simply ordered subset of S, then the union, A say, of the collection e is clearly an upper bound for t°. It is a straightforward exercise to verify that A is an ideal. Moreover, A C S, since 1 appears in no member of e and, consequently, does not appear in A. Thus, since every chain in S has an upper bound in S, Zorn's lemma may be applied to conclude the existence of a maximal element. The same argument when applied to (I C SII? J), where J is a given proper ideal, yields the existence of a maximal element which includes J. 270 Boolean Algebras! C H A P. 6 We prove next a sequence of theorems about maximal ideals of a Boolean algebra B which closely parallels that derived earlier for atoms. M |
,. If x 1, there exists a maximal ideal P with P (x) or, what amounts to the same, x C P. Proof. This follows directly from the final statement of Theorem 5.2, choosing. (x) as the given ideal. M2. For each maximal ideal P and each element x of B, exactly one of x C P and x' C P holds. Proof. We note first that for no x is x C P and x' C P, since it would then follow that 1 (= x U x') C P, which is impossible. Now assume that x (Z P, and consider the set Q of all elements of B of the form b U p with b< x and p C P. Then Q is an ideal, since (i) (b, U p,) U (b2 U P2) _ (b, U b2) U (p' U P2) = b3 U Pa, and (ii) if y C B, then (b U p) n y = (b n y) U (p n y) = b, Up,. Also, P C Q, since, clearly, P C Q and x C Q, while x (Z P. Thus, Q = B, since P is maximal. Hence, for some b < x and p C P, b U p = 1. It follows that x U (b U p) = x U 1, or x U p = 1. Then X, By the second part of the definition of an ideal it follows that x' C P. To continue with the derivation of properties of maximal ideals which parallel, in a complementary sort of way, those for atoms, we introduce the analogue of the sets A (x). If x C B, let M(x) be the set of all maximal ideals P such that x V- P or, what amounts to the same by virtue of M2, x' C P. The sets M(x) have the following properties. Ma. M(x n y) = M(x) n M(y). Proof. Let P C M(x n y). Then (x n y)' = x' U Y, C P. Since x' _ x' n (x' U y') and y' = y' n (x' U y'), it follows that x' C P and y' C P. Hence P C M(x) and P C |
M(y), or P C M(x) n M(y). Since each of these steps is reversible, the asserted equality follows. M4. M(x') = M(1) - M(x), where M(1), is the set of all maximal ideals of the algebra. Proof. We have P C M(x') iff,x' V- P iff x C P iff P C M(1) - M(x). Mb. M(x) = M(y) iff x = Y. 6.5 I Representations of Boolean Algebras 271 Proof. Assume x /- y. Then at least one of x < y and y < x is false. It is sufficient to consider the consequences of one of these. Let us say y < x is false. Then x U y' s 1, so there exists a maximal ideal P such that x U Y' C P. Now (x U y')' = x' n y (Z P, and, hence, by M3, P C M(x') and P C M(y), or P (Z M(x) and P C M(y). Thus, M(x) 54 M(y). The promised representation theorem follows easily from M,-M5. It is valid for an arbitrary Boolean algebra, but, in view of the more precise result for finite algebras, it is of interest only in the infinite case. The first proof of this result was given by the American mathematician, Marshall Stone (1936). THEOREM 5.3. Every Boolean algebra B is isomorphic to an algebra of sets based on the set of all maximal ideals of B. Proof. Let JR denote the collection of all sets of ideals of the form M(x) for some x in B. According to M3 and M,, :l is an algebra of sets. The mapping M: B - - :711 is onto by the definition of mz and one-to-one by M5. Finally, in view of M3 and M1, M is an isomorphism. With the representation theorem for the finite case in mind, it is natural to ask whether the above result cannot be sharpened to read, "Every Boolean algebra is isomorphic to the algebra of all subsets of some set." To discuss this matter we make two definitions. A Boolean algebra B is called atomic if for each nonzero element b of |
B there exists an atom a of B with a < b. A Boolean algebra B is called complete if for each nonempty subset A of B, lub A exists relative to the standard partial ordering of B. This definition has significance only when A is infinite, since in any Boolean algebra each pair, and consequently each finite set of elements, has a least upper bound. Now it is clear that the algebra of all subsets of a set is both atomic and complete. It is left as an exercise to prove that each of these properties is preserved under an isomorphism. Hence, an algebra which fails to have either property cannot be isomorphic to an algebra of all subsets of a set. Since, as noted earlier, the algebra described in Example 4.1 is not atomic, the question in mind is settled in the negative. The same conclusion is provided by the algebra defined in Exercise 2.6, since, as the reader may prove, it is not complete. The above pair of conditions which are necessary in order that a 272 Boolean Algebras I CHAP P. 6 Boolean algebra be isomorphic to the algebra of all subsets of a set are also a sufficient set. This is our next theorem. THEOREM 5.4. Necessary and sufficient conditions that a Boolean algebra be isomorphic to the algebra of all subsets of some set are that B be complete and atomic. In this event, B is isomorphic to the algebra of all subsets of its set of atoms. Proof. Since the necessity of these conditions has already been observed, we turn to a proof of their sufficiency. Suppose, therefore, that B is complete and atomic and let 7' be the set of all atoms of B. As in the proof of the finite case, let A(x) denote the set of atoms a for which a < x. Then, exactly as in the finite case, it can be proved that the mapping A on B into 01(T) has properties A3 and A4 (now, of course, property A, is an assumption). This means that A is a homomorphism on B onto an algebra of subsets of 7'. If U is an arbitrary subset of T, then, by the assumed completeness, U has a least upper bound, u say, in B. Then A(u) = U (this is a generalization of A6 for the finite case), so A is onto U'(7'). All that is needed to complete the |
proof is to show that A is oneto-one-that is, that the kernel of A is the zero ideal. This follows from the atornicity of B; if x P6 0, then A(x) s 0, so A(x) _ 0 ifl'x=0. EXERCISES 5.1. Prove property A, of atoms in a finite Boolean algebra. 5.2. Prove the Corollary to Theorem 5.1. 5.3. Referring to Example 5.1, verify that the set of divisors of 30 determine a Boolean algebra. Verify that in this algebra a U b is the greatest common divisor of a and b. 5.4. Referring again to Example 5.1, show that the set of divisors of any square-free integer determines a Boolean algebra in exactly the same way as does the set of divisors of 30. What does this result imply regarding the number of divisors of a square-free integer? 5.5. (a) Prove the converse of property M2 of maximal ideals to obtain a characterization of maximal ideals among the set of proper ideals. (b) Prove that maximal ideals can also be characterized as those ideals I of a Boolean algebra B such that B/I has just two elements. 5.6. (a) In Exercise 2.6 there is defined the Boolean algebra d of all subsets A of Z} such that either A or A is finite. Prove that the collection e of all finite subsets of _L+ is a maximal ideal of a. 6.6 Statement Calculi as Boolean Algebras 273 (b) The same collection a is an ideal of the algebra tP(Z+). Prove that e is not a maximal ideal of this algebra and determine a maximal ideal which includes C. 5.7. Devise a proof of Theorem 5.3 for the case of a denumerable Boolean algebra B that does not employ Zorn's lemma. (Hint: Prove by induction that if B is denumerable then there exists a maximal ideal which includes any preassigned ideal.) 5.8. Prove that the Boolean algebra in Exercise 5.6(a) above is not complete by showing that the collection of all unit sets of positive even integers has no least upper bound. 5.9. Prove that an isomorphic image of a complete Boolean algebra is com- |
plete and that an isomorphic image of an atomic algebra is atomic. 5.10. Prove that every ideal of a Boolean algebra B is principal iff' B is finite. (Note: The proof that B is finite if every ideal is principal is difficult.) 6. Statement Calculi as Boolean Algebras Statement calculi, as described in Section 4.3, yield models of the theory of Boolean algebras. One need merely restrict his attention to the algebraic character of a statement calculus as we now discuss it. According to Section 4.3, the core of a statement calculus is a nonempty set So of statements. This set is extended to the smallest set S of statements (that is, formulas) such that the negation of each member of S is a member of S and each of the conjunction, disjunction, conditional, and biconditional of any two members of S is a member of S. Since it was observed that the disjunction, conditional, and biconditional of two statements can be defined in terms of negation and conjunction, we may and shall assume that S is simply the closure of So with respect to these connectives. Then A takes on the role of a binary operation in S and'(which we shall use as the symbol for negation) that of a unary operation in S. In order to state precisely the structure of the system (S, A, '), that is, the set S together with its two operations, we must decide on the "natural" congruence relation for it. The obvious choice is the eq relation. With the adoption of eq as the equality relation on S we assert that (S, A, ') is a Boolean algebra. For proof we note first that eq is a proper congruence relation for the system. Indeed, we already know that it is an equivalence relation and, using truth tables, it is an easy matter to prove that A eq B implies that (A A C) eq (B A C) and A' eq B'. Moreover, it is a straightforward exercise to verify that Bt-B4 of The- 274 Boolean Algebras I CHAP. 6 orem 3.1 are satisfied; that is, (A A B) eq (B A A), and so on. The zero element of the Boolean algebra (S, A, ') is A A A' for any formula A, and the unit element is (A A |
A')'. Frequently the result which we have obtained is stated as "The statement calculus under the connectives `and' and `not' is a Boolean algebra." This is somewhat misleading, since there is a statement calculus for each set So. Actually, it is only the cardinal number of So that matters; two calculi for which the respective sets of basic statements have the same cardinal number differ only in verbal foliage. Thus, a more accurate assertion, in the sense that it recognizes the existence of different statement calculi and the congruence relation employed, is "A statement calculus under the connectives `and' and `not' is a Boolean algebra with respect to equivalence." The Boolean algebra obtained from a statement calculus by the identification of equivalent formulas will be called the Lindenbaum algebra of that statement calculus. Such algebras are discussed in the next section. 7. Free Boolean Algebras t The preceding section provides the genesis of a method for constructing, in a purely formal way, a Boolean algebra from any nonempty set. This involves the use of congruence relations in a way which extends that described in Section 4. Let us dispose of this matter first. In Section 4 the rough assertion was made that if (X,. ) is a mathematical system and p is a congruence relation for it, then, corresponding to each operation (or relation) in X, there can be defined in X/p an operation (or relation) having all the properties of the original. (This was stated precisely and proved in the case of a Boolean algebra.) Now it can happen that the resulting system with X/p as basic set has additional properties besides those inherited from the original system. Intuitively, this seems quite plausible; if X is collapsed appropriately, irregular behavior present in the original system may be smoothed out in the derived one. An instance of this occurs below; a system which has some requisites of a Boolean algebra is forced into determining one by introducing a suitable congruence relation. The system with which we begin is the abstraction of the most obvious features of an intuitive statement calculus. We proceed with its f In the remainder of this chapter there are several forward references to Section 9.2. A mere perusal of that section will suffice for an understanding of the applications to be made to Boolean algebras. 6.7 I Free Boolean Algebras 275 definition. Let So be an arbitrary nonempty set and A and'be two symbols which do not designate |
elements of So. We give an inductive definition of a set S whose elements are certain finite sequences of eleinents of So U I A, '} together with parentheses. (1) If s C So, then s C S. (II) If I C S, then (t)' C S. (III) If s, t C S, then (s) A (t) C S. (IV) The only members of S are those resulting from a finite number of applications of (1), (II), and (III). As a direct consequence of the definition of S we may regard A as a binary operation in S and'as a unary operation in S. In these formal circumstances the natural congruence relation for the system (S, A, ') is that of elements having identical form. As such, (S, A, ') is surely not a Boolean algebra. Can a congruence relation be defined for the system such that a Boolean algebra will result? On the basis of the discussion in Section 4, necessary and sufficient conditions which such a relation 0 must satisfy are that it be an equivalence relation different from the universal relation on S (the latter requirement reflects the fact that a Boolean algebra has more than one element) and that the following hold for all elements of S. Ifs0t,then sAu01Auforallu.f Ifs 01, then s' 01'. (C) sA101As. sA (t Au)0(sAt) Au. IfsA1'0uAu'forsome u,then sA10S. IfsA10s,then sA1'OuAu'forallu. In defense of our assertion we note that the first two parts of (C) are necessary and sufficient conditions that the operations in S induce operations in S/0 in a natural way, and the remaining four parts constitute a minimal set of conditions which insure that the resulting system is a Boolean algebra. Parenthetically, we remark that at times, when an equivalence relation satisfying (C) is introduced into (S, A, '), it is more natural to continue with the elements of S (instead of those of S/0) as the basic objects. This attitude is reflected in referring to the system (S, A, ') as a Boolean algebra with respect to 0. There is no question concerning the existence of equivalence relat Here we begin to follow |
the usual mathematical conventions of omitting superfluous parentheses. 276 Boolean Algebra.s I CH A P. 6 tions satisfying (C), since if members of S are interpreted as truth functions, then, as observed in the preceding section, the eq relation satisfies (C). We consider now the set e of all equivalence relations satisfying (C) and let µ denote the intersection of the collection e. It is left as an exercise to prove that µ C C and, consequently, is the smallest member of ('0, in the sense that it relates the fewest possible pairs of elements of S. The Boolean algebra S/µ is called the free Boolean algebra generated by So. In this context the word "free" is intended to suggest that the elements of the algebra are as unrestricted as is possible if they are to have the structure of a Boolean algebra. Intuitively this is clear, since the only relations which have been imposed upon them are a necessary and sufficient set to insure that they do have that structure. There are alternative definitions of a free Boolean algebra that are more exotic; our old-fashioned one has the merit that it simultaneously disposes of the existence of such algebra.-,. For an application in the next section, we note the relationship of the algebra S/0 determined by an arbitrary member 0 of e to S/µ. Since s µ I implies s 0 1, a 0-equivalence class is a union of ptequivalence-classes. Thus it is possible to define a mapping f on S/µ onto S/0 by { J [S]B. That is, the image of the µ-equivalence class determined by s is the 0-equivalence class determined by s. Clearly, f is a homomorphism onto S/0; for example, the calculation f([s]µ A [1]µ) = f([s A 1]µ) _ [s A 1]e = [s]e A [1]e shows that f preserves intersections. Since the zero element of S/µ is [u A u'],, for any u in S, the zero element of S/0 is [u A u']e. It is possible to give an interesting characterization of the congruence relation u. To this end we consider the µ-equivalence class `U = [ (u A u')'],, for some u in S. This class is independent of u |
since it includes all members of S having the same form. This follows from the fact that if 0 C 0; then (s A s')' 0 (u A u')' for all s in S, and hence (s A s')' µ (u A u')' for all s in S. Since the zero element of S/µ is [u A u'],,, `U is the unit element of S/µ. It is left as an exercise to prove that if s, I E S, then s µ 1 if (s A 1')' A (s' A 1)' C `U or, introducing s E-' 1 as an abbreviation for (s A t')' A (s' A t)', SJA 1 if SHIC`U. 6.7 I Free Boolean Algebras 277 This characterization of µ in terms of 't) is opaque until S is interpreted as the set of formulas of a statement calculus. Then it will be recognized that p is to be interpreted as the eq relation and that `U becomes the set of valid formulas of S. Finally, the characterization of u in terms of V is simply the formal version of Theorem 4.3.2 (namely, s eq t if 1= s H t). The same interpretation of S suggests, as an alternative approach to the definition of the free Boolean algebra generated by So, the introduction of the set co first, followed by the definition of p in terms of U. This is possible using some formulation of a statement calculus as an axiomatic theory. The starting point is the inductive definition of the set Sin terms of the elements of So U (A, '}, just as before. We now wish to obtain the set `U as that subset of S which, under the interpretation of S, constitutes the tautologies. This is possible using the results of Section 9.2. Introducing s -> I as an abbreviation for (s A t')', we define a subset V of S as follows. (I) Any member of S that has one of the following three forms is a member of V: (u --> (s s -, (t --->.s), (ss). ((u -' s) -, (u --, l)) (II) If s and t arc members of S such that both- s and s t are members of V, then t is a member of V. (III) An clement |
of S is a member of v iff it can be accounted for using (I) or (II). The desired conclusion, that `U = V, is then secured via the completeness theorem (Theorem 9.2.3) and its converse (Theorem 9.2.4). In terms of V, the relation µ may now be defined by soul if s - IC V. Although statement calculi served as our inspiration for introducing the concept of a free Boolean algebra, now that the latter concept has been firmly established, we may turn matters around and describe the Lindenbaum algebra of a statement calculus as simply the free Boolean algebra generated by the set of prime formulas of the calculus in question. EXERCISES 7.1. Show that the relation µ is a member of C. 7.2. Show that sµl iffsHl C U. 278 Boolean Algebras I CH A P. 6 7.3. Investigate the question of whether or not the algebra of all subsets of a set X is a free Boolean algebra. 8. Applications of the Theory of Boolean Algebras to Statement Calculi It is by way of the Lindenbaum algebra of a statement calculus that the techniques and results of the theory of Boolean algebras can be applied to the study of statement calculi. The applications include elegant characterizations of various concepts that arise in the study of statement calculi and simple proofs of important metatheorems, as we shall show in this and the next section. We begin by analyzing the theory of deducibility for statement calculi in terms of the theory of Boolean algebras. The first step is to obtain a characterization of the algebraic structure of a statement calculus when a set of formulas is singled out to serve as a set of assumptions. For this let us consider the formal analogue of a statement calculus as described in the preceding section; that is, let us consider the system (S, A, ') generated by the set So. In it we imitate the designation of a set of formulas of a statement calculus as a set of assumptions by selecting a subset r of S and adjoining to the set (C) of conditions given earlier one of the form a0(u Au')' for each element a of r. Here u is any member of S. (Notice that the interpretation of this condition is that a is "true.") Let (Cr) denote the resulting set of conditions and e, denote the set of |
all equivalence relations on S which satisfy (Cr). Further, let µr denote the intersection of er. Then '.cr C C and, indeed, is its least member. Each µr-equivalence class is the union of µ-equivalence classes. In particular, the µr-equivalence class Dr, let us call it, which includes `u, also includes r and, hence, each is-equivalence class of the form [a],, with a c r. Assuming that there are at least two µr-equivalence classes, the system S/µr is a Boolean algebra and Vr is its unit element. According to an observation made in the preceding section, S/µr is a quotient algebra of S/µ. Using the characterization given in Theorem 4.2 of the congruence relation which is determined by the associated homomorphism, we conclude that s A r t iff s + I C [u A u'],,,; that is, if (s + t)' C ` i.. In turn, this condition translates into sµrI if sue-I C'or, 6.8 1 Applications to Statement Calculi 279 which generalizes the earlier characterization of µ as s µ tiffs.-* I Cl). Before continuing we note that U,. has the following closure properties. (i) Ifs,IC`U,.,then sAICU,.. (ii) If s E `U, and I C S, then s V I E 0r To prove (i) observe that ifs, t C 'U,., then s p,. (u A u')' and I m,. (u A u')', so s A t µ,. (u A u')'. To prove (ii) let s C. 'U,. and I C S be given. Then, in turn, s µr (t A i')', s' At, t A t', s' A I'm,, I A I', and (s' A I')' fur (t A t')', which means that s V I C `or. We continue with our generalization of the results of Section 7 by showing that it is possible to reach 'U,, independently of µ,, and then define µr in terms of '0r. To accomplish this we define the subset Vr of S by modifying part (I) of the earlier definition of V |
to include I' in Vr. Then it is clear that V,, may be characterized as the smallest subset of S that includes V and r and contains the clement t whenever it contains s and s - t for some s. On the other hand, v,., as we have noticed, includes V (_ 'o) and F. Further, if s and s -, I (that is, s' V t) are in v,., then so is t V (s A (s' V t)), according to the closure properties which we derived for 'U,.. A calculation shows that l V (s A (s' V t)) µ,. 1, so we may conclude that if s, s --. l C '0,., then t C `o,.. Finally, in view of the minimality of µr (in terms of which 'U,. was defined), we conclude that U,, has exactly the same characterization as does V,.. Thereby we infer that U,. = V. It follows that µ,, may be defined (or, characterized, at one's preference) as sJ.crt ilf V. Now let us interpret the foregoing from the standpoint of the statement calculus. If we regard (S, A, ') as a statement calculus, then the role of r is that of a set of premises. Under this circumstance, the free Boolean algebra S/µ (the Lindenbaum algebra of the calculus) is supplanted by the quotient algebra S/µ,, and the set V of provable formulas is enlarged to V,., the set of all formulas which are deducible from F. The set Vr, which is the unit element of S/µ,., may be described as the smallest set which includes V and r and is closed under modus ponens. The above characterization of µ,. in terms of V,. amounts to this: Two formulas are in the same member of S/µ,. if each is deducible from the other relative to 1' as a set of assumptions. Finally, we note that a necessary and sufficient conditions that Sly,, be a Boolean algebra (an assumption which we have made) is that I' be a consistent set of formulas. Further insight into the notion of provability and the nature of so- 280 Boolean Algebras! CHAP. 6 called deductive systems at the statement calculus level can be obtained by reversing our point of view. For this our starting |
point is the consideration of a Boolean algebra (B, f1, ') whose elements are to be thought of, intuitively, as the statements of some theory. Further, assume that P is a specified noncmpty subset of B whose elements are to be regarded as the provable statements of that theory. With this interpretation of P in mind, it is reasonable to make the following assump. tions about P. If s and t arc members of P, then so is s n t (that is, "s and t") and, if s is in P, then so is s U t (that is, "s or t") for any choice of t. Nonempty subsets of a Boolean algebra which satisfy these conditions are called filters. That is, a nonempty subset F of a Boolean algebra B is called a filter if (i) x C F and y C F imply x f1 y C F, and (ii) x C F and y C Bimply xUyCF. Before considering the set P as a filter we discuss a few properties of filters. Since the defining conditions of a filter arc the duals of those for an ideal of a Boolean algebra, the term dual ideal is often used in place of filter. Filters and ideals occur in dual pairs. The pairing is easy to describe: if I is an ideal of B, then Ix C Bjx' E I} is a filter; if F is a filter, then (x C Bjx' C F} is an ideal, as is easily proved. 'T'his pairing provides a bridge for transferring observations about ideals to filters. For example, both B and (1) are filters of B. Again, if a C B, then {x E Bjx > a} is a filter; this is the principal filter generated by a. A filter of 13 which is different from 13 is called a proper filter. A proper filter may be characterized as a filter which does not contain 0. A maximal member (with respect to inclusion) of the set of proper filters of B is called a maximal filter. For example, in the Boolean algebra of all subsets of a nonempty set A, the collection of all those subsets of A that contain a fixed element of A is a maximal filter. The dual of the earlier proof, that if M is a maximal ideal of a Boolean algebra B and x C 13, then exactly one of x and x' is in M, yields the |
same conclusion about maximal filters. Proofs of the foregoing assertions are left as exercises. Finally, it is left as an exercise to prove that a filter F of B may be characterized as a subset of 13 such that 1 C F and, if x, x' U y E F, then y C F. Introducing x -4y as an abbreviation for x' U y, the latter condition may be rewritten as: if x, x --+ y E F, then y E F. We return to the discussion that we began by defining a Boolean logic 6.8 1 Applications to Statement Calculi 281 to be an ordered pair (Z, P), where 8 = (B, n, ') is a Boolean algebra and P is a filter of the algebra. The elements of B will be called statements and those of the filter P will be called provable statements. We shall abbreviate "s is in P" by "l- s." As the first logical concept that we shall introduce into a Boolean logic, we choose that of consistency. A Boolean logic (58, P) is called consistent if for no s in B both s and s' belong to P. Since P is a filter, (58, P) is consistent if P is a proper filter. Next, let us call (S$, P) negation complete if for every s in B, either s or s' is provable. We contend that (58, P) is negation complete and consistent iff p is a maximal filter. For the proof assume first that the logic is consistent and negation complete. Consistency implies that P is a proper filter and hence has a chance to be maximal. To show that it is maximal, suppose that Q is a filter which properly includes P, and let s be a member of Q that is not in P. Negation completeness implies that /C P and hence that s' C Q. But s and s' in Q imply that 0 = s n s' E Q, which means that Q = B. The converse is an immediate consequence of an earlier remark that for each element x of a Boolean algebra exactly one of x and x' belongs to a maximal filter. We state our result as the next theorem. THEOREM 8.1. A Boolean logic (53, P) is consistent and negation complete if P is a maximal filter of Z. The next logical notion that we discuss for a Boolean logic ($i, P) is that of |
deducibility. If r is a subset of B, then we shall say that a state-, u ment s of B is deducible from I' if there exists a finite sequence ul, u2, of statements of B such that u,, is s and if for each i, I < i < n, either u, is in I' or P or there exist j < i and k < i such that Uk is u, --' u:. Since P is a filter we know that 1 C P and y C P whenever x, x --+y C P. It follows that P satisfies the axioms of a statement calculus [that is, conditions (I) and (II) for V in the preceding section] and, hence, the deduction theorem (Theorem 9.2.1) in the form proved for the statement calculus is available. In the present context we may state it in the following form: If I' C B, then s is deducible from r ii' there exists a finite n rk - s. We shall subset { r,, r2j denote the set of statements deducible from I' by r; of course, r depends on both r and the choice of P. THEOREM 8.2. The set r of statements deducible from r is the, rk) of r such that i-- r, n r2 n smallest filter that includes both I' and P. 282 Boolean Algebras I CHAP P. 6. n rk - s, by the deduction theorem. Hence r, n r2 n Proof. Clearly this is true if r is the empty set, since then P = P. Suppose that r is not empty and let Q be any filter that includes r and P., rk of r such that I- r, n r2 If s C F, then there exist elements r,, r2, n n. rk C Q and rin r2 n n rk -+ s C Q, so s C Q. Thus we have proved that every filter which includes r and P also includes P. It remains to prove that P is a filter which includes both r and P. This is left as an exercise. We shall call a subset A of B a deductive system if if includes A. By the previous theorem, A C 0, so A is a deductive system if A = 0 and this implies that A is a filter including P. |
Conversely, if A is a filter that includes P, then A = 0, by the same theorem. Thus, the notion of deductive systems coincides with that of filters that include P. EXERCISES 8.1. Show that the relation µ,, is a member of Cr. 8.2. Show in detail that sµrt ifP s H I C u r. 8.3. Write an expanded version (supplying all proofs) of the paragraph in which V,. is defined and the result that tir = Vr, is obtained. 8.4. Prove the assertion in the text that two formulas are in the same member of S/µr if each is deducible from the other relative to r as a set of assumptions. 8.5. Prove the assertion in the text that S/µi is a Boolean algebra iT r is a consistent set of formulas. 8.6. Show that a proper filter may be characterized as a filter which does not contain 0. 8.7. Show that a filter F of B may be characterized as a subset of B such that1 CFandx,x'UyCFimply that yCF. 8.8. Show that a maximal filter can be characterized as a filter such that for each x exactly one of x and x' is in it. 8.9. Rewrite the proof of Theorem 5.3, using filters in place of ideals. 8.10. Complete the proof of Theorem 8.2. 9. Further Interconnections between Boolean Algebras and Statement Calculi The two-element set IT, F} determines a Boolean algebra having T as unit element and F as zero element. By a two-valued homomorphism of a Boolean algebra B we shall mean any homomorphism of B onto a two-element Boolean algebra. Since all two-element Boolean algebras are isomorphic, we may always use IT, F} in considering a two-valued 6.9 I Further Interconnections with Statement Calculi 283 homomorphism of B and, thereby, regard such a homomorphism as providing a "truth-valuation" of the elements of B. There is a natural one-to-one correspondence between the set of maximal ideals and that of maximal filters and between each of these and the set of two-valued homomorphisms of B. In fact, if I is a maximal ideal of B, then the dual of I (that |
is, the set of all a' where a E I) is a maximal filter and the formula (1) v(b) _ {T if b (Z I defines a two-valued homomorphism of B. Similarly, if F is a maximal filter of B, then the dual of F (that is, the set I of all a' such that a E F) is a maximal ideal and (1) defines a two-valued homomorphism corresponding to F. On the other hand, if v is a two-valued homomorphism of B, then the set I = {b C Blv(b) = F} is a maximal ideal and the set F = {b C Blv(b) = Ti is a maximal filter dual to I. By virtue of these natural correspondences, the following assertions are equivalent to each other. (2) For every proper ideal I there exists a maximal ideal which includes I. (3) For every proper filter F there exists a maximal filter which includes F. (4) For every proper ideal I [proper filter F] there exists a two-valued homomorphism v such that v(b) = F for b C I [v(b) = T for b C F]. Now (2) is simply our Theorem 5.2, so the validity of (3) and (4) then follow. As an application of the foregoing we analyze the nature of truth-value assignments to the formulas of a statement calculus. If So is the set of prime formulas of the statement calculus (S, A, '), then an assignment of truth values to the elements of S amounts to the extension of a given mapping on So into IT, F} to one on S onto IT, F}, in accordance with the inductive definition given in Section 4.3. Thereby it is insured that equivalent formulas are assigned the same value. Hence, the extended 284 Boolean Algebras I CH A F. 6 mapping may be construed as a mapping v on the Lindenbaum algebra (S/µ, A,') onto IT, F), and the definition of v implies that it is a twovalued homomorphism of the. Lindenbaum algebra. The kernel of v is the maximal ideal which is related to v in the natural correspondence mentioned above. On the other hand, any two-valued homomorphism v of (S/µ, A, ') (regarded as simply a free algebra |
) yields a truthvaluation of the elements of S/µ and Hence of the elements of S upon assignment of T or F to a formula according as the u-equivalence class to which it belongs is assigned T or F. It is easily shown that this is a truth-value assignment in the sense of Section 4.3. In summary, truthvalue assignments to the formulas of a statement calculus coincide with two-valued homomorphisms of the Lindenbaum algebra of the calculus. Furthermore, the existence of truth value assignments to a statement calculus is insured by the existence of maximal ideals in a Boolean algebra, and conversely. The existence of maximal ideals that include a preassigned proper ideal of a Boolean algebra also insures the existence of an isomorphic image of the algebra in the form of an algebra of sets. Indeed, the existence of such maximal ideals is the basis for the proof of Stone's representation theorem! Conversely, from the assumption that (5) For every Boolean algebra there is an isomorphic algebra of sets. may be inferred the existence of maximal ideals in Boolean algebras. This result, which is also due to Stone (1936), follows immediately from the existence of maximal ideals in an algebra of sets. To prove this, in turn, let us consider an algebra a of sets based on 11. Let V be any subset of U and let a(V) be the collection of all elements of a which are included in V. Then it is possible to prove that a(V) is an ideal of a and that a(V) is a maximal ideal of a if U - V has exactly one member. Since the proof makes an interesting exercise, we shall allow the reader to carry this out. The completeness theorem (Theorem 9.2.3) for the statement calculus can also be obtained from the theorem on the existence of maximal ideals, and hence filters, in a Boolean algebra. To show this let us consider a statement calculus e _ (S, A, ') and its Lindenbaum algebra 21 = (S/,u, A, '). In Section 9.2 we prove that the completeness theorem for e is cc,u<valent to the property that for every formula s, if (si is is consistent iff not consistent, then s is satisfiable. In turn, since {sl i- s', the consistency of (.s( means simply that s is not a member of the zero element of I. Further, in view |
of our above analysis of truth-value 6.9 I Further Interconnections with Statement Calculi 285 assignments to elements of a statement calculus, the satisfiability of s corresponds to the existence of a two-valued homomorphism v of?l such that v([sj,,) = T. Thus, the completeness theorem may be translated into the following form: For any nonzero element a of 21 there exists a two-valued homomorphism v of?1 such that v(a) = T. An equivalent statement, which results upon considering the principal filter generated by a and then the equivalence of propositions (3) and (4), is: Each nonzero element of 21 is a member of a maximal filter of W. It is this proposition which we shall take as the Boolean translation of the completeness theorem. Then the completeness theorem follows immediately from the theorem on the existence of maximal filters. We note that this derivation of the completeness theorem does not involve any restriction on the cardinality of the set of primitive symbols of the statement calculus. In particular, therefore, the set of primitive symbols may be assumed to be uncountable. Conversely, the existence of maximal filters can be deduced directly from the completeness theorem formulated in a stronger form. To be precise, we can prove the equivalence of the existence of maximal filters and the strong completeness theorem for the statement calculus (with no restrictions on the cardinality of the set of primitive symbols). For this we use the fact (see Section 9.2) that the strong completeness theorem for t is equivalent to the proposition that (6) Every consistent set of formulas is simultaneously satisfiable. Now assume that r is a consistent set of formulas of CS and let V, denote the set of all formulas which are deducible from r. Then V,./µ is a is a filter we proper filter of $l, as we shall show. To prove that use the characterization of a filter given earlier as a subset I' of a Boolean algebra B such that (i) 1 C F and (ii) if a and a -' b are in F, then so is b. In the case at hand, (i) is satisfied because the set of theorems of S is included in Vr, and (ii) is satisfied because V, is closed under modus is a proper filter. ponens. Finally, the consistency of IF implies that Next, analyzing the satisfiability of I' as we did |
above for the case of a single formula, we infer that as the Boolean translation of the strong completeness theorem we may take the statement (7) Every proper filter of the Lindenbaum algebra of a statement calculus is included in a maximal filter. Since (7) is a special case of (3), to prove the equivalence of (7) and (3) it must only be shown that (7) implies (3). For this let B be a 286 Boolean Algebras I CHAP. 6 Boolean algebra and I be some proper ideal of B. We now form the statement calculus (s, A, ') generated by a set So whose members p= are in one-to-one correspondence with the elements x of B. Now consider the mapping f on S onto B given by the following inductive definition: AM =x, As') = U(s)) ', for all s in S, f (s. A t) = f (s) A f (t), for all s and tin S. It is seen immediately that if t is a theorem of S, then f(t) = 1 and for t) = I iff f(s) = f(t). These facts imply (recalling all s and tin S, f(s Section 7) that if sit 1, then f (s) = f (l). Hence, f induces a mapping g on S/µ, the Lindenbaum algebra of S, onto B. Clearly, g is a homomorphism onto B, so B is isomorphic to a quotient algebra (S/ic)/J. Now let K denote the counterimage in S/µ of the given proper ideal I of B. Then K 2 J since I includes 10). From our assumption (7) follows the existence of a maximal ideal M that includes K, and consequently J. Now M, as an ideal, is a Boolean algebra and J is an ideal of this algebra. It is left as an exercise to prove that M/J is a maximal ideal of (S/µ)/J. But then the isomorphic image of M/J in B is a maximal ideal of B that includes I. This shows that (2), and hence its equivalent (3), holds. From the results which have been obtained it is clear that the statements (2)-(5) about Boolean algebras are equivalent to each other. Moreover, the equivalence of each pair has been established without |
recourse to the axiom of choice. On the other hand, all known proofs of (2), for example, are based upon the axiom of choice or an equivalent principle of set theory. A problem arises as to whether (2) is really. dependent on the axiom of choice. This problem has been responsible for the derivation (without use of the axiom of choice) of a great variety of statements about Boolean algebras which are equivalent to (2) and, also, the investigation of specialized forms of the axiom of choice which are consequences of (2). The most comprehensive treatment of these' matters to date is due to J. Loi and C. Ryll-Nardzewski (1954-1955). The strongest result which they found is that (2) implies the axiom of. choice for the case of a collection of nonempty finite sets. The question as to whether the axiom of choice is independent of (2) is as yet unsolved; the evidence suggests that the answer is in the affirmativej f (Added in proof.) It has just come to my attention that a further contribution to thit matter appears in J. D. Halpern (1961). There it is asserted that in certain models of sdi theories (2) is true but the axiom of choice is not. Bibliographical Notes 287 We conclude by remarking that the demonstration of the strong completeness theorem [in the form (6) ] for the statement calculus is not the end of the applications of Boolean methods to mathematical logic. Many fundamental theorems about the predicate calculus and about first-order theories can be easily proved by applying Boolean methods to appropriate "Lindenbaum algebras" associated with such theories. An outline of such applications appears in R. Sikorski (1960). EXERCISES 9.1. Show that if v is a two-valued homomorphism of the Lindenbaum algebra of a statement calculus, then it provides truth value assignments to the elements of the statement calculus in the sense of Section 4.3. 9.2. Complete the proof of the result that (5) implies (2). 9.3. Show in detail that we may take (7) as the Boolean translation of the strong completeness theorem for a statement calculus. 9.4. Fill in the details of the proof in the text that (7) implies (3). BIBLIOGRAPHICAL NOTES Sections 1-3. An introductory |
account of Boolean algebras appears in E. R. Stabler (1953). A more sophisticated treatment is to be found in P. C. Rosenbloorrr (1950). A high-level, modern treatment of the theory, which treats Boolean algebras primarily from the standpoint of a generalization of algebras of sets, has been given by R. Sikorski (1960). Another high-level account, which places more emphasis on the algebraic structure of the theory, appears in G. Birkhoff (1948). There exists a great variety of formulations of the theory of Boolean algebras. The book by Sikorski lists references to many of these. The axioms introduced in Section 3 are due to L. Byrne (1946). The same set is used by Rosenbloom in his book. Section 4. A discussion of congruence relations for Boolean algebras appears in Rosenbloorrr (1950). Congruence relations for algebraic systems in general are discussed in Section 8.1 of this book. For a more comprehensive treatment of ideals, homomorphism, and so on, Sikorski (1960) should be consulted. Section 5. An exhaustive treatment of representations of Boolean algebras by algebras of sets along with related topics is given in Stone (1936, 1937, 1938). The fundamental representation theorem and the theorem on the existence of maximal ideals have been the subjects of many papers. References to such papers and a concise presentation of Stone's work appear in Sikorski (1960). Sections 6-7. An expository account of the subject material of these sections appears in P. R. Halmos (1956). This paper also gives an introduction.to polyadic algebras, which stand in the same relationship to the pure predicate 288 Boolean Algebras I CHAP. 6 calculus of first order as do Boolean algebras to the statement calculus. Rosenbloom (1950) also discusses some of these topics. Section 9. The application of Boolean methods to mathematical logic was the subject of many papers in the early 1950's. Many of these papers were published in Fundamenta Mathemalicae. Exact references are given in Sikorski's book. CHAPTER / Informal Axiomatic Set theory THE ANTINOMIES OF INTUITIVE set theory pose the problem of providing a theory of sets which is free of contradictions. The analysis of |
the well-known antinomies (Section 2.11) for the purpose of determining possible fallacies in methods of constructing and reasoning about sets-methods which had seemed convincing before they were found to generate contradictions-has led to several reconstruction of set theory along axiomatic lines. This chapter is devoted to outlining that one known as Zermelo-Fraenkel set theory [although it would be more appropriate to call it Zermelo-Fraenkel-Skolem set theory, since it is the theory of E. Zermelo (1908) as modified by both A. Fraenkel and T. Skolcm]. In the last section fleeting contact is made with the other axiomatization of set theory with which mathematicians feel comfortable-the von Neumann-Bernays-Godel theory. Since that part of Zermclo-Fraenkel set theory which reconstructs the theory of Chapter 1 and Chapter 2, up to cardinal numbers, closely parallels the earlier intuitive development, we shall, so to speak, merely provide the axiomatic underpinnings for it. Then, for Cantor's theory of transfinite arithmetic, we substitute the theory of ordinal and cardinal numbers due to von Neumann. 1. The Axioms of Extension and Set Formation The recipe in Section 5.2 for presenting an informal theory cannot be used here since it calls for a "general theory of sets" as an ingredient. An obvious alternative, which we shall adopt, is to presuppose only a system of logic. As the primitive notions of Zermelo-Frankel set theory, which we shall symbolize by e, we take set and (the 2-place predicate) membership. We shall denote the relation of membership by "C" and, at the outset, denote sets by lower-case letters. Before describing the prime 289 290 Informal Axiomatic Set Theory I CHAP. 7 formulas of CS a decision must be reached as to whether the relation of equality shall be taken as part of the underlying logic or introduced as a defined relation of the theory; either is possible. In Example 4.7.1 the latter point of view is adopted. Here we elect the former viewpoint. This is in keeping with the procedure in Zermelo (1908). With the equality relation included in the underlying logic it is possible, in an interpretation of the theory, to admit nonsets (that is, objects which, like the empty set, have no members but are distinct |
from the empty set) in the domain of the relation assigned to E. [Such objects are commonly called individuals; Suppes (1960) and Fraenkel-BarHillel (1958) discuss this matter. ] Although we intend that in the theory which we shall formulate all variables shall denote sets, initially we shall suggest a possible distinction between sets and objects which may be members of sets by using "a," "b," to denote the former and "x," "y," to denote the latter. With equality included as part of the system of logic, the prime formulas of G have the form (1) or the form xCa a=b. (2) The first of these we shall read as "x is an element of a" or "x is contained in a." For a precise definition of a (composite) formula of Cs, we now refer the reader to the beginning of Section 4.7. However, in order to avoid completely any illusion that we are setting up a formal theory, the only symbolism that we shall employ in writing formulas is of the sort displayed in (1) and (2), along with xQa and a0b for "not (x E a)" and "not (a = b)," respectively. Thus, we shall not use the symbolism of the predicate calculus but, instead, the (meaningful) English equivalents of connectives and quantifiers. In harmony with this agreement, we shall use the word "sentence" in place of the word "formula." In particular, a formula (in the technical sense) which contains a free occurrence of x will be called a "condition on x" or a "property of x" and symbolized A(x). A statement (in the technical sense) we take to be true or false, since we assume that each prime formula is either true or false. 7.1 I The Axioms of Extension and Set Formation 291 This completes our description of the ground rules. We proceed with our first two axioms. (ZF1) (Axiom of extension). If a and b are sets and if, for all x, x E a ifl' x E b, then a = b. (ZF2) (Axiom schema of subsets). For any set a there exists a set b such that, for all x, x E b if x E a and A(x). Here, A(x) is any condition on x which (cons |
idered as a formula in the technical sense) contains no free occurrence of b. In contrast to (ZF1), which is a statement, (ZF2) is an infinite collection of statements. That is, it is a scheme for producing axioms, one for each choice of A(x). This accounts for (ZF2) being called an axiom schema. As in intuitive set theory, to indicate the way b is obtained from a and A(x) we shall write b = (x E aIA(x) }. It is an immediate consequence of (ZF1) that the axiom schema of subsets determines b uniquely. The usage of the term "subset" here anticipates the introduction of a C b (read : a is a subset of b, or, a is included in b) as an abbreviation for "all x, if x C a then x C b " At this point we might derive familiar properties of the inclusion relation and continue with the definition of proper inclusion and properties of this relation. Both here and subsequently, when we have carried a notion or topic belonging to the general set theory of Chapters 1-3 to a point where the earlier definitions and proofs are applicable, we shall drop the matter. Our emphasis will be directed principally toward notions and procedures of intuitive set theory which apparently cannot be carried out within the axiomatic framework. Our first illustration of the last remark can be given now. It is clear that (ZF2) is a substitute for the intuitive' principle of abstraction (Section 1.2) and that (ZF2) is more restrictive in this respect. Whereas the earlier principle provides a set for each condition or property, the present version only provides the existence of a set corresponding to a condition and which is a subset of an existing set. With this restrictive feature, Russell's paradox cannot be reconstructed, so far as we know. What can be produced by imitating the earlier argument is the following. According to (ZF2), with A(x) as x (4 x, for any set a, if b = (xEajx(Z x), then, for ally, 292 Informal Axiomatic Set 77teory I CRAP. 7 (3) yCbiffy Caand y(Zy. It follows that b V a. The proof is by contradiction. Assume that b C a. Now either b C b or b V b. If b C b, then |
in view of our assumption and (3), we have b (Z b and hence a contradiction. If b V b, then this and our assumption yield, in view of (3), b C b, a contradiction. The assumption that b C a having led to a contradiction, we may conclude that b (Z a. Since the set a was unspecified in reaching this result, we infer that there is no set that contains every set. In Halmos (1960) this is paraphrased as "nothing contains everything." The axiom schema of subsets is often referred to by its German name Axiom der Aussonderung (axiom of "singling out" or "separation"). This name is suggestive since it does permit us to single out or separate off those elements of a given set which satisfy some condition and form the set consisting of just those elements. Incidentally, this axiom schema may be considered as characterizing Zermelo's attitude with regard to a reconstruction of set theory which avoids the classical antinomies. His analysis of these contradictions led him to conclude that they resulted from the admission into intuitive set theory of "too large" sets. This led him to limit severely, by means of axioms, allowable methods of forming sets from existing sets and, in addition, to modify the principle that every condition determines a set. 2. The Axiom of Pairing The goal of anyone who aspires to axiomatize set theory has already been mentioned: To create a consistent theory within which as much as possible of the general set theory of Chapters 1-3 can be developed and, if a proof of consistency is not within reach, to incorporate adequate safeguards to insure that the classical antinomies cannot be derived. Axiom schema (ZF2) has both a constructive as well as a restrictive quality, the latter evidencing itself in its conditional nature. In order to imitate the intuitive set theory of Chapter 1, further means of constructing sets from existing sets must be introduced. The next three axioms are in this category. In this section we introduce one of them. (ZF3) (Axiom of pairing). a set c such that a C c and bCc. If a and b are sets, then there exists 7.2! The Axiom of Pairing 293 Using the instance of (ZF2) obtained by taking A(x) to be "x = a or x = b" and c to be a set such that a C |
c and b C c, we infer the existence of the set {xCclx=aorx=b}. Clearly this set contains just a and b and (ZF1) implies there is only one such set. We shall denote it by the symbol {a, b} and call it the (unordered) pair formed by the sets a and b. As is easily shown, an equivalent formulation of (ZF3) is the statement that for sets a and b there exists a set c such that x C c iff x = a or x = b. If we take A(x) to be condition "x = a or x = b," the foregoing remark means that we may express (ZF3) as: There exists a set d such that x E d ifl'A(x). (1) Now (ZF2), applied to a set c, asserts the existence of a set d such that xCdiff(xEcandA(x)). (2) Comparing (1) with (2) may suggest that (1) is a special case of (2) and, in turn, that (ZF3) is superfluous. This reasoning is spurious; for it is only when the existence of a set which contains a and b is assured that (2) yields (1), and it is precisely (ZF3) which gives this assurance. With the notation of intuitive set theory in mind, it seems natural to denote the set d described in (1) by {xIA(x) } ; that is, to write {a, b} _ {xJx =aorx = b}. Henceforth we shall use this symbolism when it is convenient and permissible. That is, if A(x) is a condition on x such that those x's which A(x) specifies do constitute a set, then we may denote that set by {xIA(x) }. With this convention we may rewrite {x C aIA(x) }, where a is a set, as {xJx E a and A(x) }, but we shall not do so since the latter denotation is longer than the former. If a is a set, we may form the pair {a, a). This set we denote by {a} and call the unit set of a. As an illustration of the notation agreed upon in the preceding paragraph, we may write {a} = {xIx = a |
). 294 Informal Axiomatic Set Theory I CHAP P. 7 The specialization of (ZF3) which yields the unit set of a set insures that every set is an element of some set and (ZF3) in its general form insures that any two sets are elements of some one set. Thereby, given a set a, it becomes possible to manufacture a variety of sets such as (a (a, (a) }, ((a}, ((a} 11, and so on. 3. The Axioms of Union and Power Set None of the axioms up to this point assert the existence of any sets. It will prove to be expeditious to anticipate a later axiom which does this (and more), by introducing as a temporary axiom: there exists a set. Then we can establish the existence of a set without elements. Indeed, let a be a set and take A(x) to be "x 9-6 x." Then, according to (ZF2), there exists the set (x C ajx s x}. This (uniquely determined) set has no elements. We shall call it the empty set and adopt the familiar symbol 0 for it. We now turn to the first business of this section by observing that if c 0, then there exists a set a such that is a nonempty set, that is, if c x C a if x C y for every y which is a member of c. In other words, for each nonempty set there exists a set that contains exactly those elements that belong to every member (set) of the given set. To prove this assertion, let b be any member (set) of c and define a = (xCbiforally (ifyCc,thenxCy)}. The set a is independent of the element b since it is easily shown that a = (xj for ally (if y E c, then x E y) }. The set a is called the intersection of c. For a discussion of the notation used for intersections we refer the reader to Section 1.10. Here we shall only call attention to the notation allb, where a and b are sets, for the set defined by alb=Ix CaixEb}. 7.3 I The Axioms of Union and Power Set 295 Since x C a (1 b iff' x C a and x C b, it follows that an b = {xjx |
Caandxcb}. In contrast to the situation for intersection, we require a further axiom to be able to produce in CS the notion of the union of a set. The following is a generous form of the necessary axiom. (ZF4) (Axiom of union). For every set c there exists a set a such that if x C b for some member b of c, then x C a. If c is a set and a is a set of the kind specified in (ZF4), then we may apply (ZF2) to form the set {xCal for some y(xCyandyCc)}. Clearly, for all x, x is contained in this set, which we call the union of c, if x is an element of an element of c. We may then write the union of c as {xj for somey (xCy and y E: c)). The notation a U b, where a and b are sets, will be used for the union of the set {a, b}. By virtue of the definition of the union of a set, x C a U b if x is a member of a or x is a member of b. Thus aUb= {xjxCaorxCb}. For a discussion of the notation used for unions we again refer the reader to Section 1.10. With the aid of (ZF4) it is possible to generalize pairs. For instance, the (unordered) triple formed by sets a, b, and c, symbolized may be defined by {a, b, c}, (a, b, c) = ({a) U {b}) U {c}. Then it follows easily that (a, b, c) = {xjx = a or x = b or x = c}. The extension of the notation and terminology to the case of further terms is clear. It is now possible to introduce, for sets a and b the relative comple- ment of b in a as the set a - b, defined by a-6= {xCajxiZ b}, Informal Axiomatic Set Theory I C H A P. 7 296 and in turn, the symmetric difference of a and b as the set a + b, defined by a + b = (a-b)U(b-a). At this point it is possible to derive all the results listed in Chapter 1 concerning properties of union, intersection, relative complement, |
and symmetric difference, including their interrelations. To complete the reconstruction of the intuitive theory of Chapter 1 within C5, we need the theory of relations, for which the starting point is the notion of an ordered pair. Since the (unordered) pair formed by two sets as well as the unit set of a set can be constructed, the ordered pair of sets a and b (with first coordinate a and second coordinate b) can be introduced as the set (a, b), defined by (a, b) = ( (a}, {a, b} }, just as in Chapter I. The earlier proof carries over : if (a, b) and (c, d) are ordered pairs and if (a, b) _ (c, d), then a = c and b = d. However, the existence in 6 of what we called earlier the cartesian product of two sets requires a principle of set construction which the axioms at hand do not seem to permit. We can dispose of the matter at hand as well as the existence of the power set of a set with the aid of the following axiom. (ZF5) (Axiom of power set). For each set a there exists a set b such that, for all x, if x C a, then x C b. To secure the existence of the power set of a set from this axiom is an easy matter. If a is a set and b is a set which contains all of the subsets of a as members, then we apply (ZF2) to form the set {x C bIx g a}. For all x, x is a member of this set if x is a subset of a. We call this set the power set of a, symbolized P(a). Thus, 6'(a) = {xix Cal. To establish the existence of the cartesian product of sets a and b, we notice first that if x C a and y C b, then { x } C a, l y } C b, and hence the sets (x} and {x, y} are included in a U b. In turn, {x} and {x, y} are members of (P(a U b), which implies that { {x}, {x, y} } = (x, y) is a subset of 61(a U b). It follows that (x, y) C 6(G (a U b)). We infer |
that the set we want can be obtained by an application of (an instance of) (ZF2) to tP(6'(a U b))'. The appropriate condition is quite 7.4 1 The Axiom of Infinity 297 long; for the sake of both brevity and clarity we shall write it in symbolic form. The cartesian product of sets a and b is the set a X b defined by {wE6'(6'(aUb))I(3x)(3y)(x0yAxCaAvCbA(z)(zCwE-r z = {x} V z = {x,y}))V (3x)(x C a A x C b A (z)(zC w-'z = {x)))}. Since w C a X b iff w = (x, y) for some x in a and some y in b, a X b = { w l for some x in a and some y in b, w = (x, y) 1. Defining a (binary) relation as a set each of whose members is an ordered pair, it is of importance to know that we can prove that a relation is a subset of the cartesian product of two sets. In this connection we recall Exercise 1.10.1, where it is asserted that if r is a relation, then (using notation introduced in Section 1.10) r is a subset of the cartcsian product of UUr with itself. We may apply (ZF2) to this cartesian product, taking for the condition first "for some x ((x, y) C r)," and secondly "for some y ((x, y) E r)," to produce the sets {xI for some x ((x, y) C r) } and {yJ for some y ((x, y) E r) }, which we call the domain and the range, respectively, of r. In particular, the domain and the range of a relation are sets and a relation is a subset of the cartesian product of its domain and its range. At this point it is possible to complete the reconstruction of the set theory of Chapter 1, obtaining the theory of equivalence relations, functions, and partial ordering relations found there. 4. The Axiom of Infinity Let us consider for a moment the theory of sets based on just the axioms (ZFI)-(ZF5) plus the temporary axiom that a set |
exists. The presence of the axiom of pairing makes possible the formation of an arbitrary large number of distinct (two-element) sets. We infer that the domain of any model of the theory must be infinite. On the other hand, since the union of a finite collection of finite sets and the power set of a finite set are finite sets, it does not appear that the axioms are adequate to prove the existence of an infinite set. The correctness of this surmise may be demonstrated by way of a model devised by W. Ackermann (1937). 298 Informal Axiomatic Set Theory I CHAP P. 7 The domain of the interpretation which can be shown to be a model is N. In order to define the relation of membership, we shall need the fact that a positive natural number a has a unique representation in the form a = 2=' + 2- +... + 2zr, where the x's are natural numbers and xl < X2 <. < x,.. Then, for natural numbers (that is, sets) x and a, we define x C a as true if x appears as an exponent in the representation of a in the form exhibited above. Thus, each set has only a finite number of elements. It is left as an exercise for the reader to prove that this interpretation is indeed a model of the theory under discussion. Actually, this system is a model of the theory whose axioms are all such that they will eventually be assigned to S except the axiom of infinity which is introduced below. Thereby the system provides a proof of the independence of this axiom. Ackermann, however, devised it for a more profound purpose, namely, to provide the basis for a finitary consistency proof of the theory having (ZF1)-(ZF5) together with the axiom of choice (see Section 5) as axioms. There are compelling reasons for strengthening the set of axioms introduced thus far, to provide for the existence of an infinite set. Specifically, the existence of the set of natural numbers is essential for the theory of denumerable sets and for the theory of real numbers. Although we have not as yet given a precise definition of infinity, it seems plausible that sets of the kind which are postulated by the following axiom merit being called infinite on intuitive grounds. (ZF6) (Axiom of infinity). There exists a set a such that 0 C a and, ifxEa, thenxU |
{x} Ca. Zermelo was the first to recognize the necessity of such an axiom; earlier workers regarded the existence of infinite sets as evident. He - -, which is a constructed the natural numbers as 0, 10 }, {{0 } ), satisfactory approach but one that does not generalize to the construction of infinite ordinals as easily as that adopted below. For every set x we define the successor x+ of x by x+ = x U {x}. Further, we shall say that a set,a is a successor set if 0 C a and if x+ E a whenever x C a. In this terminology, (ZF6) says that there exists a successor set. We shall now prove the existence of a unique minimal 7.4 I The Axiom of Infinity 299 successor set. It is left as an exercise to prove that the intersection of a nonempty collection of successor sets is again a successor set. So, if a is some successor set, then the intersection of the (nonempty) collection of successor sets which are included in a is a successor set which we denote by co (with the notation introduced in Section 2.6 in mind). The set w is a subset of every successor set. To prove this, consider an arbitrary successor set b. Then a n b is a successor set which is included in a. It follows that w g a (1 b, and hence w C b. In turn, the minimality of co characterizes it uniquely. For if w' is a successor set which is included in every successor set, then we have w g w' and co' C co. Then (ZFI) implies that co = co'. We now define a natural number to be an element of the minimal successor set co. Further, we define 0, 1, 2,, 9 by writing 0=0, I = 0+(= {0}), 2=1+(={0,1}), 9 = 8+(= 10, 1, 2, 3, 4, 5, 6, 7, 8}). For other natural numbers we employ the usual decimal notation. We continue by proving that (w, +, 0), where now we regard + as a function on co into w, is an integral system or, what amounts to the same, that this system satisfies Peano's axioms P1-P5 in Example 2.1.2. Since w is a successor set, 0(=Q) C co [that |
is, Pl is satisfied] and, if n C w, then n+ C w [that is, P2 is satisfied]. Moreover, n+ s 0 for all n in w, since n C n+ and n i[ 0 [that is, P4 is satisfied]. The minimality property of w can be expressed as: If a subset a of w is a successor set, then a = w. But this means that P5 is satisfied. It remains to prove that P4 (if m+ = n4", then m = n) is satisfied. This requires two preliminary results which we state as lemmas. LEMMA 4.1. No natural number is a subset of any of its elements. Proof. Let a be the set of those natural numbers that are not included in any of their elements. Thus, n C a iff n C w and, if x C n, then n!9 x. Clearly, 0 C a, since 0 has no elements. We assume next that n C a and consider n+. Since n+ = n U {n}, the elements of n+ are n n for, since n C n (and, n C a), and the elements of n. Now n'n V n. Moreover, n+ is not included in any member of n, since if n+ S x, then n C x (because n C n+), which implies (since n C a) 300 Informal Axiomatic Set Theory! C r r A P. 7 that x (Z n. Therefore, n+ is not a subset of any of its elements and consequently n+ C a. By the principle of induction (P5), it follows that a = w, and this completes the proof. In order to state the next lemma it is convenient to make a definition. A set a is called complete if each member of a is a subset of a. Expressed otherwise, a is complete if y C x and x C a imply that y C a. LEMMA 4.2. Every natural number is a complete set. A proof by induction can be supplied by the reader. We now prove that if m and n are natural numbers such that m = n+, then m = n. For this we assume that m+ = n+" and m P6 n, and derive a contradiction. From m+ = n+ it follows that m C. n+, and hence either m = n or m C n. |
Similarly, either n = m or n C m. Assuming, as we are, that m 0 n, we infer that both m C n and n C m hold. Hence, by Lemma 4.2, n C n. Combining this with the fact that n C n, we conclude that n is a subset of one of its members, which contradicts Lemma 4.1. With the proof now completed that w satisfies the Peano axiorns, the stage is set for a development of the arithmetic of w. If, as in Chapter 2, the definition of a relation that well-orders co is taken as the first order of business, then there is the following alternative to the procedure followed in Chapter 2. The first step is to prove (an exercise for the reader) the following result. LEMMA 4.3. For each pair in, n of natural numbers, either m C n or m = n, or n C m. Using Lemmas 4.1 and 4.2, it is then an easy matter to show that exactly one of these three alternatives holds. A further consequence of Lemma 4.3, in conjunction with Lemma 4.2, is stated next; the proof is left to another exercise. LEMMA 4.4. ifmCn. If m and n are distinct natural numbers, then m C n We now define m to be less than n, symbolized m < n, if m C n or, equivalently, in C n. Defining m < n in the usual way, one may then go on to show that < well-orders CO. 7.4 I The Axiom of Infinity 301 Next in order is the introduction of Theorem 2.1.2, so inductive definitions of addition and multiplication can be given. Turning to other definitions and results in Chapter 2 which pertain to natural numbers, we recall that there are several in Section 2.3 phrased in the language of cardinal numbers. All such can be handled easily in the present development in terms of the notion of the similarity of two sets (that is, the existence of a one-to-one correspondence between them) and the properties of natural numbers sketched so far. Preparatory to what we have in mind we state the following two results. Each can be proved by induction. LEMMA 4.5. Each proper subset of a natural number is similar to some smaller natural number. LEMMA 4.6. No natural number is similar |
to a proper subset of itself. We may infer from Lemma 4.6 that a set can be similar to at most one natural number. Then, defining a set to be finite if it is similar to some natural number (and to be infinite, otherwise), it follows that a finite set is not similar to any one of its proper subsets (Theorem 2.3.3) and, in turn, that w is an infinite set. Also, Lemma 4.5 implies that every subset of a finite set is finite. Once the Schroder-Bernstein theorem (Theorem 2.3.1) is proved, it can also be shown that a set a is finite if a < w. In concluding this section we note that the theory of countable sets, including Cantor's theorem (Theorem 2.3.6) stated in the forma < 61(a), could now be presented. Also it is possible to carry out the extension of w to the system of real numbers, as described in Chapter 3. EXERCISES 4.1. Prove that Ackermann's system satisfies axioms (ZF1)-(ZF5). 4.2. Show that the intersection of a nonempty collection of successor sets is a successor set. 4.3. Prove Lemma 4.2. 4.4. Prove Lemma 4.3. 4.5. Prove Lemma 4.4. 4.6. Prove that < well-orders w. 4.7. Prove Lemma 4.5. 4.8. Prove Lemma 4.6. 4.9. Let us define the number of elements in a finite set a, symbolized n(a), 302 Informal Axiomatic Set Theory I CHAP P. 7 to be the unique natural number similar to a. Prove the following statements for finite sets a and b. (a) If a C b, then n(a) < n(b). (b) The set a () b is finite and n(a () b) < n(a) and n(a f b) < n(b). (c) The set a U b is finite and n(a U b) < n(a) + n(b). (d) The set a X b is finite and n(a X b) = n(a)n(b). (e) The set ab is finite and n(ab |
) = n(a)n(b). (f) The set P(a) is finite and n(P(a)) = 2n(a). 5. The Axiom of Choice In order to clean up some details in connection with the subject matt°r sketched at the end of the preceding section and to develop a reasonable theory of cardinal numbers when they are defined as certain ordinals, the axiom of choice is required. With these applications in mind, we shall state it in the following form. An indication of a preference in this connection has no foundation, however, for within the it is possible to derive as equivalent statements those framework of c appearing in Section 2.8. (ZF7) (Axiom of choice). For each set a there exists a function f whose domain is the collection of nonempty subsets of a and, for every bCawith b76 0, f(b) C b. Concerning applications of this axiom to topics touched on in Section 4, we note first that every known proof that an infinite set is similar to a proper subset of itself (Corollary 2 of Theorem 2.9.1) requires the axiom of choice. Also we recall that this axiom was needed to prove the law of trichotomy for sets; that is, for any two sets a and b, exactly one of a < b, a N b, b < a holds. This is the content of the Corollary to Theorem 2.7.4, once the well-ordering theorem has been derived from (ZF7). Looking ahead to the theory of ordinal numbers which follows, when cardinal numbers are defined as certain ordinals, the axiom of choice is needed to show that every set has a cardinal number. 6. The Axiom Schemas of Replacement and Restriction In this section we complete the description of Zermelo-Fraenkel set theory by introducing two further axiom schemas. One of these serves to guarantee the existence of "larger" sets than can be constructed on 7.6 I The Axiom Schemas of Replacement and Restriction 303 the basis of the earlier axioms-sets which must exist if a full-blown theory of transfinite ordinal and cardinal numbers is to be possible. The other schema, whose role has not as yet been fully explored, serves to exclude the existence of certain objects as sets. To create some interest in the axiom schema of replacement consider the |
theory of sets based on just (ZFI)-(ZF7). Then, as we have seen, are sets. In w is a set. In turn, by virtue of (ZF5), 6'(w), general, defining P(w) to be w and (pk+"(w) to be P(6k(w)), each of is a set. Now, can we establish the exw, P(w), P2(w), istence of a set whose members are precisely these sets? That is, can we establish the existence of, yn(w), COI = {W, (Q(w), 192(&J), M' as a set? Since it does not appear possible to achieve this desirable state of affairs on the basis of just (ZFI)-(ZF7), a further axiom or (in order to cope with other similar situations) axiom schema is in order. A suitable candidate was first proposed by Fraenkel (1922), and independently by Skolem (1922). As modified by von Neumann (1928), it says, roughly, that if with each element of some subset of a set there is associated some one set, then the collection of the associated sets is itself a set. The instance of this schema which results upon choosing w as the initial set and associating with each n in w the set P"(w), declares that w" is a set. In the following official version of the schema in question, the hypothesis of the axiom means that for each x in a there is at most one y such that B(x, y). (ZF8) (Axiom schema of replacement.) If B(x, y) is a sentence (formula) such that for each x in a set a, B(x, y) and B(x, z) imply that y = z, then there exists a set b such that y C b if there exists an x in a such that B(x, y). It is of interest that the axiom schema of subsets, (ZF2), can be derived from (ZF8). Indeed, given a set a and a sentence A(x), take B(x,y) to be "x = y and A(x)." The hypothesis of the axiom which results is satisfied, so we may infer the existence of a set b such that y C b ill there exists an x in a |
such that x = y and A(x). That is, given a and A(x), there exists a set b such thaty E b iffy C a and A(y), which is (ZF2). The axiom of pairing, (ZF3), can also be derived from the axiom schema of replacement and (ZF5), the axiom of power set. This result 304 Informal Axiomatic Set Theory CHAP P. 7 I appears in Zermelo (1930). To prove it, let c and d be two sets whose pair is to be formed. As the set a in (ZF8) we select the power set 10, {0)) and as B(x, y) we take "x = 0 and y = c or, x = 10) and y = d." Then, for each x in 61(6'(0)) there is exactly one y such that B(x, y). Hence, by (ZF8), there exists a set b such that y C b if there exists an x in 61(c>'(0)) such that x = 0 and y = c or x = {0) and y = d. Thus, b is the set having just c and d as members. Next let us indicate how we can prove the existence of w, as a set with (ZF8). The intuitive idea, as we have already noted, is to replace the element n of w by 6'n(w) for n = 0, 1, 2,. A suitable choice for B(x, y) in (ZF8) is the following formula, which, for the sake of clarity, we will write in terms of the symbolism of the predicate calculus: (u) (((0, w) E u A (v) (w) ((v, w) C u -' (v, 61(w)) E u)) -* (x, y) E u). The reader may ponder our contention that this is a suitable choice for B(x, y). In contrast to the axiom schema of replacement which, as we shall show later, provides for the existence of enough sets to reproduce all of Cantor's theory of transfinite arithmetic, the final axiom schema has a restrictive character. Since the theory based on axioms (ZFI)-(ZF8) appears to be sufficiently comprehensive for mathematics, it is natural to consider the inclusion |
of an axiom which would serve to limit the theory to the minimal extension embracing these axioms. There are reasons to believe that this is too ambitious a goal. However, various axioms of a restrictive nature suggest themselves if it is desired to exclude as sets certain models of (ZF1)-(ZFB) having features that run counter to the intuition. One such feature is the possibility of a set which is a member of itself or, more generally, a collection of n sets a,, az,, an such that a,Can,anCan-i, The existence of such collections-even that of an infinite descending sequence of sets (that is, a sequence such that a i l, E a; for i = 1, 2, ) -is consistent with the theory having (ZF1)-(ZFB) as axioms. It is possible to prevent finite cycles of membership as well as infinite descending sequences of sets by means of an axiom. Such an axiom was initially proposed by D. Mirimanoff (1917) as a consequence of his discovery that descending sequences of the type just mentioned might exist. It is an instance of the axiom schema which we adopt. Von Neumann (1925) was the first to introduce it. 7.6 ( The Axiom Schemas of Replacement and Restriction 305 (ZF9) (Axiom schema of restriction). Let A(x) be any condition on x which (considered as a formula) has no free occurrences of y or z. If there exists an x such that A(x), then there exists a y such that A(y) and, for all z, if z C y then it is not the case that A(z). If we take A(x) to be "x C a," where a is a set, the resulting axiom, which is called the axiom of regularity is : Every nonempty set a contains an element b such that a n b = 0. This axiom is due to Zermelo (1930); it is a simplified version of an essentially equivalent axiom given in von Neumann (1929). The axiom of regularity is sufficient to exclude phenomena of the type mentioned above. We substantiate, in part, this claim by deducing from it the following two results. LEMMA 6.1. For eachseta, a iZ a. Proof. Assume, to the contrary, that a is a |
set such that a E a. Then, on the one hand, a C {a} n a (1) since a C (a). On the other hand, by (ZF9), there is a member of {a} whose intersection with {a} is the empty set. Since the only member of {a} is a, it follows that {a} n a = 0, which contradicts (1). LEMMA 6.2. For no two sets can each be a member of the other. Proof. Assume, to the contrary, that a and b are sets such that a C b and b C a. Then aC {a,b} nb and bC {a,b} na. (2) The axiom of regularity implies the existence of an element x in (a, b) such that (a, b } n x= 0. But since we must have either x = a or x = b, it follows that either { a, b } n a = 0 or (a, b } n b = 0, which contradicts (2). In order to give an application of the axiom schema of restriction of a different nature, we recall that prior to the statement of (ZF8) we mentioned that there appears to be no way to obtain wi as a set on the basis of (ZFI)-(ZF7). When these axioms are augmented with (ZF9) it can be proved, by way of a model, that wi cannot be shown to be a set; this was done first by von Neumann (1928). For convenience in discussing this matter, let us denote the theory whose axioms are those of e, except for (ZF8), by Coo. Consider the interpretation of to, whose domain is the union of wi. We contend that it is a model of C5o. First, it is clear that (ZF1) and (ZF4)-(ZF6) are satisfied and, since (ZF2) requires 306 Informal Axiomatic Set Theory I CHAP. 7 only the existence of certain subsets of a given set, it also is satisfied. To prove that (ZF3), the axiom of pairing, is fulfilled, consider two members a and b of Uw,. Then there exist m and n such that a C pm(w) and b C 61"(w) Hence, both a and b are members |
of 61"'+"(w). Thus {a, b} is a member of p-+"+1(w) and, therefore, a member of Uw,. A proof that the interpretation under consideration satisfies (ZF7) is complicated and we omit it. To prove that (ZF9) is satisfied we assume, to the contrary, that it is not fulfilled and derive a contradiction. So, by hypothesis, there exists a condition A(x) such that (i) there is an x such that A(x) holds, and (ii) for all y, if A(y) holds, then there is a z such that z C y and A(z) holds. Let xo be an x which satisfies (i) and take it as y in (ii). Let x, be a set which satisfies (ii) ; hence x, C xo and A(xi) holds. Thus, by (ii) again, there exists an x2 such that x2 C x, and A(x2) holds. Continuing in this fashion yields a sequence xo, x,, x2,, x2 C x1, x, C xo. Now there exists an n such that xo C 61"(w). It follows that, in, x" C w. Finally, we conclude that turn, x, C 4'"-'(w), x2 C ps-2(w), for some m, x"+. C 0, which is impossible. such that Now we raise the question of whether Uw1 is a set in this model. The answer is "no" by virtue of Lemma 6.1. Therefore, since there is a model of Coo in which Uw, is not present, Coo is not sufficiently strong for proving the existence of co, as a set. Furthermore, it follows that 6 is a stronger theory since, as observed earlier, we can prove the existence in it of to,. In conclusion, we call attention to Section 9.11, wherein appear some remarks about Zermelo-Fraenkel set theory when formulated as a formal axiomatic theory. EXERCISES 6.1. By imitating the proof of Lemma 6.2, prove the nonexistence of three sets a, b, and c such that a C b, b C c, and c C a. 6.2. Use the axiom of regularity to prove that if a is a set such that a |
C a X a, then a = 0. 6.3. Prove Lemma 6.1, using the instance of (ZF9) corresponding to the condition, "there exists an x such that x C x." 7. Ordinal Numbers In this and the following section we shall outline the theory of ordinal numbers due to von Neumann (1928a) as simplified by R. M. Robinson 7.7 I Ordinal Numbers 307 (1937). We shall presuppose familiarity with several definitions and theorems in Chapter 2. The definitions that we have in mind are those of a well-ordered set, an initial segment of a well-ordered set, and ordinal similarity (symbolized =),of two well-ordered sets. The results which we shall presuppose are (i) for each set there is a relation which wellorders it, (ii) the principles of proof and definition by transfinite induction, (iii) the existence of exactly one isomorphism between ordinally similar well-ordered sets, (iv) a well-ordered set is not ordinally similar to any of its initial segments, and (v) for well-ordered sets a and fl, exactly one of the following hold: a is ordinally similar to an initial segment of X13, a = j9, S is ordinally similar to an initial segment of a. Also we shall use the fact that if a is a well-ordered set, then a+ = a U { a } is a well-ordered set when we order the elements of a in the given way and, further, require that t < a for all t in a. In the von Neumann theory, an ord}nal number is a specific wellordered set of a particular kind. Thelleby the concept of order type (which, at best, is a hazy notion) is avoided completely. The defining property of those well-ordered sets which are called ordinal numbers may be thought of as qualities which well-ordered sets should have if they are to serve as "counting numbers" in the sense that the natural numbers serve this end. We begin by calling attention to several properties of natural numbers, relative to the ordering relation <, which culminate in one observation which is crucial for the generalization in mind. A natural number n is a set whose members are natural numbers; indeed, n = {x C cwjx < n}, since x < n means x C n. In |
particular, as a subset of the well-ordered set co, n is a well-ordered set. Suppose that in C n. Then the initial segment s(m)t of n which is determined by m is {x C njx < m } = m. That is, a natural number is a well-ordered set such that the initial segment determined by each of its elements is equal to that element. This is the property on which the extended counting process is based. We now define an ordinal number as a well-ordered set a such that for all >; in a, s(t) = t. In addition to the natural numbers qualifying as ordinal numbers, - are ordinal numbers, since we w does also. Moreover, w+, (w+)+, can prove that if a is an ordinal number then so is a+. The proof goes as follows. If >; C a+, then either t E a, in which case s(s) = i;, by assumption, or else i; = a, in which case s(s) = a; that is, s(a) = a, t This notation for initial segments of well-ordered sets is better suited for our present exposition. We may recall at this point that an element of the initial segment determined by a member f of a well-ordered set is called a predecessor of f. - - 308 Informal Axiomatic Set Theory I CHAP P. 7 by the definition of order in a+. Anticipating notation from ordinal arithmetic, we shall denote the ordinal numbers w, w+, (w+)+, by w,w+1,w+2, Applying (ZF8) with a as co and B(x, y) as y = co + x we may infer that w, co + 1, co + 2, form a set. The union of this set and w we shall denote by w2. Is w2 an ordinal number? The answer would appear to depend on the choice of the definition of order in w2. Actually, the question is settled automatically without any human intervention. The facts are these. The condition that a well-ordered set a must satisfy in order to qualify as an ordinal number, namely, s(E) = t for each t; in a, serves to specify the collection of initial segments determined by the elements of a. But, as the reader can easily show, even a simple ordering relation in a set is |
uniquely defined by the collection of initial segments determined by the elements of the set. (That is, if < and <' are simple orderings of a set S and, for each x in P, the initial segment determined by x relative to < is equal to that determined by x relative to <', then < = <'.) Hence, since s(E) = t; means that the set of predecessors of l; must be the ele, the only possible ordering of a which can lead to the conments of clusion that a is an ordinal number is the relation < such that for all t and 'j in a, t < 71 iff t E 'n Now either this relation is a well-ordering of a such that s(s) = t for each t in a or it is not. In the first case a is an ordinal number and in the second case it is not. In particular, it is now an easy matter to see that w2 is an ordinal number. After the ordinal number w2 comes its successor w2 + 1, followed by the successor w2 + 2 of w2 + 1, and so on. Next, after all terms of the sequence with this beginning comes w3; this set is secured by the application of another axiom of replacement. There follows in turn w3 + 1, w3 + 2, and immediately after these comes w4. In this. Then with the application manner we get successively co, w2, w3, of another axiom of replacement we get an ordinal number which follows the members of this sequence in the same sense that w follows the natural numbers. This ordinal number is w2. Continuing in this manner (and, continuing to anticipate the notation of ordinal arithmetic) we can secure all "polynomials" in to as ordinal numbers, in a manner parallel to that discussed in Example 2.7.10. We derive next several basic properties of ordinal numbers. 7.7 1 Ordinal Numbers 309 LEMMA 7.1. Each element of an ordinal number is itself an ordinal number. Proof. Let i be an element of the ordinal number a. Then i is a subset of a, since from the fact that s(s) = i it follows that an element of l is a predecessor of t, and hence an element of a. Therefore, as a subset of a well-ordered set, t is well-ordered. Now consider an element |
'7 of E. The initial segment determined by rt in i; coincides with the initial segment determined by rt in a and, since the latter is equal to rr, so is the former. Thus, in t, s(j) = n for all ri. If two ordinal numbers are ordinally similar, then LEMMA 7.2. they are equal. Proof. Let a and 0 be ordinal numbers and suppose that f is an ordinal similarity on a onto P. It is left as an exercise to prove by transfinite induction that f Q) = t for each t in a. This implies that a=P. The next result asserts that every set of ordinal numbers is wellordered. As in Chapter 2, we first prove that any two ordinal numbers are comparable. If a and (3 are ordinal numbers, then, as well-ordered sets, either they are ordinally similar or one is ordinally similar to an initial segment of the other. In the first case a = /3, by Lemma 7.2. To examine the consequences of the other possibilities, assume that a is ordinally similar to an initial segment of P. Now an initial segment of 0 is an element of /9 and hence an ordinal number, by Lemma 7.1. Using Lemma 7.2 again, it follows that a is an element of fl; so we may write a<P. Similarly, if /3 is ordinally similar to an initial segment of of, then (3 < a. Thus, for ordinal numbers a and /3, exactly one of a = (3, a < (3, (3 < a holds. Moreover, the conditions a C P, a C /3, and a < I3 are equivalent to each other. Since the proof of the well-ordering of any set of ordinal numbers parallels that given for the earlier statement of this result (Theorem 2.7.6), we shall leave it as an exercise. For completeness we record the conclusion again. LEMMA 7.3. Any set of ordinal numbers is well-ordered. To establish the next property of ordinal numbers it is convenient to make a definition in connection with well-ordered sets. If a and b 310 Informal Axiomatic Set Theory I CHAP P. 7 are well-ordered sets, we shall call b a continuation of a if a is an initial segment of b and if the ordering of the |
elements in a is the same as their ordering in b. For example, if a and P are distinct ordinal numbers, then one of them is a continuation of the other. Now let C be a collection of well-ordered sets such thaf for each distinct pair of elements of C, one. is a continuation of the other. This condition may be expressed by saying that e is a chain with respect to continuation. It is a straightforward exercise to prove the following property of such a chain. LEMMA 7.4. Let a be a collection of well-ordered sets that is a chain with respect to continuation. Then there exists a unique wellordering of c, the union of C, such that c is a continuation of each set (other than c) in the collection C. LEMMA 7.5. Every nonempty collection of ordinal numbers has a least upper bound. Proof. Let C be a collection of ordinal numbers. Then e satisfies the hypothesis of Lemma 7.4, as noted above. Hence the union y of C is a well-ordered set such that y is a continuation of each t in C, other than y itself. Actually, y is an ordinal number, since the initial segment determined by an element of y is equal to the initial segment determined by that element in whatever set of C it occurs. If E E C, then E < y, which means that y is an upper bound for C. Indeed, y is the least upper bound for C, since if S is an upper bound for C then t C S whenever i; C C and, therefore, y C S. As in the case of the Russell -paradox, the Burali-Forti paradox is avoided in Zermelo-Fraenkel set theory by our ability to prove that the troublesome set of the intuitive theory is not a set of the axiomatic theory. In the present case we can argue that if there were a set whose members consisted of all ordinal numbers, then we could form its least upper bound. That ordinal number would be greater than or equal to every ordinal number. But for each ordinal number there exists a greater one-its successor, for example. This contradiction rules out the existence of the proposed set. In our concluding result we bring the present theory in still closer agreement with the intuitive theory. LEMMA 7.6. Each well-ordered set is ordinally similar to exactly one ordinal number. 7.7 I Ord |
inal Numbers 311 Proof. The uniqueness is clear, since, for ordinal numbers, ordinal similarity is the same as equality. The major step in proving the existence of a suitable ordinal number for a given well-ordered set is the preparation for an application of the principle of transfinite induction to show that each initial segment of a well-ordered set is ordinally similar to some ordinal number. Let a be a well-ordered set and suppose that c is an element of a such that the initial segment determined by each predecessor of c is ordinally similar to some ordinal number. There exists a set e whose members are precisely all such ordinal numbers (that is, which are ordinally similar to the initial segment determined by some element of s(c). This follows from the axiom of replacement corresponding to the set s(c) and the sentence B(x, a), which says, "a is an ordinal number and s(x) = a." [This sentence does satisfy the hypothesis of (ZF8) in view of Lemma 7.2.1 Now either c is the immediate successor of one of its predecessors or c = lub s(c). If the first possibility is true and c is the immediate successor of d, then s(c) = S+, where b is the ordinal number to which is ordinally similar. If the remaining possibility is true, then s(d) lub e. Therefore, in every case, s(c) is ordinally similar to an s(c) ordinal number. Now consider the well-ordered set i of all initial segments of a (that we may do this follows from Lemma 7.4), and let j be that subset consisting of those initial segments which are ordinally similar to some ordinal number. Then the result obtained above comes to this: If x is a member of i such that s(x) C j, then x C j. By the principle of transfinite induction we then have j = I. That is, each initial segment of a is ordinally similar to an ordinal number. From the axiom of replacement corresponding to the set a and the sentence B(x, a) used above, it follows that there exists a set D whose members are precisely those ordinal numbers which are similar to an initial segment of a. Then it is an easy matter to justify the conclusion that a is ordinally similar to an ordinal number by the same argument employed above to show that s |
(c) has the same property. For a well-ordered set a we shall symbolize the unique ordinal num- ber which is ordinally similar to a (that is, its ordinal number) by If a is finite then ord a is the same as the natural number n(a) defined in Exercise 4.9. The natural numbers are, of course, the finite ordinal ord a. 312 Informal Axiomatic Set Theory I e }r A P. 7 numbers; the others are called transfinite. As in Chapter 2, those ordinal numbers which have an immediate predecessor (as is the case for each finite ordinal number other than 0) are called ordinal numbers of the first kind and those (like co) which do not are called ordinal numbers of the second kind or limit ordinals. EXERCISES 7.1. Prove the assertion made in the text that an ordering relation in a set a is uniquely determined by the collection of initial segments of the members of a. 7.2. Complete the proof of Lemma 7.2. 7.3. Prove Lemma 7.3. 7.4. Prove Lemma 7.4. 8. Ordinal Arithmetic There are two standard approaches to definitions of arithmetical operations for ordinal numbers: one relies on set theory and the other on the principle of definition by transfinite induction. The set-theoretical approach is based on formulating arithmetical operations in terms of operations of set theory; illustrations are provided by the definitions in Section 2.6 of addition and multiplication for order types. The inductive approach follows the pattern we employed to define operations for natural numbers with the principle of definition by induction replaced by that of definition by transfinite induction. Whichever approach one elects, the definitions in one can be proved as theorems in the other. Illustrations are suggested by results which are at hand. For example, from the inductive definitions of addition and multiplication for natural numbers, the reader proved in Exercise 4.9 that the number of elements in the cartesian product of two finite sets a and b is equal to n(b). This result could be used instead to define multiplication n(a) of natural numbers. That is, for natural numbers r and s we could define their product by choosing sets a and b such that n(a) = r and s = n(a X b). Since we wish to maintain as n(b) = s |
and writing r close contact with intuitive set theory as possible, we shall emphasize the set-theoretical approach. As a preliminary to defining operations for ordinal numbers, we recall the technique introduced in Section 2.5 for obtaining from two given sets a and b, possibly not disjoint, sets which have the same structural features as a and b and which are disjoint: replace a by a X 101 7.8 Ordinal Arithmetic 313 and b by b X 111. The obvious one-to-one correspondence which exists between such pairs as a and a X (0) may be used to transfer whatever structure is assigned to a to its replacement. This leads to the conclusion that if we are given two sets having possibly some structure we may assume at the outset, without loss of generality, that they are disjoint. This conclusion can be generalized to arbitrary families of sets. If {a:lx C i } is a given family, then replace each ax by ax X {x} to obtain a disjoint family which may be assigned all features of the original. The definition of addition for ordinal numbers follows the same pattern as that given in Section 2.6 for addition of order types. Let a and b be disjoint well-ordered sets. In their union a U b we define an ordering relation as follows : Pairs in a and pairs in b are ordered according to the given orderings in a and by respectively, and each element of a precedes each element of b. The assumption that a and b are wellordered implies that a U b is well ordered. This well-ordered set we call the ordinal sum of the well-ordered sets a and b. The concept of the ordinal sum of two well-ordered sets extends directly to an arbitrary (well-ordered) family of well-ordered sets. First, a word about the notation for such families. In view of Lemma 7.6 we may take the indexing set to be an ordinal number. We shall do this and use notation like {ael E X} for such a family. If, then, { aEI C X) is a disjoint family of well-ordered sets, indexed by (the ordinal number) It, we define its ordinal sum as U tat ordered as follows: If x and y are members of the union and in the same set at, then the order in at prevails; if x E at and y C all, where |
z; < 77, we take x < y. To define the sum of ordinal numbers a and ft we introduce disjoint well-ordered sets a and b such that ord a = a and ord b = ft. Let c be the ordinal sum of a and b. The sum, a + I3, is defined to be ord c. It is left as an exercise to prove that a + 3 is independent of the choice of the sets a and b (provided, of course, that each has the correct ordinal number). Analogues of this remark hold for the other arithmetic operations for ordinal numbers; they will be omitted. The definition of sum extends without difficulty to an arbitrary family {atlE C X} of ordinal numbers. Let {atlt C X} be a disjoint family of well-ordered sets at such that ord at = at for each E and let a be the ordinal sum of { atl i E It I. The sum Ztat is defined to be ord a. 314 Informal Axiomatic Set Theory I CHAP P. 7 The ordinal product of two well-ordered sets a and b is defined to be the cartesian product a X b ordered as follows: (x, y) < (x', y') iffy < y' or y = y' and x < x'. It is left as an exercise to prove that an ordinally similar set (and hence an alternative definition of the ordinal product of a and b) can be obtained as follows. Let a,, = a X { y } for each y in b, and order a,, in the obvious way. Then the family {avIy C b} is disjoint and its ordinal sum is ordinally similar to a X b. This approach to the ordinal product of a and b has intuitive appeal since it corresponds to adding a to itself b.times. To define the product of ordinal numbers a and lg we introduce well-ordered sets a and b such that ord a = a and ord b = /3. Let c be the ordinal product of a and b. The ordinal product, a#, is defined to be ord c. For properties of finite products and sums of ordinal numbers we refer the reader to Sections 2.6 and 2.7. Since it is not necessary, for the definition of the product a# of the ordinal numbers a and #, to employ disjoint well-ordered sets |
whose ordinal numbers are a and P, it is permissible to choose the most easily available well-ordered sets whose ordinal numbers are a and #-namely, a and S! Similarly, for the definition of product for a family of ordinal numbers it is not necessary to use disjoint "representatives." We take advantage of this fact by choosing ordinal numbers to be their own representatives. The first step in defining the product of a family {aelt C X} of ordinal numbers is to form the cartesian product of this family of well-ordered sets. [We recall that an element of this set is a function f on X such that f Q) C at. I Let 61 be the subset of this cartesian product, which consists of all functions which have only a finite number of values different from 0. We order P in reverse lexiographical ordering. Let f and f' be two distinct members of 6'. Then they take different values for only a finite number of arguments, and hence there exists a last argument, to, for which f(l o) 0 f'(to). If f (i o) < f'(o), then we set f < f'; if f'(to) < f(Eo), then we set f' < f. It is left as an exercise to prove that this is a well-ordering of 61. We now define the product IItat to be ord 61. Among the immediate consequences of this definition we note that if X (the indexing set) is the empty set, then IItat = 1, since the Cartesian product of the family is {0 1. Vprther, the product of a nonempty set of ordinal numbers is equal td O if at least one of the factors is equal to 0. 7.8! Ordinal Arithmetic 315 Finally, we define exponentiation as iterated multiplication. If a and 13 are ordinal numbers, then we set ap _ lEE$aE, where at = a for all E E 13. That is, aP is the ordinal number of the set of all functions on (3 into a which assume only a finite number of values different from 0, ordered in reverse lexiographical ordering. Among the laws of exponents which hold there are the following: a' = a ao= 1 0P=0 1P = I aP+r = agar aar = (as)Y for all a, for |
all a, forall13> 1, for all fl, for all a, for all a, y, y. From the first and the fifth of these properties it follows that a.a.... a (n factors) = an for each natural number n (including n = 0). Since multiplication of ordinal numbers is not commutative, no analogue from elementary arithmetic to the identity (ab)c = a°b° can be expected. A comparison of (w2)2 = w(2w)2 = w22 and w222 = w24 settles the matter. We conclude our introduction to the theory of ordinal numbers by listing a "few" of them in order. Each number which appears immediately after a sequence of three dots is the least upper bound (indeed, the limit, in the sense explained in Example 2.7.11) of those which precede it; the letters of the English alphabet which appear denote finite ordinals. The creation by Cantor of this so-called series of ordinals certainly ranks as an outstanding achievement: 0, 1,...,n,... w,w+1, --,w+n, -- w2, w2 + 1,.. w3, wn+m,... w2,...,w2+wn+m,... w2n,... wa,... wn, Wm,... WWn,... WW+l Wnrnn + wn-lmn_1 +... + m0, (WW)n,... ((WW)W)W, (WW)W,...... The next ordinal number after all of these is usually denoted by ea. It may be "reached" more directly as the least upper bound of the sequence 1, W, wW, (wW)W,... ; the proof that it is a set is left as an exercise. Further ordinal numbers, beginning with to, include to, to + 1,. 0 + w,... to + w2,... E0 + W2,... to + Co.,.... e02,... eow,... cow,,... eo....... Cantor called any solution of the equation we = e an epsilon number. It is left as an exercise to prove that to is the least epsilon number. 316 EXERCISES Informal Axiomatic Set Theory I C Ii A P. 7 8 |
.1. Show that the definition of order which was adopted for the union of well-ordered sets a and b may be stated as follows. If p and a are the given wellordering relations in a and b, respectively, then order a U b by p U a U (a X b). 8.2. Show that the sum a + Q of two ordinal numbers is independent of the choice of well-ordered'sets a and b such that ord a = a and ord b = Q. 8.3. Show that the ordinal product of two well-ordered sets a and b, as defined in the text, is ordinally similar to the ordinal sum of the family {a,,I y C b}. 8.4. Prove that the ordering assigned to the subset tp of the cartesian product of a family {atiE C X} is, in fact, a well-ordering. 8.5. Prove each of the laws of exponents displayed in the text. 8.6. Show that e0 is the least epsilon number. 9. Cardinal Numbers and Their Arithmetic Although in Section 2.3 we gave a definition of the concept of a cardinal number, we emphasized there that we would rely on only that consequence of the definition to the effect that card a = card b iff a r b. Using just this property of cardinal numbers it is possible to reproduce, with the framework of Zermelo-Frankel set theory, (i) the definition of the order relation < for cardinal numbers, the proof (after the Schroder-Bernstein theorem is established) that card a < card b if a < b, and that of Cantor's theorem; (ii) the definitions of addition and multiplication for cardinal numbers (Section 2.5) and the proofs of the properties of these operations stated in Theorems 2.5.1 and 2.5.2; (iii) the definition of exponentiation and the proofs of those properties stated in Theorem 2.5.3. Defining a cardinal number as finite if it is the cardinal number of a finite set and as infinite if it is the cardinal number of an infinite set, we may continue by proving the following results: The arithmetic of finite cardinal numbers is the familiar finite arithmetic and, if u is an infinite cardinal number, then u u = u and u + u = u. We now consider a suitable definition of the cardinal number of a set. From earlier results we know that every |
set is similar to some ordinal number. In general, a set is similar to many ordinal numbers. The result 7.9 ( Cardinal Numbers and Their Arithmetic - 317 on which the von Neumann definition of the cardinal number of a set leans is that for each set a, the ordinal numbers which are similar to a form a set. We begin the proof by observing that it is possible to find an ordinal number greater than all ordinal numbers similar to a. An ordinal number t9 which is similar to 6'(a) will serve. Then, for each ordinal number a similar to a, the set a is less numerous than the set 9, and hence card a < card p. Hence, it is not the case that 0 < a, and therefore a < P. In turn, this means that a C S. Thus, P is a set that contains every ordinal number similar to a and the existence of such a set implies that the ordinal numbers similar to a form a set. In view of this result, a natural choice for card a is the least ordinal to which a is similar. This is the motivation for a consideration of the following definition : A cardinal number is an ordinal number a such that if 0 is an ordinal number similar to a, then a < 16. That is, a cardinal number is an ordinal number which is not similar to any smaller ordinal number. If a is a set, then card a, the cardinal number of a, is the least ordinal similar to a. That this definition is satisfactory follows from the fact that we can prove that card a = card b if a '' b. Indeed, since each set is similar to its cardinal number, it follows that if card a = card b, then a - b. For the converse, we assume that a - b and infer that card a = card b. Since card a is the least ordinal similar to a, certainly card a < card b and, upon interchanging a and b in this argument, also card b < card a. Hence, card a = card b. Since a finite ordinal number (that is, a natural number) is not similar to any different ordinal number, the set of ordinal numbers similar to a finite set is a unit set. Hence, the cardinal number and the ordinal number of a finite set are the same. Notice that we are now entitled to infer from the similarity of Y(a) and 2a |
, where a is a set, that card P(a) = 2a, since we now know that 2 is a cardinal number. Also, we may state Cantor's theorem in its familiar form: a < 2a. The above inequality brings to mind one of the last two questions which should be raised regarding the definition of cardinal number. We recalled at the beginning of this section that on the basis of the identity that card a = card b if a r., b, an ordering relation can be defined for cardinal numbers and that it follows from this definition that card a < card b iff a < b. Now ordinal numbers have already been outfitted with an ordering relation. Fortunately, there is no collision of the two possible meanings of card a < card b, since they coincide. We leave the details as an exercise. The other question concerns the status of Cantor's 318 Informal Axiomatic Set Theory I CHAP P. 7 paradox. Its fate is settled in much the same way as the Burali-Forti paradox. Every set of cardinal numbers, as a set of ordinal numbers, is well-ordered. Moreover, we know that every set of cardinal numbers has an upper bound and that for every set of cardinal numbers there is a cardinal number greater than each member of the set (see Section 2.9). It follows that there is no largest cardinal number or, what is equivalent, there is no set that consists precisely of all the cardinal numbers. As the smallest transfinite ordinal number, co is a cardinal number and, when playing the role of a cardinal number, is denoted by bto. Since Theorem 2.9.3 (every set of cardinal numbers is well-ordered), holds in Zermelo-Fraenkel set theory, we can define the alephs in general as in Section 2.9. The immediate successor, Ki, of 14o in the ordering of cardinal numbers may be described as the least uncountable ordinal number, or as an uncountable well-ordered set each of whose initial segments is countable. It may come as a surprise to learn that this ordinal number is greater than all of those explicitly named in Section 8, for they are all countable! EXERCISES 9.1. Give definitions of addition and multiplication for an arbitrary family of cardinal numbers by imitating corresponding definitions for ordinal numbers. 9.2. Show that the two possible meanings of card a < card b coincide. 10. The von |
Neumann-Bernays-Godel Theory of Sets In this section we shall describe the theory in question (and, for brevity, refer to it simply as von Neumann set theory) only to the point where we can indicate the essential differences between it and ZermeloFraenkel set theory. The original version of von Neumann set theory appeared in von Neumann (1925, 1928a, 1929), and in simplified form in R. M. Robinson (1937). Since a distinguishing feature of this original version was its adoption of the notion of function, rather than that of set, as primitive, it differed considerably from other axiomatizations of set theory. In a series of seven papers, beginning in 1937 (see References), P. Bernays formulated a modification of the von Neumann approach which brought it in much closer contact with Zermelo set theory. In turn, in Godel (1940) the theory is further simplified. One essential difference between the von Neumann theory and the 7.10 ( The von Neumann-Bernays-Godel Theory of Sets 319 Zermelo-Fraenkel theory reflects a difference in attitudes toward the question of how to cope with the "too large" sets of intuitive set theory. In the Zermelo theory it is possible to prove the existence of most of the sets which are necessary for mathematics, but the axioms which are concerned with the existence of sets are so designed that it seems impossible to construct any "troublesome" sets. In brief, the theory S is a conservative one! The von Neumann theory, on the other hand, reflects the attitude that it is not the existence of too large sets as such which leads to contradictions but rather their being taken as members of other sets. In the von Neumann theory a technical distinction is drawn between sets and classes. Every set is a class, but the converse is not true. Those classes which are not sets are called proper classes and their distinguishing feature is that they are not members of any other class. The class of all ordinals, for example, exists, but it is a proper class. Thus the Burali-Forti paradox cannot be constructed, since it requires that the class of all ordinals be a member of a class. The other paradoxes meet with a similar fate. In Godel (1940) three primitive notions are adopted: class, set, and the binary relation of membership. A slight modification of the theory allows one to reduce |
the number of primitive notions to one-the binary relation C. Then elements of the union of the domain and the range of C are called classes and elements of the domain are called sets. The axioms of the theory, as stated in Gi del (1940), fall into several groups. The first consists of the axioms of extension and that of pairing. Using lower-case letters as set variables and capital letters as class variables, the axiom of extension is (u)(uEXHuCY)-4X=Y. The axiom of pairing provides for the existence of the set whose members are just the sets x and y. This is formulated as (x)(y)(3z)(u)(). The eight axioms of the second group are concerned with the existence of classes. These axioms, which are due to Bernays, replace an axiom schema in the original von Neumann theory. From them Bernays proved the general existence theorem (a metatheorem), which asserts that for any formula F(x) which contains no bound class variables there exists a class Y that contains just those x's which satisfy F(x). This result, which is referred to as the class theorem, bears a strong resemblance to the principle of abstraction of intuitive set theory; the sole difference is 320 Informal Axiomatic Set Theory I CHAP. 7 that "defining conditions" determine classes and not necessarily sets. The class theorem yields as a by-product the fact that classes in the von Neumann theory play the role that formulas do in the ZermeloFraenkel theory. The remaining axioms of the von Neumann theory coincide with the remainder of those for S [that is, (ZF4)-(ZF9) ], with the one important distinction that none of the former are axiom schemas. For example, in place of (ZF8), the axiom schema of replacement, there is the axiom of replacement which is the formula (x) (y) (z) (((x, y) E X A (x, z) C X) --*y = z) --> (3y)(x)(x Ey - (3w) (w E z A (w, x) E X)) Thus, by way of the theorem schema described above, this axiom yields all instances of the axiom schema (ZF8). This brings us to the second, and last essential difference between the two theories: von Ne |
umann set theory is finitely axiomatized. That is, no axiom schema of set construction is required; instead, a finite number of specific set and class constructions is adequate. BIBLIOGRAPHICAL NOTES In Fraenkel (1961) general set theory is developed at a level which is between that of Chapters 1 and 2 and that of this chapter. Fraenkel's excellent book, Abstract Set Theory, is a thoroughly revised (and greatly improved) edition of an earlier book. The book by Fraenkel and Bar-Hillel (1958) complements Abstract Set Theory in the same way that the present chapter complements our earlier coverage of intuitive set theory. In addition, it considers other approaches (for example, Quine's New Foundations) to set theory. Zermelo-Fraenkel set theory is also expounded in Suppes (1960). An interesting feature of this book is an elegant unorthodox treatment of finite sets by means of Tarski's definition (see Exercise 2.3.14), which allows Suppes to develop the theory of finite sets before the theory of finite ordinals. Another treatment of Zermelo-Fraenkel set theory appears in Halmos (1960). This is a beautiful presentation. An outline of von Neumann-Bernays-Godel set theory is given in Godel (1940) and in Bernays and Fraenkel (1958). The latter book presents a modification of the system developed by Bernays in the series of seven papers mentioned in the text. For a high-level development of transfinite arithmetic beginning with the theory of ordinal numbers, H. Bachmann (1955) should be consulted. CHAPTER 8 Several Algebraic Theories I T I S PER H A P S in algebra that the axiomatic method has scored its greatest successes. The majority of axiomatic theories which are regarded as belonging to algebra are noncategorical. This is by design, since the goal of algebra is a systematic analysis of various combinations of central features common to a variety of specific algebraic systems. This modern approach to algebra yields theorems which not only illuminate a multitude of classical examples by displaying them in the most general light without foreign hypotheses, but also it contributes formalism and powerful tools which are indispensable to a large part of mathematical research, including that in the theory of numbers, algebraic geometry, functions of several complex variables, integration theory, and topology. Thus, algebra is not |
merely a branch of mathematics, for it plays within mathematics a role analogous to that which mathematics itself has played with respect to physics for centuries. As is the case with most branches of mathematics, it is foolhardy to attempt a definition of algebra. It is possible, however, to suggest a characterization by describing basic features of those theories which may be called "algebraic theories," that is, axiomatic theories which, it is generally agreed, belong to the province of algebra. Some such features are discussed in Section 1. The theory of Boolean algebras qualifies as an example and serves to illustrate some of the concepts introduced. The brief introduction to semigroups which appears in Section 2 is included simply because this theory can be used as a vehicle to introduce a variety of definitions that are applicable to the algebraic theories with which the remainder of the chapter is concerned. Each of these theories, apart from that of groups, had its origin in one of the number systems constructed in Chapter 3. That is, each is founded' on the basic properties of one of the system of integers, the system of rational numbers, the system of real numbers. When it is realized that these theories 321 322 Several Algebraic Theories I CHAP. 8 form the backbone of modern algebra, the fundamental role played by the familiar number systems in stimulating the development of modern algebra becomes apparent. Exposing the role of the familiar number systems as a source for algebraic theories is one goal of this chapter. The other is to provide vyays and means for characterizing, in turn, these number systems as models of certain algebraic theories. These characterizations are presented in the last three sections of the chapter. 1. Features of Algebraic Theories Ordinarily, algebraic theories are presented as informal theories within the context of set theory. That is, as explained in Section 5.3, an algebraic theory is formulated in terms of a nonempty set X and certain constants associated with X. These constants may be of various types : elements of X, subsets or collections of subsets of X, unary operations on X (that is, functions on X into X), binary relations or operations in X, and so on. Collectively, the constants serve as the basis for imposing a certain structure on X. The structure is given in the axioms-that is, the properties assigned to X and the constants. It is principally the form of the axioms that distinguishes algebraic theories among axiomatic theories in |
general. The axioms pertaining to binary operations imitate, in part at least, the basic properties of addition and multiplication and include, possibly, the existence of interrelations such as distributive laws. Those pertaining to any binary relations present may imitate properties of "less than" for number systems. If unary operations are present, they are often called (left or right) operators. As an indication of the form that axioms pertaining to operators might have, the properties of scalar multiplication in an elementary treatment of vector algebra are suggestive. These include a(a + f) = as + a$, (a+b)a=act +ba, a(ba) = (ab)a for all vectors a and i8 and all scalars (real numbers) a and b. The first of these is a property of individual scalars (left operators). In contrast, the others are interrelations between combinations of operators and combinations of vectors and, as such, presuppose the existence of operations for the set of scalars. In general, a set of operators may or may not have some assigned structural features. The theory of Boolean algebras qualifies as an algebraic theory. If 8.1 I Features of Algebraic Theories 323 the theory is formulated as in Theorem 6.3.1, then the constants associated with the basic set B are one binary operation and a single operator. We turn next to the description in general terms of two notions which occur so consistently in algebraic theories that they may be considered as serving to further delineate algebraic theories. If we agree that by an algebra is meant any model of some algebraic theory, then one of the notions is that of a subalgebra of an algebra. This requires two preliminary definitions. Let f be an operation in a set X and A be a nonempty subset of X. Then A is said to be closed under f if the restriction off to A X A is an operation in A or, in other words, the range of fjA X A is included in A. If A is closed under f, the operation f IA X A is said to be that induced in A by f. Although f jA X A Of (assuming that A C X), if instead of "f " a familiar symbol like "-I-" or " " is used for the initial operation, it is customary to designate that operation which it may induce in a subset by the same symbol. Next, let g: X -+- X |
and A be a nonempty subset of X again. We shall say that A admits g if g [A19 A. Now suppose that (X, ) is an algebra having X as its basic set and that A is a nonempty subset of X which admits each operator on X and is closed under each operation in X. Then it may be the case that A, together with the constants induced in it by those of X, is a model of the theory of which (X, ) is called a subalgebra of (X, ). ) is a model. In this event, (A, According to the foregoing, if (X, ) is a model of an algebraic theory, then subsets of X which are closed under the operations in X and so on provide a potential source of further models of the same theory. Another possible means for deriving further models from given models of an algebraic theory is by way of congruence relations. This notion for an arbitrary algebra is a direct generalization of that given in Section 6.4 of congruence relations for Boolean algebras. A congruence relation on an algebra (X, ) is an equivalence relation 0 on X such that if * is a binary operation in X, then for all a, b, and c in X, (CI) a9bimpliesc*aOc*banda*cOb*c and, if f is an operator on X, then for all a and b in X, (C2) a 0 b implies f(a) Of(b), and, if < is an ordering relation, then for all a, b, c, and d in X, (C3) a 0 b, c 0 d, and a <c imply b <d. Upon reviewing the discussion of congruence relations for Boolean algebras it should be clear that requirements (Cl)-(Ca), whenever applicable, 324 Several Algebraic Theories I CHAP P. 8 are sufficient (and, indeed, necessary) conditions that each operation and so on defined for X induces a corresponding constant for X/0 by way of representatives of 8-equivalence classes (that is, if * is a binary operation in X, defining ff * b to be a _*b, and so on). If 0 is a congruence ), then it may be the case that X/0, relation on the algebra ( X, together with the constants induced by those associated |
with X in the way described, is a model of the theory at hand. In this event, (X/9, ) is called a quotient algebra of ( X, ). In conclusion, it will do no harm to rephrase for algebras in general a remark made earlier for Boolean algebras. Namely, the description ) includes (usually implicitly) an equality relation of any algebra (X, on X and this is taken to be a congruence relation on X. That is, equality is assumed to satisfy whichever of (C1)-(C3) are applicable. 2. Definition of a Semigroup A semigroup (with neutral element) is an ordered triple (X, *, e), where X is a set, * is an associative binary operation in X, and e is a member of X such that e* x= x* e = x for all x in X. Our sole purpose in touching on this theory is to derive a few basic properties and introduce some terminology and notations. This will prove to be efficient, since we shall find a variety of applications for these items later. It is with the diversity of the applications in mind that we have adopted the neutral symbol "*" for the operation in X. The property enjoyed by the element a of the semigroup (X, *, e) characterizes this element, since if e' * x = x * e' = x for all x, then e' * e = e and e' * e = e', whence e = e'. We shall call e the neutral element for the operation in X. EXAMPLES 2.1. If A is a nonempty set, then (P(A), U, 0) and (6'(A), 0, A) are semi- groups. 2.2. If, as usual, N is the set of natural numbers, then 0) and (N,, 1) are semigroups. 2.3. If A is a nonempty set, then (AA, o, iA) is a semigroup. An algebra (X, ) is often identified by merely its basic set, if no confusion can arise. For example, we shall often use the term "the semi- 8.2 ( Definition of a Semigroup 325 group X" in place of "the semigroup (X, *, e)." If there is need to mention the operation, "X is a semigroup under *" |
may be used in place of "(X, *, e) is a semigroup." For example, we may say "the set Z of integers is a semigroup under addition" in place of "(Z, +, 0) is a semigroup." In subsequent instances of semigroups the notation for the composite of a and b will usually be a + b (read: the sum of a and b) or ab (read: the product of a and b). In the first case we say that we have an additive operation and in the latter case, a multiplicative operation. The neutral clement for an additive operation is always denoted by "0" and called the zero element of the semigroup; the neutral element for a multiplicative operation is usually designated by "I" and called the unit or identity element of the semigroup. One theorem that we have already proved for a semigroup is the general associative law (Theorem 2.2.2), which asserts that all composites that can be associated with, an) of elements of a sernigroup are the same clean n-tuple (a,, a2, ment of the semigroup. For an additive operation this element is denoted by +an or 2;;'_j"r ac while for a multiplicative operation it is denoted by a,+a2+ a,a2... a or II,""_, a;., an are all equal to the same element a, then the composite If a,, a2,, an) is denoted by "na" and "a"" in the additive and multiof (a,, a2, plicative cases, respectively. For n = 1 we agree that both na and a" are simply a. We extend the definition of na and an to all natural numbers by defining Oa to be 0 and a° to be 1-that is, the neutral element in each case. Then, for all natural numbers m and n and all elements a of a semigroup X, (1) Oa = 0, 1a = a, (m + n) a = ma + na, (mn)a = m(na), if the operation in X is additive. If the operation is multiplicative, then a° = 1 a' = a, am+n = ama" amn = (a-) n. (2) These formulas follow from our definitions and the general associative law. A semigroup (X, |
*, e) is commutative or Abelian iff a * b = b * a for all a and b in X. For commutative semigroups we have the general commutative law stated in Exercise 2.2.4: If a,, a2, ", a" are elements 326 Several Algebraic Theories I CH A P. 8 of a commutative semigroup and if 1', 2', of the numbers 1, 2,, n, then, n' is some rearrangement a, * a2 *... *a, = a,. *a2 *. *a",. From this it follows easily that for a commutative semigroup the string of formulas (1) may be supplemented by n(a+b) =na+nb, (3) and those in (2) by (4) (ab)" = a"b". Further notation and computational rules enter in connection with our next definition. An element a of a semigroup X is invertible if there exists an element a' of X such that a * a' = a' * a = e. In that event there is just one such element a' with this property. For if with a" we can also demonstrate that a is invertible, then all =all *e=all *(a* a')=(all *a)*a'=a*a'=a'. The element a' is the inverse of a. If a is invertible and a' is its inverse, so that a * a' = a' * a = e, then these equations demonstrate that a' is invertible and that a is its inverse. Another important property of invertible elements is proved next. THEOREM 2.1. If a and b are invertible elements of a semigroup (X, *, e), then a * b is invertible. If a' and b' are the inverses of a and b, then b' * a' is the inverse of a * b. to show that (a * b) * (b' * a') = e and Proof. (b' * a') * (a * b) = e. The first of these, for example, is shown as follows : sufficient It is (a*b)*(b'*a') = a*(b*b')*a' = a*e*a' = e. COR |
OLLARY. If a,, a2, group and a,', a2, vertible and aR * aa_, * an their inverses, then al * a2 * a" are invertible elements of a semi* a" is in- * al' is its inverse. The notation involved in discussing further properties of invertible elements is sufficiently different in the additive and multiplicative cases as to warrant separate treatments. Let us consider an additive notation first. If a is an invertible element of a semigroup (X, +, 0), then negative multiples of a can be defined. Namely, we observe that if a' is the inverse of a (thus, a + a' = a' + a = 0); then (5) ma = (m + 1)a + a' 8.2 Definition of a Semigroup 327 for all nonnegative m. This equation we take as the basis for an inductive definition of ma for negative m. Then we observe that the third formula in (1) above is true for any fixed m and n = 0; it can be proved for all natural numbers n by induction from n to n + 1 and for all negative n by induction from n + I to n, using the following consequence of (5) : (m + 1)a = ma + a. One instance of the formula thus obtained is na + (- n)a = Oa = 0 = (- n)a + na for an.arbitrary n. This means that for all n, na is invertible and (-n)a is its inverse. It follows that m(na) and (mn)a are defined for every m. The equality of these two elements for arbitrary m and n can then be proved by the two inductions used before. Thus the fourth formula in (1) and thereby all formulas in (1) hold for all integers m and n. If a is an invertible element of (X, +, 0), then, according to (5), (-1)a is the inverse of a. We abbreviate "(-1)a" by "-a" and call it the negative of a. The earlier result that the inverse of the inverse of a is equal to a then takes the form -(-a) = a and Theorem 2.1 translates into -(a + b) _ (-b) + (-a) for invertible elements a and b. For |
an arbitrary b and an invertible element a of X, b + (-a) C X; this element will be designated by b - a. Thus, (b - a) + a = b. Further, the element (-a) + b will be denoted by -a + b, so that a + (-a + b) = b. These definitions lead to the following computational rules which are easily verified : -(a - b) = b - a, -(-a + b) = -b + a. Finally, if the semigroup is commutative, (3) holds for arbitrary n. All the foregoing definitions and results have multiplicative analogues. The starting point for their derivation is the observation that if a' is the inverse of a, then (6) a?%+Ia. ain = for all nonnegative m. This equation we take as the basis of an inductive definition of a'" for negative m. It is left as an exercise to verify that the third and fourth equations in (2) above are true for arbitrary integers m and n. According to (6), a-' is the inverse of a. Moreover, (a-')-' = a, (ab)-' = b-'a-1, 328 Several Algebraic Theories I CH A P. 8 and, if the semigroup is commutative, (4) holds for all integral values of n. EXAMPLES 2.4. The semigroup 0) is commutative; 0 is the only invertible ele- ment. In contrast, each element of the semigroup (Z, +, 0) is invertible. 2.5. In the multiplicative semigroup Z the only invertible elements are 1 and -1. 2.6. Let A be a nonempty set. Then ((P(A), +, 0), where + is the symmetric difference operation, is a commutative semigroup. Each element B is invertible; indeed. - B = B. 2.7. The semigroup of all mappings on a set of at least two elements into itself (see Example 2.3) is not commutative. The invertible elements are the one-to-one and onto mappings. EXERCISES 2.1. Let be an associative operation in a nonempty set X. An element a in X such that x a = x for all x is a |
right identity element. (a) Give an example of such a system that has more than one right identity element. (b) Show that if more than one right identity element is present in X, then no identity element is present. 2.2. In a nonempty set X introduce the operation (a, b) -} ab = a. Show that this is an associative operation and that every element is a right identity. When is X a semigroup? 2.3. Show that (N, *, 0), where a * b = a + b + ab, is a semigroup. 2.4. We define NO) to be the set of all objects of the form (a d) where a, b, c, d C N. A multiplication is defined for these elements as follows:: b a a' b' _ aa' + bc' ab' + bd' (c d) (c' d') - (ca' + dc' cb' + dd' Show that N(" is a semigroup under this multiplication. What elements are invertible? Defining an element x of a semigroup as idempotent iff x' = x, determine the idempotents of N('). 2.5. Establish each of the identities appearing in (1) and (2) in the text for natural numbers m and n. 2.6. Establish the identities (3) and (4) for commutative semigroups. 2.7. Give a detailed account of the extension of the identities in (1) to the case of arbitrary integers m and n. 2.8. Give a detailed account of the extension of the identities in (2) to arbi- trary integers m and n. 8.3 I Definition of a Group 329 3. Definition of a Group In spite of the repetition which results, we start afresh with the theory of groups for the sake of completeness. Our initial formulation is the one appearing in Exercise 5.4.15. A group is an ordered triple (G,, e), where G is a set, is a binary operation in G, e is a member of G, and the following axioms are satisfied. is an associative operation. G,. G2. For each a in G, e G3. For each a in G there exists a = a. a' in G such that Two properties of a group follow directly from the axi |
ble. In accordance with conventions introduced for semigroups, if multiplicative notation is used for a group operation we shall write "1" for the identity element and "a-"' for the inverse of a. If additive notation is used instead, then "0" and "-a" will be used in place of "1" and "a-'." In either case the definitions and properties pertaining to powers and multiples given in Section 2 are available for use. The converse of Theorem 3.1 is obviously true and consequently another formulation of the theory of groups is at hand: A group is a semigroup such that each element is invertible. We prefer the initial one, however, since it is clearly a weaker formulation, which means 330 Several Algebraic Theories i CHAP. g that there are fewer steps in the verification that a given system is a group. The set of axioms in the explicit formulation of the theory of groups as a semigroup in which each element is invertible is simply the result of supplying the "left-right" symmetry which the initial formulation lacks. This symmetrical set of axioms is, like {G1, G2, G$}, independent. A third formulation in which symmetry is an inherent part is given next. THEOREM 3.2. An ordered pair (G, ), where G is a set and is a binary operation in G, defines a group if Go. G is nonempty, - is associative, G1. each of the equations a x = b and y a = b has a solution G4. in G for all elements a and b in G. Proof. Assume that (G, -, 1) is a group. Then obviously (G, ) satisfies Go and G1. Moreover, G.1 is valid since, for given elements a and 6 in G, a(a 'b) = b and (bar')a = b. For the converse, let (G, ) be a system satisfying Go, G1, and G4. According to Go there exists an element c in G. According to G4 there exists an element e in G such that ec = c. Moreover, by G4, if a is any element of G, then there exists an element d in G such that cd = a. Hence ea = e(cd) = (ec)d = cd = a, so e satisfies Ga. As for G3, it is a |
consequence of the solvability of. xa = e for each a. Hence, (G,, e) is a group. Each of the equations ax = b and ya = b has a unique solution in a group. This is an immediate consequence of THEOREM 3.3. For all elements a, b, and c in a group, each of ab = ac and ba = ca implies that b = c. Proof. Assume, for example, that ab = ac. Then a'(ab) = a '(ac), whence b = c. If finiteness is assumed for the set G in Theorem 3.2, then G4 can be replaced by the, in general, weaker cancellation laws. THEOREM 3.4. A pair (G, ), where G is a finite set and is a binary operation in G, defines a group iff 8.3 I Definition of a Group 331 G is nonempty, is associative, Go. G,. G5. a = c a implies that b = c. each of a b = a c and b In view of Theorem 3.2 it is sufficient to prove that G5 implies Proof. G9 in the presence of Go and G1. Let a be an element of G and consider the mapping fa: G ; G such that fa(x) = ax. By G5, fa is oneto-one and hence onto, since G is finite. That is, for each b in G, ax = b has a solution in G. The solvability of ya = b is shown similarly. We forego giving examples of groups until we have given several more definitions. If for group elements a and b, ab = ba, then a and b commute; if every pair of elements of a group commute, then the group is called commutative or Abelian. Examples that we shall encounter will demonstrate not only the consistency and independence of the set of axioms for a group but also the independence of the set of axioms for a commutative group. If the elements of a group are finite in number then the group is finite and the number of elements is the order of the group. If a group is not finite, then it is infinite. Finally, we mention that analogous to the convention introduced for semigroups, we shall frequently use "G" as a name of the group (G,, 1) if the operation involved is unambiguous. EXAMPLES |
3.1. If A is a nonempty set, then the set of all one-to-one mappings on A onto itself, symbolized G(A), together with function composition and the identity map iA, is a group. This conclusion simply summarizes basic properties of one-to-one correspondences. We shall call this group the group of one-to-one transformations on the set A. 3.2. If n is a positive integer, then congruence modulo n is a congruence relation on the additive group of integers. Consequently an operation + is defined in Z., the set of equivalence classes a, by choosing a + b to be a _+b. It is an easy matter to prove that (Z., +, 0) is a commutative group of order n. 3.3. Congruence modulo n is also a congruence relation on the multiplicative semigroup of integers. This leads to the commutative semigroup (Za, -, 1) where, by definition, a - b = ab. The identity element for the operation is 1 and since 0 has no inverse, the semigroup is not a group. Discarding 0 does not always overcome the difficulty, since the resulting set may not be closed under multiplication; for example, in Z5, 2 3 = 0. This difficulty is absent if n is a prime p, since then as - b = 0 implies in turn that ab 0(mod p), p divides a or b, either a orb is equal to 0. That is, multiplication is an operation in Z; = 332 Several Algebraic Theories I CHAP. 8 Zn - {0}. With I the identity element, to conclude that the system is a group, it remains to prove that each element has an inverse or, in other words, that the equation ax with a P` 0 has a solution in Z. Now 331 = 3 with a P 0 is equivalent to ax = 1(mod p) where p does not divide a. If p does not divide a, then a and p are relatively prime and there exist integers r and s such that ra - sp = 1. But then ram I (mod p) or ra = 1, and a has an inverse. Thus, (Zp,, i) is a commutative group. 3.4. For groups of small order a multiplication table, as described earlier for Boolean algebras, is a practical |
device for exhibiting the group operation, inverses, and so on. As an illustration, consider the set F of six functions fl, f2,, f6 of a complex variable z, where fl(Z) = Z, f2(z) = I 1 z' fa(z) = z I' z f4(Z) = z' f6 (Z) = I - Z, fe(z) _ Z z - I' with the composite of fi and f; taken to be f; o f;. Since f, is an identity element f,) is certainly a semigroup. The for the operation and a is associative, (F, following multiplication table shows that actually it is a group and, further, that the group is noncommutative. fl f2 fa f4 f6 Is flf2f3f+f6f6 f2 fa f4' f6 f3 fl f5 f6 f1 f2 f6 f4 f6 f6 fl f3 f4 f6 f2 ft f6 f4 f3 f2 ft f2 f3 f4 f5 f3 f2 f, f4 f6 f6 f6 There is also the possibility of using this device to concoct groups of sma'l, k, which ar order. For this we start with a nonempty set S of letters a, b, to be the group elements, and fill out a multiplication table in such a way that all the group axioms are fulfilled. The table will exhibit an operation in S ig each entry is a member of S. A much stronger requirement is given by condition G4 in Theorem 3.2. The unique solvability of ax = b for all a and b in S means that in each row of the multiplication table each element of S must appear exactly once. Similarly, the unique solvability of ya = b implies that each column in the table is simply S in some order. A table whose rows and columns fulfill these conditions defines a group if the operation is associative. Unfortua nately, it is not easy to check the associative law directly from a multiplication table unless special preparations are made. 3.5. Let C be the set of all rotations about the origin in a Cartesian plane_ An element of G is a mapping of the form (x, y) -+- (x', y'), where x' = x cos 0 - y sin |
0, y' = xsin0+ycos0. 8.4 ` Subgroups 333 Here B is the angle of rotation. Then (G, o, :), where o is a function composition and i is the identity map, is a group. EXERCISES 3.1. (a) For the real number a, let ta: R -} R be such that xta = x + a for each real number x. Show that (T, i), where T = {t.1a E R-}, - is function composition, and i is the identity map on R, is a group. (b) For the real number a, let sa: R --- R be such that xsa = xa for each real number x. Show that (S, a, :), where S = {s,I a E R - (0)), o is function composition, and i is the identity map on R, is a group. 3.2. For real numbers a and b with a -' 0, let [a, b] be the mapping on R into itself such that x[a, b] = xa + b. Show that A = {[a, b]ia, b C B. and a,E 0} is a group under function composition. 3.3. Show that {(1 + 2m)/(1 + 2n)lm, n E Z} is a group under ordinary multiplication. 3.4. Show that {cos r + i sin rjr C (9} is a group under ordinary multipli- cation. 3.5. Write out a multiplication table for Z,'. 3.6. An operation in {e, f} may be defined as follows: ee = fe = e, of = ff = f. Show that this system satisfies the group axioms G, and G2, but not G3. Construct two other systems to complete the proof of the assertion that the set of axioms for a group are independent. 3.7. In the text an Abelian group is defined to be a group having the further property that ab = ba for all a and all b. Prove that an Abelian group can be characterized as an ordered triple (G,, ') where G is a nonempty set, is a binary operation in C,'is a unary operation in G, and the following property holds: if (aa')b |
' _ (rs')t', then b = (tr')s. 4. Subgroups A group H is a subgroup of a group G if H e G and the restriction of the operation in G to H X H is equal to the operation in H. In other words, the subgroups of a group G are the closed subsets that satisfy the group axioms. Let H be a subgroup of the group G and 1' and 1 be the identity elements of H and G respectively. Then V- 1' = 1' and t' 1' = 1', so 1'- 1' = 1.1'. By the cancellation laws it follows that 1' = 1 ; thus the identity element of a group G is the identity element of any Subgroup H of G. This result is a consequence of the following necessary I111c1 sufficient conditions that a subset of a group determine a subgroup. We have derived it independently in order that it be available for use fr the proof. 334 Several Algebraic Theories I CHAP. a THEOREM 4.1. A nonempty subset H of a group G determines a subgroup of G if (i) H is closed, and (ii) the inverse (in G) of each member of H is a member of H. Proof. Let H be a nonempty subset of G having the two stated prop. erties. Then there is in H an element a of G and hence aa'' = 1 is in H by (i) and (ii). Since lx = x for x in G, 1x = x for x in H and for each a in H there is in H an element a', namely a-', such that aa' = 1. Thus H satisfies G2. Since H is closed under the operation in G, that operation restricted to H X H is certainly an associative operation in H. Hence, H is a group. Conversely, if H is a nonempty subset of G which determines a subgroup of G, then (i) must hold. Since 1 C H, as observed above, the equation ax = I has a solution in H. Since the only solution of this in all of G is a ', (ii) must hold for H. COROLLARY. A nonempty subset H of a group G determines a subgroup of G if for all a and b in H, ab-' is in H. THEOREM 4.2. A nonempty |
subset H of a finite group G deter. mines a subgroup of G if H is closed. Proof. This follows from the definition of a subgroup and Theorem 3.4. THEOREM 4.3. The intersection of a nonempty collection of sub, groups of a group G is a subgroup of G. The proof is left as an exercise. Every group G includes two subgroups, namely G and [ 1 } ; these are the improper subgroups of G. Any other subgroup of G is a proper subgroup. Proper subgroups can usually be obtained by the following technique. Let S be a subset of a group G. Then the intersection of all subgroups of G which include S is a subgroup of G which includes S. This is called the subgroup of G generated by S and is symbolized by [S]. The set [S] has the following properties: (i) it is a subgroup of G, (ii) it includes S and (iii) is included in any subgroup of G that includes S. It is easily seen that these three properties characterize [S]. This characterization can he' used to obtain an explicit description of a (n arbitrary), where the elements of [S] as the finite products ala2 335 I Subgroups 8.4 a; C S or a; is the inverse of an element of S. To prove this assertion, let H be the set of such products. In view of Theorem 4.1, H is a of G and, clearly, H Q S. If K is a subgroup of G that subgroup includes S, then K contains each member and the inverse of each member of S. Hence, K Q_ H. Thus H satisfies the properties which characterize (S], whence [S] = H. The subgroup generated by the unit set (a} will be called the subgroup generated by a and symbolized by [a]. It consists of all integral powers of a; a° is the unit element and a '° is the inverse of a'. The group [a] is commutative since ama" = a'+n = a"am. A group C is called a cyclic group if there exists an element a of C such that C = [al. For example, the additive group of integers, (Z, +, 0), and the additive group of integers modulo r, (Zr, +, 0), are cyclic groups; the first |
is generated by I and the second by 1. The multiplicative group (ZD,, 1) is also cyclic, but to prove this requires a few facts, exhaust from number theory. The cyclic groups Z and ZT, r = 1, 2, the collection of all essentially different cyclic groups in a sense which we now explain. An isomorphism of a group G onto a group G' is a one-to-one map- ping f on G onto G' such that for all x and y in G, f(xy) = f(x)f(y) where, on the left, the operation in G is in force while on the right it is that in G'. Thus, a one-to-one mapping on G onto G' is an isomorphism if the image of a product is the product of the images. If there exists an isomorphism f of G onto G', then G' is called an isomorphic image of G. In this event it is clear that f'is an isomorphism of G' onto G so that if G' is an isomorphic image of G, then G is an isomorphic image of G'. We say then that G and G' are isomorphic groups. For example, the mapping f : R+ -} R where f (x) = logio x is well known to be one-to-one and onto and, since log10 Ay = logio x + logio y, it is an isomorphism of the multiplicative group of positive real numbers onto the additive group of real numbers. Isomorphism is an equivalence relation on any collection of groups and, from the standpoint of group theory, members of an isomorphism-equivalence class are indiscernible., yield all The sense in which the cyclic groups Z and Z, r = 1, 2, cyclic groups can be inferred from the following theorem. 336 Several Algebraic Theories I CHAP P. 8 THEOREM 4.4. An infinite cyclic group is an isomorphic image of the additive group of integers and a cyclic group of order r is an isomorphic image of the additive group of integers modulo r. If C = [a] is a cyclic group, then the mapping f: Z -} C such Proof. that f(n) = a" is onto C. If it is not one-to-one, then ar = as for some |
distinct pair of integers r and s. We may assume that r > s. = 1, so there exists a positive integer p such that ap = 1. Then ar Let n be the smallest positive integer such that all = 1. Then 1 = a0,., a"-1 are distinct from each other, since ar = as with 0 < s, r < n a, implies that ar = I with 0 < r - s < n, which contradicts the choice of n. Moreover, all distinct powers of a appear among a0,, a"-'. For since any integer m can be written in the form a, m = nq + r, 0<r<n, we may conclude that am = an4+r = (an)9ar = ar. Thus, if f is not one-to-one, then C has finite order. It follows that K C has infinite order, then f is one-to-one and onto C. Finally, since f(m + n) = am+n = f(m)f(n), we have shown that an infinite cyclic group is an isomorphic image of Z. Next assume that C = [a] has order r. According to the preceding part of the proof, r is the least positive integer such that ar = 1 and C = 11, a,, ar-' 1. It is left as an exercise for the reader to com= plete the proof by proving that C is an isomorphic image of Zr. The notion of a cyclic group provides one means of classifying the elements of any group G. If a C G, then a is of infinite order or finite order r, according as [a] is infinite or is finite of order r. In the first 1 if n is any nonzero integer; in the second case, ar = I and case, a" r is the least positive integer such that ar = 1. By virtue of the simplicity of cyclic groups it is possible to determine all subgroups of a cyclic group in a straightforward way. We discuss this next. Let C = [a) and let H be a subgroup different from 111. Then H contains a power am of a, where m 0 0. Since, if am C H, then a m C H, it follows that there exists a positive integer m such that am C H. Let s be the smallest positive integer such that as C H. We shall show that H = [as] |
and that the mapping g on the set of all sub- 8.4 Subgroups 337 groups H 0 [ 11 into _Z+ such that g(H) = s is one-to-one. To prove the first assertion let am be any element of H and write m in the form m=sq+u, 0<u<s. am(al)_Q C H, and hence, by the minimality of s, u = 0. Then a" = Thus am = (a')°. Since; on the other hand, any power of all is in H, H = [as]. That g is one-to-one is clear, because if g(H) = s = g(H'), then H = [as] = H'. To complete the investigation of the subgroups of C = [a), we consider separately the cases where C has infinite order and has finite order. If C is infinite, then the mapping g is onto Z+, because if s C Z+, then g [a' ] = s, since the smallest positive power of a in [a-') is s itself. If C has finite order r, then g is onto the set of positive divisors of r which are less than Y. To prove this we observe that 1 = a' E H and then repeat an argument used above to conclude that r is a multiple of s; that is, s divides r. On the other hand, let s be any positive divisor of r which is less than r. If r = st, then (a')' = 1 and (a')" 0 1 if o < t' < t. Hence t is the order of [as]. If g[a'] = s', then [as"] = [a'], and hence [as"] has order t. It follows in turn that as" = 1, s't > r = st, and s' > s. Since s' < s by the definition of s, we have s' = s. If C is infinite, the one-to-one correspondence g can be extended to one between the set of all subgroups and the set of natural numbers by choosing 0 as the image of [ 1). If C has finite order r, then g has a corresponding extension whose range is the set of all positive divisors of r upon choosing r as the image of [ 11. In the finite case, if H corresponds to s, so that H |
= [as], then the order of H is r/s. Hence another oneto-one correspondence between the subgroups of C and the positive divisors of r results if with each subgroup we associate the order of that subgroup. We summarize our results in the next theorem. THEOREM 4.5. A subgroup H of a cyclic group C is cyclic. If C = [a] and H 7-1 111, then H = [as], where s is the least positive integer such that a' C H. If C is infinite, then the subgroups [as] of C are in one-to-one correspondence with the set of natural numbers. If C is finite of order r, its subgroups are in one-to-one correspondence with the positive divisors of r. Alternatively, in the finite case the order of a subgroup is a divisor of r.and corresponding to each divisor t of r there is exactly one subgroup of order t; it is generated by a'". A subgroup of the group of one-to-one transformations on a set A is called a transformation group on A. Since the theory of groups had 338 Several Algebraic Theories I C H A P. 8 its origin in the study of certain groups of this type, a representation problem arises : Is every group isomorphic to a transformation group? This question has an affirmative answer, which was first supplied by Cayley. We state it as our next theorem. THEOREM 4.6. For every group G there is an isomorphic trans- formation group. Proof. As the set on which the transformations shall be defined, we take the set G itself. Consider the mapping ta: G -'- G defined by the group element a as ta(x) = ax for all x in G. Since the equation ax = b has a solution in G for given a and b in G, this map is onto G. Since the cancellation laws hold, to is one-to-one. Thus, to is a member of the group of one-to-one transformations on the set G. We show now that { tala E G } is a transformation group L on G. Since Q. ° tb)(x) = ta(tb(x)) = ta(bx) = a(bx) = tab(x), to o tb = tab and L is closed. Further, t.-' C L |
, since it is easily shown that to 1 = to Hence L is a group by Theorem 4.1. Next we prove that L is an isomorphic image of G under the correspondence a -'-.. By definition of L, this map is onto L. It is oneto-one since, if a and b are distinct elements of G, then al 0 bl, and hence to 0 tb The validity of the relation to G tb = tab completes the proof. EXERCISES 4.1. Prove the Corollary to Theorem 4.1. 4.2. Find two proper subgroups of each of the groups defined in Exercise 3.1. 4.3. Prove Theorem 4.3. 4.4. Complete the proof of Theorem 4.4. 4.5. Let G be the subset of the set A in Exercise 3.2, consisting of those map- pings with a= f 1 and b C Z. (a) Show that G determines a subgroup of A. (b) Is G Abelian? Is G cyclic? (c) Determine the orders of [1, 1) and [-1, -1 ]. (d) Specify all values of a and b fqr which [a, b] is a member of H, the sub- group of G that is generated by [1, 2] and [-1, 0]. (e) Specify two members of G which, taken together, generate G. 8.5 1 Coset Decompositions and Congruence Relations 339 4.6. Show that a group of even order has an odd number of elements of order 2. 4.7. Show that if a, b and ab are group elements each of order 2, then ab = ba. 4.8. Prove that if a and b are elements of a group, then ab and ba have the same order. 4.9. Let a and b be elements of a group such that ba = ambn for integers m and n. Show that the elements ambn-Q, am_sb", and ab-' have the same order. 4.10. Let a and b be elements of a group such that b-'ab = ak for some in- teger k. Show that b-'a'br = a°k'. 4.11. Show that in an Abelian group the product of an element a of order n and an element b of order |
m is an element of order mn, provided that m and n are relatively prime. 5. Coset Decompositions and Congruence Relations for Groups Let G be a group and H a subgroup. A subset of G of the form {gh1h C H), where g is a fixed element of G, is abbreviated to gH and called a left coset of H in G. Left cosets, along with their "right" analogue, are distinguished types of subsets of a group, as we shall show. Their basic properties include the following. (I) For any subgroup H of 0, each element of G is a member of a left coset of H. Two left cosets of H are either disjoint or equal. (II) All left cosets of H have the same cardinal number as the set H. To prove (I) we observe first that, since the unit element I of G is in H, an element g of G is a member of the left coset gH. Next, suppose that two cosets aH and bH have a common element c. Then c = ah, = bh2, and hence a = bhs, where h8 C H. Hence ah C bH for all h in H, which means that aH a bH. Reversing the roles of a and b gives bH S aH and hence aH = bH. Property (II) is established by the mapping h --i- gh on H into gH. From (I) it follows that there exists a family {g;Hhi C I) of left cosets of H that is a partition of the set G. This is the left coset decomposition of G modulo H. Clearly the set G is the union over a left coset decomposition of G. The cardinal number of the left coset decomposition of G modulo H is the index of H in G, symbolized (G: H). In view of (II) we have the following relation among the cardinal numbers and (G: H) : 0 = (G: H) I. 340 Several Algebraic Theories I CHAP. 8 Now the cardinal number of any group G may be written as an index, indeed (G: { 1 D. This is usually shortened to (G: 1); With this notation the above relation may be written as (G: 1) = (G: H) ( |
H: 1). It is left as an exercise to prove the following generalization : If G is a group, H is a subgroup of G, and K is a subgroup of H, then K is a subgroup of G and (G: K) = (G: H) (H: K). If G is a finite group of order n and H a subgroup of order m, we have n = (G: H)m, which implies that m divides n. This is a famous result due to Lagrange. We state it along with two immediate consequences as our next theorem. THEOREM 5.1. The order of a subgroup of a finite group divides the order of the group. COROLLARY 1. The order of an element of a finite group divides the order of the group. COROLLARY 2. A group whose order is a prime is cyclic. If G is a group and H a subgroup, then a subset of G of the form {hgIh E H} where g is a fixed element of G is abbreviated to Hg and called a right coset of H. Properties (I) and (II) above hold for right cosets. The family {Hg;l j C J} of right cosets of H that is a partition of G is the right coset decomposition of G modulo H. It is left as an exercise to show that the set of inverses of the members of a left coset of H is a right coset of H and that, consequently, the left and right coset decompositions of G modulo H are similar sets. Therefore the index (G: H) can also be determined from the right coset decomposition. For later applications we introduce some notation which extends that used for cosets. Let A and B be subsets of a group G. By AB we shall mean labia C A and b C B. If one of these subsets, for instance A, is simply { a }, then we shall write aB instead of { a } B. The extension of this notation to more than two subsets is clear. In additive notation we shall write A + B in place of AB. In particular, a left coset modulo a subgroup H will be written as a + H and a right coset as H + a. 8.5 I Coset Decompositions and Congruence Relations 341 EXAMPLES |
5.1. Referring to the group Fwhose multiplication table is given in Example is a subgroup. The left coset decomposition of F modulo H is 3.4, H {H, f2H, faH) _ { {fl, f4}, {fs, f6}, if., f6) ) and the right coset decomposition modulo H is {H, Hfz, Hfa) = { {h, f4}, {f2, ffi}, {f3, f6} }. It should be observed that these are different partitions of F. In a commutative group, the left cosets and right cosets of a subgroup are identical, of course. For example, in (Z,2, +, 0) the left and right coset decomposition modulo the subgroup H= {0,4,$} is {H,1 +H,2+H,3+H}. 5.2. In the multiplicative group C* of nonzero complex numbers rei6 (r > 0, 0 real), the subset R+ of all positive real numbers is a subgroup. The coset decomposition of C* modulo R+ can be described geometrically as the collection of rays, with initial point deleted, issuing from the origin in the complex plane. If instead of R+ we start with the subgroup U of all complex numbers such that r = 1, then the coset decomposition of c* modulo U can be described geometrically as the collection of all circles with positive radii and centered at the origin in the complex plane. Given a group G and a subgroup H, let 0 be the equivalence relation on G corresponding to the left coset decomposition of G modulo H. Thus, by definition, a B b if a and b are in the same left coset of H or, what is easily proved to be the same, if a 'b E H. The relation 0 has the further property that a Ob implies that ca 0 cb for all c in G. That is, 0 satisfies one of the two requirements [see (C1) in Section 11 for a congruence relation on G. We shall call 9 a left congruence relation on this account. How left congruence relations on G and subgroups of G are related is described next. LEMMA 5.1. Let G be a group and |
B be a left congruence relation on G. Then H=Ix E G(x 0 l} is a subgroup of G and a 0 b iff a 'b C H (or, alternatively, if a and b are members of the same left coset of H). Conversely, if H is a subgroup of G, then the relation B such that a 0 b if a -'b C H is a left congruence relation on G. The correspondence of subgroups to left congruence relations is a one-to-one correspondence between the set of left congruence relations on G and the set of subgroups of G. Proof. Let 0 be a left congruence relation on G and consider H = {x E Gjx 6 11. Since 1 E H, this set is nonempty. Assume that a, 342 Several Algebraic Theories i CHAP. 8 b E H. Then b 0 1 and hence ab 0 a. Since a 0 1 and 0 is transitive, it follows that ab 0 1, whence H is closed. If a C H, so that a 8 1, then a-'a 8 a-', whence a' 0 1 or a' C H. Therefore H is a subgroup. Next, if a 0 b, then, in turn, a -'a 0 alb, a 'b 0 1, a -'b C H. Each of these steps is reversible, so that a 0 b if a-'b C H. Turning to the converse, the fact that a 'b E H if a and b are in the same left cosct of H, coupled with the fact that the left coset decomposition of G modulo H is a partition of G, implies that the relation 0 defined in the lemma is an equivalence relation on G. That, in addition, a 0 b implies ca 0 cb, is a consequence of the identity a 'b = (ca)-'(cb). The proof of the last assertion of the lemma is left as an exercise. The preceding lemma has an analogue for right congruence relations (that is, equivalence relations 0 such that if a 0 b then ac 0 bc) for a group G. They determine and are determined by right coset decompositions of G modulo subgroups H of G. Now let 8 be a congruence relation on G (that is, simultaneously a left and right congru |
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