text stringlengths 270 6.81k |
|---|
ence relation). As a left congruence relation, 0 determines a subgroup H of C such that the equivalence class determined by an element g in G is gH. As a right congruence relation, 8 determines the same subgroup H (note that II is defined independently of left congruency) and the equivalence class determined by g is Hg. Hence, for all g in G, Jig = gH or, what is equivalent, g'Hg = H. A subgroup II of C such that g -111g = H for all g in G is a normal or invariant subgroup of G. Thus a congruence relation 8 on a group G determines a normal subgroup H of C. Indeed, from Lemma 5.1 it is immediate that the congruence relations on G are in one-to-one correspondence with the normal subgroups of C. If to the congruence relation 0 on G corresponds the normal subgroup H of G, it is customary to denote the quotient set G/8 by G/H. We shall do this. Further, we shall often write the element gH of G/II as ff. We already know (see Section 1) that an operation is defined by G/II by the rule and proceed to show that (G/H, -, 1) is a group, the quotient or factor group G modulo H. The associativity of the operation in C/II is inherited from that of the operation in G, the element 1 is clearly an identity clement, and, finally, a-'-' is a solution of the equation TO = I. The operation in G/H admits of an alternative description. Goscts of H 8.5 I Coset Decomposition and Congruence Relations 343 are subsets of G, and hence can be composed using the operation in G as described prior to Example 5.1. With H normal the product (aH) (bH) is equal to abH, as the reader can prove. But the element abH of G/H is the product of the elements aH and bH of G/H. Thus, the operation in G/H may be interpreted as one for (restricted) subsets of G. EXAMPLES 5.3. Suppose that C is an additive commutative group. Then our foregoing results take the following form. If 0 is a congruence relation on G, then |
H = {a C Gja0O} is a'subgroup of G and aOb if -a + b (or, equivalently, a - b) is in H. Conversely, if His a subgroup of G, then the relation 0 such that aOb if a - b C H is a congruence relation on C. In the quotient group G/H (or what is often called the difference group, G - H, in this case) the operation reads (a+H)+(b+H) = (a+b)+H. 5.4. To assist the reader in acquiring familiarity with the additive notation introduced in the preceding example, we reestablish the fact that 0) (see Example 3.2) is a group. The normal subgroup corresponding to congruence modulo n in the additive group of integers is the cyclic group [n]. Its cosets are [n],1+[n],...,(n-1)+[n] and these are the elements of (Z/[n], +, [n]). 5.5. It is left as an exercise to show that the intersection of a collection of normal subgroups of a group is a normal subgroup. For a group G we may then define the normal subgroup generated by a subset S as the intersection of all the normal subgroups that include S. It is left as another exercise to prove that the normal subgroup generated by S is the subgroup generated by the subset T of G consisting of all elements of the form g'sg for some g in G and some s in S. To describe the relationship of a quotient group G/H to G, a definition is needed. A homomorphism of a group G onto a group G' is a mapping f on G onto G' such that for all x and y in G, f(xy) = f (x)f (y). That is, a homomorphism onto differs from an isomorphism onto only in that a homomorphism need not be one-to-one. If there exists a homom )rphism of G onto G', then G' is called a homomorphic image of G. By virtue of the definition of the operation in a quotient group it is clear that if G is a group and G/H a quotient group, then G/H is a homomorphic image of G under the natural mapping on G onto G/Hthat is, |
the mapping p on G onto G/H such that p(x) = xH. We con- 344 Several Algebraic Theories I CHAP. 8 sider next the converse situation. Let G' be a given homomorphic image of G and f the accompanying homomorphism. Then the equivalence relation 0 on G associated with f, namely, a 0 b iff f(a) = f(b), is a congruence relation on G. The corresponding normal subgroup K of G, namely, {a C Gjf (a) = 11, is called the kernel of the homomorphism f. The quotient group G/K is isomorphic to G'. Indeed, the relation g, which we define as is a function on G/K onto G' such that {(x, f(x))Ix C G/K}, g(xy) = g( ) = f(xY) = f(x)f(Y) = g(x)g(y) That is, g is an isomorphism. Further, if p is the natural mapping on G onto G/K, then f = g c p. That is, any homomorphic image of a group G can be duplicated to within an isomorphism by some quotient group of G. We state our results in our next theorem. THEOREM 5.2. If G is a group and K a normal subgroup, then the quotient group G/K is a homomorphic image under the natural mapping on G onto G/K. Conversely, if the group G' is a homomorphic image of G, then those elements which are mapped onto I determine a normal subgroup K of G and G/K is isomorphic to G'. If f : G - G' is the given homomorphism, then f = g o p where p is the natural mapping on G onto G/K and g is an isomorphism of G/K onto G'. EXAMPLES 5.6. We illustrate the above theorem by using it to derive again Theorem 4.4 concerning the structure of cyclic groups. Let G be a multiplicative cyclic group generated by a. The mapping m a°' is a homomorphism of the additive group of integers onto G. Hence C is isomorphic to Z/K, where K is the kernel of the homomorphism and, in particular, a subgroup of Z. Now |
it is easily proved that the only subgroups of Z are the cyclic groups [n]. If K = [0], then m -} a'" is an isomorphism and G is isomorphic to Z. Otherwise G is isomorphic to Z/[n], a cyclic group of order n. It follows immediately that two cyclic groups are isomorphic if they have the same order. For this reason it is common to speak of "the" cyclic group of infinite order and "the" cyclic group of order n. 5.7. Every subgroup of a commutative group is normal and consequently determines a quotient group. Thus, the subgroup R+ of all positive real numbers of the multiplicative group C* pf nonzero complex numbers determines a quotient group; C*/R+ is isomorphic to the additive groups of real numbers. Again, the quotient group of C* modulo U, the subgroup of complex numbers 8.5'Coset Decomposition and Congruence Relations 345 of absolute value 1, is isomorphic to the multiplicative group R* of nonzero real numbers. 5.8. We note that with the above theorem a homomorphism can be shown to be an isomorphism by proving that its kernel is {1}. 5.9. If f : G -+- G' is a homomorphism of the group G onto the group G', then f(l) = 1', the identity element of C' and f(a ') = (f(a))-'. 5.10. Suppose that G is a group, G' is a set in which a binary operation is defined, and f is a mapping on G onto G' such that f(ab) = f(a)f(b). Then G' is a group. EXERCISES 5.1. Verify the relation (G: K) = (G: H)(H: K), given in the text. 5.2. Establish the two Corollaries to Theorem 5.1. 5.3. Prove the assertion made in the text that if G is a group and H is a subgroup, then there exists a one-to-one correspondence between the left coset decomposition of G modulo H and the right coset decomposition of G modulo H. 5.4. Let G be a group and H and K be subgroups of finite orders. Show that if these |
orders are relatively prime, then H (1 K = {1}. 5.5. Let G be a group having H and K as subgroups. Show that any left coset of H (1 K is the intersection of a left coset of H and one of K. Use this to deduce that if H and K have finite index in G then so has H (l K. 5.6. Let H and K be two finite subgroups of a group G. Show that the subset HK of G contains precisely (H: 1)(K: 1)/(H (1 K: 1) distinct elements. 5.7. Let G be a group having H and K as subgroups. Show that HK is a sub- group iff HK = KH..5.8. Supply the missing part of the proof of Lemma 5.1. 5.9. Let G be a group and H a subgroup. Under what circumstances is xH - - Hx a mapping on the left cosets of H onto the right cosets of H? 5.10. Show that if for a subgroup H of a group G, g -'Hg C H, for all g in C, then H is normal in C. 5.11. Show that if H is a subgroup of a group G, then g'Hg, for g C G, is a subgroup isomorphic to H. Let N = n {g'Hg!g C G} and show that N is a normal subgroup of C, indeed the largest normal subgroup of G included in H. then 5.12. Prove that if H is a normal subgroup of a group G, (aH) (bH) = abH. 5.13. Establish the assertions made in Example 5.5. 5.14. Let if be a collection of distinct subsets Si of a given group G with the following properties. (a) Every element of G is in at least one Si. (b) No Si is a proper subset of an S,. (c) The product of any two members of if is included in a member of if. Show that if is the coset decomposition of a normal subgroup of G. 346 Several Algebraic Theories I CHAP. 8 5.15. Let C* be the set of nonzero complex numbers z = re2:ir, U be the set of complex numbers of absolute value |
1, and R, C, and R* have their usual meanings. Investigate each of the following mappings-deciding which are homomorphisms, which are isomorphisms, and so on., 1) - - (R, +, 0) (a) f : (R+, (b) f: (C*,, 1) -} (U,, 1) (c) f : (R, +, 0) -*- (U,, 1) (d) f: (C*,, 1) ->- (R+,, 1) 5.16. If G is a group, elements of the form x.ty =1xy are called commutators. Prove that the subgroup C generated by the set of all commutators of G is a normal subgroup, that G/C is Abelian, and, if N is any normal subgroup of C such that GIN is Abelian, then C C N. where f (x) = In x, wheref(z) = eYrip, where f (,p) = esrip, wheref(z) _ IzI*' 6. Rings, Integral Domains, and Fields A ring (with identity element) t is an ordered quintuple (R, -}-,, 0, 1), are binary operations in R, 0 and I are dis- where R is a set, + and tinct members of R, and the following conditions are satisfied. Rt. ring). R2. (R, +, 0) is a commutative group (the additive group of the (R,, 1) is a semigroup with identity element (the multi- plicative semigroup of the ring). Rs. The- following distributive laws hold : a(b + c) = ab + ac, (b + c)a = ba + ca. EXAMPLES 6.1. The statement that (Z, +,, 0, 1) is a ring summarizes many of the basic properties of the system of integers. To be precise, it is a concise formulation of properties (1)-(5), (7), and (8) in Theorem 3.3.1 of this system. 6.2. The system of rational numbers and that of the real numbers provide further models of the theory of rings. 6.3. The set Z[V'5] of all real numbers of the form m + |
n1'5, where m, n C Z, together with the familiar operations and 0 and 1, is a ring. 6.4. (Zr, +,, 0, 1) (see Examples 3.2 and 3.3) is an example of a finite ring, that is, a ring such that the basic set has a finite number of elements. t The usual definition of a ring does not require the existence of an identity element. However, since those rings which interest us have an identity element, we have incorporated this requirement into our definition at the outset. The assumption that 0 and I are distinct elements of R serves to rule out the extreme and trivial case of a ring such that R consists of a single element. 8.6 Rings, Integral Domains, and Fields 347 6.5. If (B, (, ', 0, 1) is a Boolean algebra, then it is possible to introduce operations in B such that the resulting system is a ring. For addition in B we choose the symmetric difference operation; that is, if a, b C B, we define a + b = (a ( b') U (b ( a'). For multiplication in B we take (1 and henceforth use the customary ring notation ab for a (l b. Then (B, +,, 0, 1) is a ring. The reader is asked to prove this and derive properties of such a ring in the exercises for this section. Many of the computation rules of ordinary arithmetic carry over to arbitrary rings. First of all, the definitions and properties in Section 2 pertaining to powers of an element and those pertaining to multiplication apply to the additive group and the multiplicative semigroup, respectively, of any ring. In addition to the earlier rules for multiples we have the rules (1) n(ab) = a(nb) = (na)b. These follow from the general distributive laws aE;- ibi = E;-iabj, (X:=i b:) a which, in turn, are easily proved by induction. We call attention to the fact that the multiple na of a ring element a should not be confused with a ring product. However, since we are assuming that a ring always has an identity element, we can write na = la + la + and the last is a product. + 1a(n summands) = (1 + 1 + + 1)a = (nl)a The distributive laws hold |
for subtraction in a ring: (2) a(b - c) = ab - ac, (b - c)a = ba - ca. To prove, for example, the first of these, we must show that a(b - c) + ac = ab. But this follows directly from the first distributive law in Ra, since (b - c) + c = b. For b = c identities (2) yield the following important properties of the ring element 0: (3) aO = Oa = 0, for all a in R. In particular, Oa is equal to the ring element 0 whether "0" in Oa is the ring element or the natural number zero. If in (2) we set b = 0, we get a(-c) = -ac, (-c)a = -ca, 348 Several Algebraic Theories I CHAP. 8 and if in the first of these identities we replace a by -a we obtain (-a)(-c) _ -(-a)c = (- -a)c, whence (-a)(-c) = ac. An element a of a ring Rt is called a left (or right) zero-divisor if there exists in R an element b 0 0 such that ab = 0 (or ba = 0). By (3) the element 0 is both a left and right zero-divisor, since by assumption our rings contain more than one element. A proper zerodivisor is a zero-divisor which is different from 0. A ring has a proper zero-divisor if it contains a pair a, b of nonzero elements such that ab = 0. We shall say that a ring is without zero-divisors if it has no proper zero-divisors. Since an element of the ring R is an element of the semigroup (R,, 1), the definition of an inverse of a ring element is at hand. A ring element is called a unit if it has an inverse. According to Section 2, if a has an inverse, it is unique; the inverse of a will be denoted by a-'. Again according to Section 2, if a and b are units then also a-' and ab are units, which implies that the set of units of a ring form a group. The element 0 is not a member of the group of units of a ring since for every element a in R,aO |
=Oa=0F& 1. Various specialized types of rings are obtained by imposing conditions on the multiplicative semigroup at hand. For example, a ring is said to be commutative if its multiplicative semigroup is commutative. A commutative ring R (with identity element) having no proper zerodivisors is called an integral domain. The latter condition means simply that the set R* of nonzero elements of R is closed under multiplication. A ring R is called a division ring (or skew field) if R* is closed under multiplication and (R*,, 1) (where now the domain of is restricted to R* X R*) is a group. Finally, a division ring is called a field if multiplication is a commutative operation. Referred back to the definition of a ring, the field (R, +, -, 0, 1) is a ring such that the set R* = R - {0} is closed under multiplication and (R*,, 1) is a commutative group. EXAMPLES 6.6. For any ring R we now define the ring Rc2) of 2 X 2 matrices with ele- ments in R. The elements of R(2) are all arrays or matrices (a) _ (au al) ali a22/ f Henceforth we shall often call the ring (R,, -h, 0, 1) simply "the ring R." 8.6 I Rings, Integral Domains, and Fields 349 of two rows and columns with elements a;; in the ring R. The element a;; located at the intersection of the ith row and jth column of (a) will be called the i,j-element of (a). Two matrices (a) and (b) are defined to be equal if aq = b11 for all i and j. Addition of matrices is defined by the formula au azj aiz1/ ass/ /(bit \b21 b1z _ all + bu a1 + b12 bzz) azl + bat 1722 + b22 It is easily proved that (R(2), +, 0), where 0 is the matrix all of whose elements are 0, is a commutative group. The negative of (a) is the matrix having -a;; as its Q -element. Multiplication of matrices is defined by the formula (au a12\ (b11 |
azz/ \b21 \a21 b12\ = anbll + alzbzl as,bnl + azzb21 bzz/ anb12 + alzbzz azlblz + azzbn>. That is, the i, j-element of the product is the sum of the products of the elements of the ith row of (a) and the corresponding elements of the jth column of (b). The matrix (1 0) `0 1 is an identity element for this operation and (R(2),, 1) is a semigroup (see Exercise 2.4). Further, the distributive laws R3 hold, so Rc2 is a ring. This ring is not commutative, since 1(0 00)(0 0)=(0 0) and (0 0) (0 0O)=(O 0)' The second equation exhibits two proper zero-divisors in R(z). 0 6.7. A characterization of integral domains among commutative rings can be given in terms of the (restricted) cancellation law for multiplication: ac = be and c 0 0 imply that a = b. Indeed, if for elements a, b, and c in any ring without zero-divisors, ac = be and c # 0, then (a - b)c = 0 where c 0 0. It follows that a - b = 0, whence a = b. Conversely, if the above cancellation law holds in a ring, then ab = 0 and b 0 0 imply that ab = Ob and b 0 0, whence a = 0. In summary, a commutative ring is an integral domain if the cancellation law for multiplication holds. The system of integers is an integral domain. This statement summarizes parts (1)-(9) of Theorem 3.3.1. 6.8. A ring R such that every element is idempotent (a'- = a) is commutative and each element is equal to its negative. To prove this we notice that for all elements a and b of such a ring a+b = (a+b)z=az+ab-l--ba+bz=a+ab+ba+b, whence (4) ab + ba = 0. 350 Several Algebraic Theories I CHAP P. 8 Setting b = a in this identity yields the identity a$ + a$ = 0. Since as = a, it follows that a + a = |
0 or, in other words, each element is its own negative. In particular, the negative of ab is ab and this fact, together with (4), implies that ab = ba, thereby completing the proof. 6.9. The ring Z. of integers modulo n is a field if the modulus is a prime (see Example 3.3). 6.10. According to Theorem 3.4.1 the system of rational numbers is a field. This is a restatement of parts (1)-(10) of that theorem. According to Theorem 3.6.1, the system of real numbers is a field. EXERCISES 6.1. Show that in the definition of a ring R (with identity element 1) the requirement that 0 0 1 may be replaced by the requirement that R contain an element different from 0. 6.2. Prove that the set Z[V'5J of all real numbers of the form a + b/ where a, b E Z, together with addition, multiplication, 0, and 1, is a ring. 6.3. Which of the following sets, together with addition, multiplication, 0, and 1, is a ring? (a) The set of all real numbers of the form a + (b) The set of all real numbers of the form a + b/ +cY where a, b, b E Z. cCZ. (c) The set of all rational numbers which can be expressed in the form m/n, where m is an integer and n is a positive odd integer. 6.4. Suppose that (R, +,, 0, 1) is a ring and that in R we introduce new operations 0 and 0 by way of the following definitions. aO+ b=a+b - 1,aOb=a+b-ab. Show that (R, O, 0, 1, 0) is a ring. Describe the ring which results from this ring if new operations are introduced in R by repeating the same definitions. 6.5. Referring to Example 6.5, prove that (B, +,, 0, 1) is a ring, all of whose elements are idempotent. 6.6. By a Boolean ring is meant a ring (with identity), all of whose elements are idempotent. According to Exercise 6.5, a Boolean algebra determines a Boolean ring. Using the results in Example 6.8 |
, show that, conversely, a Boolean ring determines a Boolean algebra upon defining a U b = a + b + ab, anb = ab. Further, show that the processes of deriving a Boolean algebra from a Boolean ring and of deriving a Boolean ring from a Boolean algebra are inverses of each other. Thereby a one-to-one correspondence between Boolean algebras and Boolean rings is established, a result which was first proved by Stone (1936). 6.7. Prove that a finite integral domain is a division ring. 8.7 ( Subrings and Difference Rings 351 6.8. Referring to Example 6.6, prove that if R is a commutative ring and (a)(b) = 1 for (a), (b) C Rte>, then (b)(a) = 1. 6.9. We assume it known that the set Q of complex numbers forms a field. Show that the set of all matrices of Q(2) having the form ( ab al where X is the complex conjugate of x, forms a division ring which is not a field. 6.10. If a is a ring element, then an element b of that ring, such that ab = 1, is called a right inverse of a. Prove that the following conditions on a are equivalent. (a) a has more than one right inverse. (b) a is not a unit. (c) a is a left zero-divisor. 6.11. Prove that if a ring element has more than one right inverse, then it has infinitely many. (Hint: Consider the set of ring elements b + (I - ba)a", where ab = I and n = 0, 1, 2,..) 6.12. Prove that a ring R is an integral domain if for all a, b, and c in R and b -' 0, ba = cb implies that a = c. 7. Subrings and Difference Rings A ring S is a subring of a ring R if S C R and the restriction of addition and multiplication in R to S X S are equal, respectively, to addition and multiplication in S. Having chosen to restrict our attention to rings with an identity element, we shall insist further that a subring S of a ring R contain an identity element. It follows that if a subset S of a ring R is a subring, then (sec the Corollary |
to Theorem 4.1) it must satisfy the following conditions. (i) If a, b C S, then a - b C S. (ii) If a, b C S, then ab C S. (iii) There exists an element 1, in S such that 1,x = A. = x for all x in S. Conversely, it is clear that these conditions are sufficient to insure that a subset S of a ring R form a subring. It is possible for the identity element 1, of a subring S to be different from the identity element 1 of R (see Example 7.4 below). In that event 1, is a zero-divisor of R. For by assumption there exists in R an element a such that 1,a = b 96 a. Since 1,b = 1,(1,a) = 1,a = b, 352 Several Algebraic Theories I CHAP P. 8 it follows that l,a = 1,b, and hence l,(a - b) = 0. Thus, 1, is a (proper) zero-divisor of R since a 54- b. As a corollary there is the fact that if R is an integral domain, then the identity element of a subring is necessarily the identity element of R. A field S is a subfield of a field F if S C F and the restriction of addition and multiplication in F to S X S are equal, respectively, to addition and multiplication in S. Since a field is an integral domain, the identity element of S is the identity element of S. This also follows from the fact that the multiplicative group of S must be a subgroup of the multiplicative group of F. This condition, together with the condition that S be a subgroup of the additive group of F, characterizes the notion of a subfield. Hence, S is a subfield of F if the following conditions hold. (i) a, b C S imply that a- b C S. (ii) a, b C S and b s' 0 imply that ab-I E S. EXAMPLES 7.1. The set of all matrices in R(2 (see Example 6.6) of the form C 0\ determines a subring of RO). 7.2. The field of rational numbers is a subfield of the field of real numbers. 7.3. The intersection of any nonempty |
collection of subfields of a field F is a subfield of F. 7.4. Let A and B be rings with identity elements IA and IB, respectively, and let R be the set of all ordered pairs (a, b) where a C A and b C B. We define operations in R as (a, b) + (a', b') _ (a + a', b + b'), (a, b)(a', b') _ (aa', bb'). It is an easy calculation to prove that R is a ring having (1A, IB) as identity clement. Further it is clear that RA = {(a, 0)Ia C A} is a subring of R having (1A, 0) as identity element. Thus the identity element of RA is distinct from that of R. The definition of a congruence relation for an algebra (Section 1) takes the following form in the case of a ring. A congruence relation 0 on a ring R is an equivalence relation on R such that for all a, b, and c in R, a 0 b implies that c + a,O c + b, (Cm1) a 0 b implies that ca 0 cb, (Cmr) a 0 b implies that ac 0 bc. 8.7 Subrings and Difference Rings I 353 The right-hand analogue of (C,) is superfluous since addition is cornmutative. The condition that multiplication preserve equivalent elements has been written in two parts for easy reference. Now (Ca) means that 0 is a congruence relation on the group (R, +, 0). Hence, (i) 0 determines (and, is determined by) the subgroup S = { s C Ris 0 0 } of R (see Example 5.3), (ii) a 0 b if a - b C S, and (iii) 0-equivalence classes are left (= right) cosets a + S of S in R. Addition can be defined in R/S in terms of representatives [that is, (a + S) + (b + S) = (a + b) + S] and (Cmt) and (Cmr) are additional necessary and sufficient conditions that multiplication can be defined similarly. Let us translate these into conditions for S. From (Cmi) we infer that if r E R and s C S, then rs C S, since s C S means s |
0 0, and hence rs 0 rO or rs 0 0. Conversely, if a subgroup S' of R has the property that r C R and s' C S' imply that rs' C S', then the additive congruence relation 0' which S' determines satisfies (Cmi), since if a 0'b then, in turn, a - b C S', c(a - b) C S', ca - cb C S', ca 0 'cb. Similarly, (Cmr) holds for a relation 0 which satisfies (CB) if the subgroup S corresponding to 0 has the property that r C R and s C S imply that sr C S. There follows the existence of a one-to-one correspondence between the congruence relations on R and the subgroups S of the additive group of R such that r E R and s C S imply that rs, sr C R. A subset S of a ring R such that S is a subgroup of the additive group of R and, for all r in R and s in S, both rs and sr are in S, is called an ideal of R. Thus, a nonempty subset S of R is an ideal if (i) sCSandICSimplythats - ICS, (ii) s C S and r C R imply that rs, sr C S. The results obtained above may now be summarized by the statement that the congruence relations on R are in one-to-one correspondence with the ideals of S. As one might suspect, ideals are the analogue for rings of normal subgroups for groups. Every ring R has at least two ideals, namely, the entire ring and {0}. The ideal R of R corresponds to the universal relation on R and the ideal 101 corresponds to the equality relation on R. EXAMPLES 7.5. If R is a commutative ring (with identity element) and a C R, then Ra = {ralr C R} is an ideal called the principal ideal generated by a. Since R = Rl and {0} = R0, both R and {0} are principal ideals. 7.6. A field F has only two ideals, F and {0}, for if I is an ideal of F and 354 Several Algebraic Theories ( CHAP. 8 I s {0}, then I contains a nonzero element a and hence I contains a -'a = 1, whence I = F. 7 |
.7. If a commutative ring R has only two ideals, then it is a field. For let a be a nonzero element of R and consider Ra. This principal ideal contains la = a and hence is different from {0}; hence, it is equal to R. But this implies that the equation xa = 1 has a solution for every a 0 0. 7.8. We recall that in Section 6.4 we defined the notion of an ideal of a Boolean algebra. It is left as an exercise to prove that the ideals of a Boolean algebra B coincide with the ideals of the corresponding Boolean ring B (see Exercise 6.6). Now we can get to the whole point of this discussion. Let R be a ring and S be an ideal of R which is distinct from R. Then we know that operations can be introduced in R/S, the collection of cosets a + S of S in R (that is, the 0-equivalence classes where 0 is the congruence relation corresponding to S) by the following definitions: (a+S)+(b+S) =(a+b)+S, (a + S) (b + S) = ab + S. Further, we know that (R/S, +, S) is a commutative group. Also, since S 0 R by assumption, 1+ S 0 S and 1+ S is an identity element for multiplication. Finally, it is a straightforward exercise to prove that (R/S, +,, S, 1 + S) is a ring, the so-called difference (quotient, residue class) ring of R modulo the ideal S. EXAMPLES 7.9. It is an easy matter to determine all ideals of the ring Z of integers. Since an ideal of Z is a subgroup of (Z, +, 0) it has the form [r], that is, the set of all multiples of r (see Section 4). But it is clear that each such subset is an ideal, indeed, the principal ideal Zr. (That is, Zr in ring notation is [r] in group notation.) Since Zr = Z(-r), it follows that Zr for r = 0, 1, 2, exhaust the ideals of Z. The difference ring Z/Zr is Z if r = 0. Since Z1 = Z we exclude the value I for r. If r > 2, Z/Zr |
has r elements 0=Zr,I =1+Zr,,1=r-1+Zr. This is the ring we denoted by Z. earlier. If r is a composite number, say r = mn with m > 1 and n > 1, then m # 0 and n s 0, but MR = i = 0. This shows that Z/Zr is not an integral domain if r is composite. On the other hand, if r is a prime then we know (see Example 3.3) that Z/Zr is a field. 7.10. Some properties of rings carry over to each of their difference rings. For example, if R is a commutative ring then R/S is commutative. But if R is an 8.7 I Subrings and Difference Rings 355 integral domain, then the same need not be true of a difference ring, as the preceding example shows. A homomorphism of a ring R onto a ring R' is a mapping f on R onto R' such that for all x and y in R, f(x + y) = f(x) +f(y'), f(xy) = f(x)AY) If there exists a homomorphism of R onto R', then R' is called a homomorphic image of R. A homomorphism of R onto R' which is one-toone is called an isomorphism and R' is called an isomorphic image of R. If f is an isomorphism of R onto R', then f-1 is an isomorphism of R onto R' and hence each ring is an isomorphic image of the other. In this event we shall refer to R and R' simply as isomorphic rings. By virtue of the definition of operations in a difference ring it is clear that a difference ring R/ S of a ring R is a homomorphic image under the natural mapping a --} a + S on R onto R/S. We go on to show next that, conversely, every homomorphic image of a ring R is isomorphic to a difference ring of R. Let f : R --} R' be a homomorphism of the ring R onto the ring R'. Then f is a homomorphism of the additive group R onto the additive group R', and hence (Theorem 5.2) if S is the kernel of f (thus S is the inverse image of the zero element of R'), |
f = g o p, where p is the natural map on R onto (the additive group) R/S and g is an isomorphism of R/S onto R' [indeed, g(a + S) = f (a) ]. The further property off, that it preserves multiplication, implies that S is an ideal of R. Indeed, if r E R and s C S, then f(rs) = f(r)f(s) = f(r)O' = 0', whence rs C S. Similarly, if r C R and s C S, then sr C S. Hence, g establishes R' as an isomorphic image of R/S. We summarize our results in the next theorem. THEOREM 7.1. The difference ring R/S of the ring R modulo the ideal S of R is a homomorphic image of R. Conversely, any homomorphic image of R is isomorphic to the difference ring R/S, where S is the kernel of the homomorphism regarded as a homomorphism of the additive group R. We conclude this section with the introduction of some terminology which will have applications later. A ring R is said to be imbedded in a ring S if S includes an isomorphic image R' of R. If R is imbedded in S then S is called an extension of R. If R is imbedded in S it is possible to construct a ring isomorphic to S which actually includes R as a subring. One rarely bothers to do this, however, since usually it is not 356 Several Algebraic Theories I CHAP P. 8 necessary to distinguish between isomorphic rings. Instead, one "identifies" R with R' which, practically speaking, means that henceforth one regards S as actually including R. Alternatively, one can think of discarding R, using R' in its place, and appropriating the names of elements of R for use as names of the respective image elements in S. It was this latter point of view which was adopted in Chapter 3 in the successive extensions of the natural number system to the real number system. EXERCISES 7.1. Prove that the intersection of a nonempty collection of subfields of a field F is a subfield of F. 7.2. If a and b are distinct elements of a field F, we define a new addition 8 and a new multiplication O in F as x©y= |
x+y-a,x 0 y = a + (x - a)(y - b)(b - a)-'. Prove that (F, (D, (D, a, b) is a field. 7.3. Prove the assertion made in Example 7.8. 7.4. Prove that under a homomorphism the zero and identity element of a ring map onto the zero and identity element, respectively, of the image ring and that negatives map onto negatives. Remark. For the remaining exercises assume that the definition of a ring is modified by discarding the requirement that an identity element be present. Then, for example, the set of even integers forms a ring. Further, assume that by an integral domain is meant simply a ring (in the above sense) with no proper zero-divisors. 7.5. Show that a ring A can be imbedded in a ring with identity element. Hint: In B = Z X A introduce the operations (m, a)+(n,b) _ (m+n,a+b), (m, a)(n, b) _ (mn, na + mb + ab), where na and mb are the nth multiple of a and the mth multiple of b, respectively. Prove that B with these operations forms a ring having (1, 0) as identity element and that A is imbedded in B. 7.6. If the ring A of Exercise 7.5 is an integral domain, then the ring B need not be an integral domain. Establish this fact by taking for A the ring of even integers. Remark. The next three exercises are devoted to proving that it is possible to imbed an integral domain in an integral domain with an identity element. 7.7. Let A be an integral domain, containing elements a and b, with b, 0, such that ab + mb = 0 for some integer m. Prove that ca + me = 0 = ac + me for all c in A. 8.8 1 A Characterization of the System of Integers 357 7.8. Let A be an integral domain and let B be the ring obtained from A and Z by the construction of Exercise 7.5. The mapping on A into B such that a -* (0, a) demonstrates that A is imbedded in B and the mapping on Z into B such that m -3- (m, 0) demonstrates that |
Z is imbedded in B. Let us identify A with its image and Z with its image. That is, we shall write simply a for (0, a) and m for (m, 0). Then, by virtue of the definition of addition in B, B= {m+almCZand aCA}. Show that C= {bCBiba=Ofor all aEA} is an ideal of B and that B/C is an integral domain with identity element. 7.9. Prove that the set A' = (a + C E B/Cia C A) forms a subring of B/C isomorphic to A. 8. A Characterization of the System of Integers The statement that the system of integers is an integral domain summarizes many properties of, but does not characterize, this system. The latter assertion is substantiated by the existence of finite integral domains (see Example 6.9). An additional property of Z, which one might at least suspect would serve to distinguish it among integral domains in general, is the presence of a simple ordering relation which is preserved under addition and under multiplication by positive integers. Since this ordering relation can be formulated in terms of the set of positive integers it is natural to consider integral domains which include a distinguished subset having properties (11)-(13) of Theorem 3.3.1, in connection with an attempt to characterize the system of integers. This is the motivation for our next definition. An ordered integral domain is an integral domain D which includes a subset D+ with the following properties. If a,bCD+,then a+bCD+. If a,bCD+, then abC D. O,. 0. Os. For each element a of D, exactly one of a = 0, a C D+, -a C D+ holds. The elements of D+ are called the positive elements of D. The elements a such that -a C D+ are called the negative elements of D. Further, the members of D+ U {01 are called the nonnegative elements of D. The relation less than, symbolized by <, is defined in an ordered domain by a<biffb-aCD+. As usual, a < b means that a < b or a = b and b > a means that a < b. It is clear that a > 0 iff a C D+ and that a < 0 if -a C D+. In 358 Several Algebraic Theories I CHAP P |
. 8 terms of less than, properties 0,-03 of D can be restated in the following form: 01. 02. 03. Ifa>0andb>0,then a-{-b>0. Ifa > O and b > 0, then ab > 0. If a E D, then exactly one of a = 0, a > 0, a < 0 holds. Additional properties of less than include the following. 04 If a < b and b < c, then a < c. 05 For all a and b in D, exactly one of a < b, a = b, b < a holds. 06. 07. O6. If a < b, then a + c < b + c. Ifa < b and c > 0, then ac < bc. If a 96 0, then a2 > 0. To prove 04 let us assume that a < b and b < c. Then b - a and c - b are positive, and hence, by 01, so is their sum c - a. But this means that a < c. Proofs of 05-07 are left as exercises. To prove O6 let us assume that a 76 0. By 03, either a > 0 or a < 0. If a > 0, then a2 > 0 by 02. If a < 0, then -a > 0 and (-a)2 > 0, by 02. But (-a)2 = 0. for any ring element. So, in all cases, if a 5& 0 then a2 > 0. From 04 and 0s it follows that less than is irreflexive and transitive and hence < is a partial ordering relation. Supplementing this observation with O6, 06, and O7, we infer that < is a simple ordering relation which is preserved by addition and by multiplication with positive elements. It is left as an exercise to show that, conversely, if D is an integral domain which is endowed with a simple ordering relation < which is preserved under addition and under multiplication by elements a such that 0 < a, then D is an ordered domain. At this point the reader who has studied Chapter 3 will recognize that we have established for the ordering relation in an arbitrary ordered domain all but one of the properties which we proved for the ordering relation in Z. The exception is concerned with the well-ordering of the nonnegative elements. We continue to imitate the developments in Chapter 3 by defining the absolute value of an element |
x of an ordered domain as x, if x > 0, -.-x,ifx<0. It is left as an exercise to prove that the absolute value function on an Ixl _ I A Characterization of the System of Integers 8.8 359 arbitrary ordered domain has all the properties which hold in the case of familiar ordered domains (see Theorem 3.4.4). If D and D' are ordered domains and f is a one-to-one mapping on D onto D' which preserves addition and multiplication and maps positive elements onto positive elements, then f is called an orderisomorphism of D onto D'. It is left as an exercise to prove that an order-isomorphism f of D onto D' does preserve ordering, that is, x < y iff f (x) < f(y), and that f-' is an order-isomorphism of D' onto D. If there exists an order-isomorphism of D onto D' we shall say that D is order-isomorphic to L)' or that D and D' are order-isomorphic. Illustrations of orderisomorphisms occur in Chapter 3, where we proved that Z is orderisomorphic to a subset of 0 and, in turn, that Q is order-isomorphic to a subset of R. Further, if we stretch the basic definition under consideration a little, we can reformulate Theorem 2.1.8 in terms of an order-isomorphism. We turn now to the derivation of certain structural properties of ordered domains which yield as a by-product a characterization of the ordered domain of integers. In preparation for the first result the reader should review the discussion of integral systems in Section 2.1. THEOREM 8.1. An ordered domain D includes a unique subset consisting of 0 and positive elements which, together with the function s such that xs = x + 1 and 0, forms an integral system. Proof. Setting Do = 101 U D+, we note that (i) 0 E Do, (ii) x E Do implies that xs C Do, (iii) xs v-1 0 for all x in Do, and (iv) xs = ys implies that x = y. Hence, (Do, s, 0) is a unary system which satisfies condition I, (that is, s is a one-to-one mapping on Do into Do - {01) for an integral system. Hence |
the collection a) of all subsets of Do, which together with s and 0 satisfy I,, is nonempty. Let ND be the intersection of the collection D. Then (ND, s, 0) is a unary system satisfying I,. We claim, further, that (ND, s, 0) satisfies 12, and therefore is an integral system. To prove this, consider any subset M of No such that 0 E M, and if x C M then xs E M. Clearly, M E a) and therefore No C M, whence M = No. To prove the uniqueness of ND suppose that I is a subset of D consisting of 0 and positive elements and such that (I, s, 0) is an integral system. Then 1 E 2) and so Nn e I. Since 0 E ND and x C No implies that xs E ND, it follows that ND = I. 360 Several Algebraic Theories I CHAP. 8 Since for the integral system (ND, s, 0) defined in the above theorem, addition, multiplication and less than satisfy the defining properties of addition, multiplication, and less than, respectively, in N, it follows from Theorem 2.1.8 that an ordered domain D includes a unique subsystem which is order-isomorphic to the system of natural numbers. This is not the end of the matter. In order to state the final result it is convenient to make a definition. If D is an integral domain and E is a subring of D, then it is clear that E is an integral domain which we shall call a subdomain of D. If D is ordered, then so is E. The refinement of the preceding theorem can now be stated as THEOREM 8.2. An ordered domain D includes a subdomain order-isomorphic to Z. Proof. Let ND be the subsystem of D which is order-isomorphic to the system of natural numbers. If a, b C ND, then D contains a - b, the solution of x + b = a. Let ZD = (a - bla, b C ND). Then for all a - b, c - d C ZD, a - b=c - d if a+d=b+c, (1) (a - b) + (c - d) = (a + c) - (b + d), (2) (a - b) (c - d) = (ac + bd) - |
(ad + bc). (3) 0<a-b if a - bCND - (0). (4) Recalling the definition of an element of Z (see Section 3.3), it follows a' and b -- b' under the isomorfrom (1) that if a, b C ND and a phism between ND and N, then the correspondence a - b -} [(a', b')J; is a mapping on ZD into Z. Indeed, it is seen immediately that this is a one-to-one and onto mapping. Moreover, (2) and (3) imply that this mapping preserves operations, and (4) implies that positive elements map onto positive elements, whence order is preserved. In summary, ZD is order-isomorphic to Z. THEOREM 8.3. An ordered domain D with the property that the set Do of nonnegative elements of D is well-ordered is orderisomorphic to Z. Proof. Again let ND be the subsystem of D which is order-isomorphic to N. We shall prove first that, by virtue of the added assumption, ND exhausts the set Do of nonnegative elements of D. Indeed, assume to the contrary that Do - ND 96 0. Then this is a set of positive elements and has a least member a. Now a Pp 1 (since 1 C ND), so a > 1 since 1 is the least positive element in D (see Exercise 8.5 in this section). Then a - 1 E Do - ND, since if a - 1 E ND then 8.9 I A Characterization of the System of Rational Numbers 361 (a - 1) + I = a C ND, contrary to the choice of a. However, since a = (a - 1) + I and 1 > 0, it follows that a - 1 < a and this contradicts the fact that a is the least element of Do - ND. Thus, the assumption that Do - ND is nonempty leads to a contradiction, so we may conclude that Do = ND. According to Theorem 8.2, D includes along with ND an ordereddomain ZD which includes ND and is order-isomorphic to Z. Our proof is completed by showing that ZD exhausts D. For this we use the fact that if d C D, then exactly one of d = 0, d > 0, d < 0 holds. In the first two cases d C ND while in the last -d C |
ND, and therefore - (-d) = d C ZD. Thus, D = ZD. As we learned in the foregoing theorem, the system of integers may be characterized to within isomorphism as the only ordered domain with the property that the set of its nonnegativb elements is wellordered. What amounts to the same, the fourteen properties of Z listed in Theorem 3.3.1 characterize Z to within an order-isomorphism. EXERCISES 8.1. Prove properties 05-07 of the ordering relation in an ordered domain. 8.2. Let D be an integral domain in which there is defined a simple ordering relation < such that if a < b then a +c < b + c and if a < b and c > O then ac < bc. Prove that D is an ordered domain. 8.3. Let D be an ordered domain. Prove the following properties of the absolute value function on D. Ia + bI 5 IaI + Ibl. (i) (ii) Iabl = IaIIbI 8.4. Prove that if D and D' are ordered domains and f is an order-isomorphism of D onto D', then f[D+] = (D')+, f preserves ordering, and f-' is an order-isomorphism of D' onto D. 8.5. Let D be an ordered domain whose nonnegative elements form a well- ordered set. Prove that I is the least positive element of D. 8.6. Prove that a$ = b2 implies that a = b in an ordered domain. 8.7. Prove that the cancellation law for multiplication can be deduced from the other assumptions for an integral domain if the domain is ordered. 8.8. In an ordered domain prove that a2 - ab + b2 >0 for all a and b. 9. A Characterization of the System of Rational Numbers In Section 6 a field was defined as a ring F such that the' set F* of nonzero element is closed under multiplication and (F*,, 1) is a com- 362 Several Algebraic Theories I CHAP. g mutative group. The latter condition implies that each equation of the form bx = a, with both a and b nonzero has a unique solution, namely b-'a (= ab-'). We shall also designate this element by a b or a/b. The |
equation bx = 0 with b P` 0 also has a unique solution, namely, x = 0, since b is not a zero-divisor. For this reason we make the definition b(=0/b) =0 ifb00. Computations with field elements written in the form alb may be caried out exactly as with elements of the field of rational numbers. For example, a ac c d -bd The first of these, for instance, is simply the identity (ab-1)-1 = a 'b written in the new notation. b Another important rule is the following: a __ c b iffad =bc. To prove this let us assume first that alb = c/d, that is, that ab-' = cd-1. Multiplication by bd yields ad = be. Conversely, if ad = bc, then multiplication by b-'d-' gives ab-' = cd-' or, otherwise expressed, alb = c/d. Since our only concern with the theory of fields is to obtain a characterization of the field 0 of rational numbers, we turn directly to a consideration, in abstract form, of the relationship of 0 to the ring Z which was used to construct 0. The obvious feature of this relationship is that 0 is an extension of Z in which division by nonzero elements can be carried out (that is, the equation bx = a has a solution for b P` 0). What conditions if any, we ask, must a ring R satisfy in order that there exist an extension of R in which division by nonzero elements can be carried out? In other words, what rings can be imbedded in some field? 8.9 I A Characterization of the System of Rational Numbers 363 Obvious necessary conditions are that the ring be commutative and that it have no proper zero-divisors. Collectively, these conditions mean that the ring is an integral domain. We shall prove that, conversely, these conditions are sufficient. Although there is no reason to separate the finite case from the infinite one in proving that an integral domain can be imbedded in a field, it is worthy of note that there is nothing to prove in the finite case, since a finite integral domain is a field (see Exercise 6.7). Further, we argue, the proof which must be supplied in the infinite case has already been given. Indeed, if the construction in Section 3.4 of the field of rational numbers from Z is reviewed |
, suppressing all mention of positive elements and positiveness, it will be found that only properties of Z as an integral domain are employed. That is, the construction described in Section 3.4 may be carried out starting with any integral domain D and the result is a field QD [that is, a system having properties (1)-(10) of Theorem 3.4.11, which includes an isomorphic image of D. We interrupt our discussion to state this as our next theorem. THEOREM 9.1. An integral domain can be imbedded in a field. The extension QD of an integral domain D which is secured by the construction in Section 3.4 is called the field of quotients (or quotient field) of D. An element of QD is an equivalence class of ordered pairs (a, b), where a, b C D and b 0 and the subset of QD which is isomorphic to D consists of those equivalence classes having representatives of the form (a, 1). The isomorphism in question maps a onto [(a, 1)]. We shall identify a and [(a, 1) ] which implies, since an arbitrary element [(a, b) ] of QD can be written as [(a, 1)][(b, 1)1-', that the elements of QD consist of all quotients alb where a, b C D with b s 0 and alb = c/d if ad = bc. The field QD is the smallest field in which D is imbedded, in the sense that any field Fin which D is imbedded includes a subfield isomorphic to Qo. To prove this, let us assume that D is imbedded in F. We shall prove that QD is also imbedded in F. Let D' be the isomorphic image of D in F and consider the subset F' of F where F' = {a'(b')-'Ia', b' C D' andb' ; 01. It is a routine exercise to prove that F' is a subfield of F. Assuming that this has been done, we go on to show that F' is an isomorphic image of Qo under the mapping f on QD onto F' such that f(a/b) = a'(b')-', 364 Several Algebraic Theories I CHAP P. 8 where x' is the image in D' of x in D under |
the given isomorphism of D onto D'. From the definition of F', f is onto P. Further, f is one-to-one since if a'(b')-' = c'(d')-' then, in turn, a'd' = b'c', ad = bc, alb = c/d. Finally, we note that alb + c/d = (ad + bc)/bd ->- (ad + bc)'((bd)')-' _ (a'd' + = a'(b')-' + b'c')(b')-'(d')-l c'(d')-' and (a/b)(c/d) = ac/bd -+- (ac)'((bd)')-' = a'c'(b')-'(d')-' (a'(b')-') (c'(d')-'). Hence, f is an isomorphism of QD onto P. A field is said to be an ordered field if, when considered as an integral domain, it is an ordered domain. In the event that an integral domain D is ordered, then its field of quotients, QD, is an ordered field. That is, QD includes a subset QD which is closed under addition and multiplication and has the property that if x C QD, then exactly one of x = 0, x C Qv, -x E Qn holds. Our candidate for Qn is {a/b C QDjab > 01. It is closed under addition since if a/b, c/d C Q,, then (ad + bc)bd = abd2 + b2cd > 0, since ab > 0, cd > 0, and so on, whence alb + c/d E Q+. It is closed under multiplication, since if ab > 0 and cd > 0, then abcd > 0. Finally, it is immediately seen that if alb C QD, then exactly one of ab = 0, ab > 0, ab < 0 holds. Hence, QD has the three required properties and the field QD is ordered. We note that what we have done is to make use of the given ordering of D to define an ordering of its quotient field. Since we have identified the element a in D with the element all of QD, it is clear that a is a positive element of D if a is a positive clement of QD. That |
is, our ordering of the quotient field is an extension of the given ordering of D. We can prove further that the ordering which we have introduced for QD is the only ordering which extends that of D. For this we recall that in an ordered domain a nonzero square is always positive. If the quo tient alb is positive, then the product (a/b)b2 = ab must be positive) and conversely. Hence, in any ordered field, alb > 0 if ab > 0. This completes the proof of 8.9 I A Characterization of the System of Rational Numbers 365 THEOREM 9.2. The quotient field QD of an ordered integral domain D is ordered upon defining alb as positive if ab is a positive element of D. This is the only way in which the ordering of D can be extended to an ordering of Qn. In an ordered field the relation of less than is defined as in any ordered domain; that is, a < b if b - a is positive. In addition to the properties 0; Os in Section 8, there are the following for the ordering relation of an ordered field. 0<1/aiffa>0. alb < c/d iff abd2 < b2cd. 0<a<bimplies 0<1/b<1/a. a <b <0implies 0> 1/a> 1/b. ai+a2--.. +an>0. Our next theorem yields a characterization of the field of rational numbers. THEOREM 9.3. An ordered field F includes a subfield order- isomorphic to the field of rational numbers. Proof. Since an ordered field is an ordered domain, Theorem 8.2 is applicable and we may conclude that an ordered field F includes a subdomain D order-isomorphic to Z. From the argument after Theorem 9.1 it follows that F includes an isomorphic image of the quotient field of Z; that is, F includes an isomorphic image of Q. This result gives a characterization of Q as the smallest ordered field (to within isomorphism, naturally). The statement that 9 is an ordered field summarizes properties (1)-(13) of Theorem 3.4.1. The "smallness" of 0 is the content of (14) of that same theorem. If F is an ordered field, then the ordered subfield of F which is isomorphic to 9 |
is called the rational subfield of F. It should be clear that it consists of just those elements of F having the form ml /ni, where 1 is the identity clement of F and m and n are integers with n$0. We conclude this section with the introduction of one further notion for ordered fields. The ordering of an ordered field F is said to have the Archimedean property if for every pair a, b of elements of F with 366 Several Algebraic Theories I C H A P. 8 a > 0, there exists a positive integer n such that na > b. The origin of this definition is the property of the ordered field 0 which is stated in Theorem 3.4.3 and of the ordered field R stated in Theorem 3.6.3. Although in the statement of Theorem 3.4.3, "nr" is interpreted to be a product of field elements, such a product has an interpretation in any field as an nth multiple, and this is the interpretation intended in the general case. Since in the case of 0 the interpretation of nr as a field product and as the nth multiple of r coincide, the ordering of the field of rational numbers has the Archimedean property in the sense of the general definition. If the ordering of an ordered field F has the Archimedean property, we shall refer to F as an Archimedean-ordered field. If F is Archimedcan-ordered, then its rational subfield is dense in F in the same sense that Q is dense in R (Theorem 3.6.2). We prove this next. THEOREM 9.4. If F is an Archimcdean-ordered field and a and b are in F and a < b, then there exists an element c of the rational subfield Q of F such that a < c < b. Proof. Consider first the case where a > 0. Since b - a > 0, there exists a positive integer n such that n(b - a) > 1, so nb>na+1. (1) Also, there exists a positive integer m such that ml > na. Supposing m to be the smallest such positive integer, ml >na> (m-1)1, since 1 is positive. In view of (1) it follows that nb> (m-1)1 +1 =ml >na. Hence, b > ml/nl > a |
, which is the desired conclusion. If a < 0, then there exists a positive integer p such that pl > -a, and then a + pl > 0. By the first part of the proof, there is an element c in Q such that a + pl < c < b + pl. Hence, a < c - p1 < b, where c - p1 E Q. EXERCISES 9.1. Prove those properties of less than which are stated immediately follow- ing Theorem 9.2. 9.2. Prove that the positive elements of an ordered field are not well-ordered by the given ordering relation. 9.3. Let P be the set of all sequences (ak) = (ao, a1,.., a,,...) 8.10 I A Characterization of the Real Number System 367 of rational numbers having only a finite number of nonzero members. We define (ak) = (bk) if ak = bk for all k. We introduce operations into P by the following definitions: (ak) + (bk) = (Sk) where Sk = ak + bk, (ak)(bk) = (Pk) where pk = E aib;. i+j-k (a) Prove that., 0, (1, 0, (P, 0, 1), where 0 = (0, 0,..., 0,...) and 1 = ), is an integral domain. (b) Defining P+ to be the set of all elements (ak) of P such that the last nonzero member of (ak) is a positive rational, show that P is an ordered domain. (c) Using Theorem 9.2, the quotient field Qp of P is an ordered field. Show that this ordering does not have the Archimedean property by proving ), then for no positive integer n is ni > x. that if x = (0, 1, 0, 9.4. Prove that the ordering of an ordered field F has the Archimedean property if for each element a of F there exists a positive integer n such that nl > a., 0, 9.5. Prove that if F is an Archimedean ordered field, then for each element a in F there exists a positive integer n such that -nl < a and there exists a positive integer n such that 1/nl < a if |
a is positive. 10. A Characterization of the Real Number System An ordered field F is called complete if every nonempty subset of F which has an upper bound has a least upper bound. According to Exercise 1.11.15, an ordered field F is complete if every nonempty subset of F which has a lower bound has a greatest lower bound. Thus the notion of completeness takes a symmetric form which is seemingly lacking in its definition. According to Theorems 3.6.1 and 3.6.4, the real number system is a complete ordered field. In this section we shall prove that these properties of R characterize it to within isomorphism. As the first step in this direction we prove three results about complete ordered fields. THEOREM 10.1. If F is a complete ordered field, then the order- ing has the Archimcdean property. Proof. Assume to the contrary that there exists a pair a, b of elements of F with a > 0 such that for all positive integers n, b > na. Then b is an upper bound of { na C Fln C Z } 1. Since F is complete, this set has a least upper bound c. Then every positive multiple of a is less than or equal to c, so that (m + 1)a < c for every positive integer m. 368 Several Algebraic Theories I CHAP P. 8 This implies that ma < c - a, so c - a is an upper bound for Ina C Fln C Z+}. Since c - a < c, this contradicts the property of c of being the least upper bound. COROLLARY. If F is a complete ordered field, then its rational subfield is dense in F. Proof. This follows from Theorem 9.4. THEOREM 10.2. Let F be a complete ordered field and Q be its rational subfield. For a member c of F let Ac _ {a C Qla < c} and B,, _ {b E Qjb > c}. Then both the least upper bound of A, and the greatest lower bound of B, exist and lub A. = c = glb Bc. Proof. By the Corollary above there is in Q an element a such that c - I < a < c, so A. is nonempty. Also, c is an upper bound for A., and hence the least upper bound of A. exists and is less than |
or equal to c. To prove equality we assume that lub A. < c and derive a contradiction. If lub A, < c, then there exists an a' C Q such that lub A. < a' < c. This is a contradiction since, on one hand, it implies that a' E A, and, on the other hand, it asserts that a' > lub A,. The proof that the greatest lower bound of B, exists and is equal to c is similar. We are now in position to prove the main theorem of this section, namely, that to within isomorphism there is only one complete ordered field. THEOREM 10.3. Any two complete ordered fields are order- isomorphic. Proof. Let F and(F' be complete ordered fields and Q and Q' their respective rational subfields. Then Q and Q' are order-isomorphic since each is order-isomorphic to the field of rational numbers. If f is the isomorphism of Q onto Q' we shall write x' for f(x) and X' for, f [X] if X C Q. Further, we shall denote members and subsets of Q' by primed letters and their counterimages in Q by the same letters. without primes. The strategy of the proof is to define an extension of f having F. as domain and which can be proved to be an order-isomorphism of 8.10 1 A Characterization of the Real Number System 369 onto P. To this end, consider an element c of F. Defining A. and B. as in Theorem 10.2, we know that lub A. = c = glb B0. If b' E B'C, then for each a' E A',, a' < b' since a < c and c < b. Hence, b' is an upper bound for A', so the least upper bound of A' exists and is less than.or equal to Y.,Since this holds for each b' in B',, lub A., is a lower bound for B',, and then the greatest lower bound of B', exists and lub A', < glb B'. We establish equality here by showing that the other possibility leads to a contradiction. Indeed, the assumption that lub A', < glb B', implies that there exists a d' in Q' such that lubA' <d' <gibB: |
. If c C Q, so that c' would be a possible choice for d', we select a d' different from c'. It follows that for the corresponding element d of Q we have a<d<b (1) for every a in A., and every b in B.. Since either d < c or c < d, either d C A, or d C B., which, in view of (1), yields the contradiction d < d. Thus, we have proved that lub A' = glb B". In case c C Q, it is clear that lub A,' < c' < glb B', and hence tub A', = c' = glb B. (2) In case c C F - Q, we define c' by (2). It is this extension of f which we shall prove is an order-isomorphism of F onto F. We show first that this mapping preserves ordering. Let cl, c2 C F and cl < c2. Then there exist a, b E Q such that cl<a<b<c2, whence a C B0, and b C A,,, so that a' C B, and b' C A'. By (2), cl = glb BB, and ca = lub Aa, so ci<a'<b'<ca. Hence, cl < cs implies that cl' < ca. As a by-product of this we have the result that the mapping c - - c' is one-to-one. The proof that it is also onto F' is left as an exercise. We show next that the mapping in question preserves addition; that is, if cl, c2 C F, then (cl + c2)' = + ca. Let a', and as be members of A', and A,,, respectively. Then al and as are members of Q such that al < cl and as < C2. Further, al + as < cl + as < cl + cz, so that al + as < cl + c2. Hence c1' 370 Several Algebraic Theories i CHAP. 8 ai + a' = (a, + a2)' < (c, + c2)' and, consequently, Since ai is an arbitrary element of A' we infer that a', < (c1 + c2)' - a2'- c; = lub A,', < (c, + c2)' |
- a$) which implies that a2 < (c, + C2)' - Ci for all as in A. Hence, in turn, c$ = lub A' < (c, + c2)' - ci, c' + Cl' (Cl + C2)'- A similar argument, in which cl and cs are interpreted as greatest lower bounds, establishes the reverse inequality. Thus, we have proved that Ci + C2 = (CI + c2)'. The proof that the mapping c -'- c' preserves multiplication is somewhat more complicated. We consider first the case of positive elements. Suppose that c, and c2 are positive elements of F and let a; and as be positive elements of A', and A', respectively. Then at and at are positive elements of Q such that a, < c, and as < c2. Further, a,a2 < C1C2 < tics, so that alas < c,c2. Hence a',a2' = (alas)' < (cic2)'. Thus, for each positive element as in A', a, < (CIC2)'(a2') for all positive ai in A. Hence c; = lub A',, < (c,c2)'(a2)-1, which implies that for all a2' in Ate, and then as < (clc2)'(4)-' c2 = lub A'C. < (c,ct)'(c;) Thus, C;cg < (c,c2)'. A similar argument, in which c; and c2' are in. terpreted as greatest lower bounds, establishes the reverse inequality Thus (3) where c, > 0 and C2 > 0. elcs = (C1C2)', 8.10 f A Characterization of the Real Number System 371 Finally, we extend (3) to all cl and C2. If one or both of c1 and c2 is equal to 0, then (3) is true trivially. If c, > 0 and c2 < 0, then the restricted version of (3) applies to c, and -c2. This, together with the c' is an isomorphism of the additive group F onto the fact that c additive group F', justifies the following computation: 414 = Ci(-(-C8)) _ -(C'(-Cz)) - |
(C1(-C2))' _ - (- (C1C2))' = - (- (CIC2)') = (C1C2)'- The proof of (3) for the case c, < 0 and c2 < 0 is left as an exercise. There are other characterizations of R ; these stem from other methods of extending Q. to obtain a system with the least upper bound property (that is, the existence of least upper bounds for nonempty sets having an upper bound). Before describing one of these we call attention to the point of view adopted in the constructions of Chapter 3. There, in order to correct a "deficiency" of N, of Z, and of Q, we constructed in turn a new system designed to avoid the deficiency at hand and simultaneously to include a subsystem isomorphic to the parent system. The characterizations of Z, 0, and B. obtained so far in this chapter establish the fact that in each case we obtain a minimal extension with the desired property (as asserted in the introduction to Chapter 3). An alternative point of view for these constructions includes taking into account from the outset the desired feature of minimality of the extensions. For instance, in the extension of N to Z, this point of view manifests itself by adjoining to N a suitable disjoint set to serve as the negatives of the nonzero natural numbers. Similarly, the third extension is approached as the problem of constructing a minimal extension of 0, considered merely as a dense chain, having the least upper bound property. The first step in the solution is the construction of an extension of Q (that is, a dense ordered chain which includes 0 and which preserves the given ordering of the elements of Q), having the least upper bound property. The second step is the proof that, within isomorphism, there is only one such extension E which is a part of any suitable extension and which has the following two properties : (i) an element of Q which is a least upper bound of a subset S of 0 continues to be a least upper bound of S in E, and (ii) every element of E is a least upper bound of some set of rationals having an upper bound in Q. Finally, such a minimal set is selected and the operations of addition and multiplication extended to it from its subsystem 0. The result is a complete ordered field and, hence, R. 372 Several Algebraic Theories I CHAP P. 8. Apart from this approach leading to, what is |
from our viewpoint, a characterization R, it is of interest that there exist extensions of Q which lack either property (i) or (ii) above. Such extensions when equipped with operations become fields which fail to have the Archi. medean property (and so are called non-Archimedian ordered fields). EXERCISES 20.1. Prove that every Archimedean-ordered field is isomorphic to a subfield of R. 10.2. Supply the missing parts of the proof of Theorem 10.2. BIBLIOGRAPHICAL NOTES Section 2. A more comprehensive introduction to the theory of semigroups appears in C. Chevalley (1956). Sections 3-5. There are several excellent textbooks devoted to group theory. W. Ledermann (1953) is an introductory account of the theory of finite groups, More complete accounts of the entire theory appear in M. Hall, Jr. (1959) and A. G. Kurosh (1955). Sections 6-7. Accounts of the topics treated appear in every textbook of modern algebra. Sections 8-10. Most of the notions discussed are treated in textbooks devoted to modern algebra. The proof of Theorem 10.3, that any two complete ordered fields are order-isomorphic, is taken from E. J. McShane and T. A. Botts (1959). CHAPTER 9 First-order Theories IN THIS CHAPTER we give an introductory account of modern investigations pertaining to formal axiomatic theories-that is, axiomatic theories in which there is explicitly incorporated a system of logic. particular attention is paid to those theories for which the logical base is the predicate calculus of first order. These are described in Section 4 after disposing of a necessary preliminary in Sections 2 and 3, namely, an axiomatization of the first-order predicate calculus. Section 7 gives an account of the notions of consistency, completeness, and categoricity for first-order theories, using results obtained in Section 6. After a brief introduction to recursive functions in Section 8, the notion of decidability for first-order theories is examined in Section 9. In this section there is sketched a proof of the famous theorem, due to Church, which asserts the unsolvability of the decision problem for the first-order predicate calculus. In Section 10 appear two other famous theorems about formal axiomatic mathematics. These are the Godel theorems of 1931. One |
asserts that a sufficiently rich formal theory of arithmetic is either inconsistent or contains a statement that can neither be proved nor refuted with the means of the theory. The other asserts the impossibility of proving the consistency of such a theory, if, indeed, it is consistent. Such results may be interpreted as establishing definite limitations for the axiomatic method in mathematics. Section 11 is concerned with a brief discussion of the Skolem paradox for a formulation of set theory as a first-order theory. 1. Formal Axiomatic Theories In order to achieve precision in the presentation of a mathematical theory, symbols are used extensively. A formal theory carries symbolization to the ultimate in that all words are suppressed in favor of symbols. Moreover, in a formal theory the symbols are taken to be merely marks which are to be manipulated according to given rules which depend only on the form of the expressions composed from the symbols. Thus, 373 374 First-order Theories I CHAP. 9 in contrast to the usual usage of symbols in mathematics, symbols in a formal theory do not stand for objects. One further distinguishing feature of a formal theory is the fact that the system of logic employed is explicitly incorporated into the theory. We require additional properties of the formal theories which we shall discuss. These involve an auxiliary notion which we dispose of first. In nontechnical terms, an effective procedure is a set of instructions that provides a mechanical means by which the answer to any one of a class of questions can be obtained in a finite number of steps. An effective procedure is like a recipe in that it tells what to do at each step and no intelligence is required to follow it. In principle, it is always possible to construct a machine for the purpose of carrying out such instructions. The formal theories with which we shall be concerned are axiomatic theories. In such theories formulas are certain strings (that is, finite sequences) of symbols. We require the following properties of formulas. (I) The notion of formula must be effective. That is, there must be an effective procedure for deciding, for an arbitrary string of symbols, whether it is a formula. (II) The notion of axiom must be effective. That is, there must be an effective procedure for deciding, for an arbitrary formula, whether it is an axiom. (III) The notion of inference must be effective. That is, there must be an effective procedure for deciding, for an arbitrary finite sequence of formulas, whether each member of the sequence may be inferred from one or more of those preceding it |
by a rule of inference. In such a formal axiomatic theory the notion of proof is effective; that is, there is an effective procedure for deciding, for an arbitrary finite sequence of formulas, whether it is a proof. Such an effective procedure does not furnish a method for discovering proofs. It merely enables one to decide whether a purported proof is, in fact, a proof. We do not require the notion of theorem to be effective. If there can be found for a theory an effective procedure for deciding, for an arbitrary formula, whether it is a theorem, the theory often loses its appeal to mathematicians. For the implication of the notion of theorem being effective is that one can devise a set of preassigned instructions for a 9.2 1 The Statement Calculus 375 machine such that it could check formulas of the theory to determine whether they are theorems. Mathematical logicians have shown that for many interesting axiomatic theories the notion of theorem is not effective. We emphasize that this means the nonexistence of effective procedures for "theoremhood" has been proved for some theories and not merely the nondiscovery to date of effective procedures. It follows that human inventiveness and ingenuity is necessary in mathematics. A problem which must be faced in presenting a formal axiomatic theory is how to specify the system of logic to be used. One obvious way is to give the rules of inference. In all interesting systems the set of rules is infinite, and there arises the problem of how to specify the set in such a way that one can determine whether a particular rule is in the set. The solution we shall employ calls for specifying a finite set of rules of inference and adding logical axioms to those of the axiomatic theory for the purpose of generating theorems which express further logical principles. That is, the solution calls for the fusion of an axiomatized system' of logic with an axiomatic theory to produce a formal axiomatic theory. Of the systems of logic which might be used in this connection, we shall choose the predicate calculus of first order. Our justification for this choice is that it formalizes most of the logical principles accepted by most mathematicians and that it supplies all the logic necessary for many mathematical theories. In the next two sections we describe an axiomatization. 2. The Statement Calculus as a Formal Axiomatic Theory In view of the role of the statement calculus in a theory of inference (Section 4.4), the goal of an axiomatization is |
a formal axiomatic theory in which the theorems are precisely the tautologies. This was first achieved by Frege, in 1879. Since then, many formulations have appeared. That which we shall present is the simplification of Frege's formulation due to -.ukasiewicz. The primitive symbols (or formal symbols) are a B e a1 B1 e1... The symbols in the second row are called statement variables. The three dots, which are not symbols, indicate that the list continues without end. We define formula inductively as follows. 376 First-order Theories I CHAP. 9 (I) Each statement variable alone is a formula. (II) If A and B are formulas, then (A) - (B) is a formula. (III) If A is a formula, then -1 (A) is a formula. (IV) Only strings of primitive symbols are formulas. A string is a formula only if it is the last line of a column of strings, each either a variable or obtained from earlier strings by (II) or (III). As in the definition of formula, we shall use capital English letters as variables for arbitrary formulas. It can be proved that the notion of formula is effective. In applications the statement variables are replaced by the prime formulas, and hence are interpreted as designating the values of the prime formulas (that is, the truth values T and F). In terms of the definitions made in Section 4.3, a truth value may be assigned to any formula A for a given assignment of values to the variables of A. When writing formulas, the conventions described earlier regarding the omission of parentheses will be followed. Also, we introduce the following abbreviations for certain formulas: AV B for A A B for A- B for -,A->B, (A ---). --I B), (A-'B)A(B-'A). The axioms for the theory are the following formulas, where A, B, and C are any formulas: (PC1) A -' (B -' A), (PC2) (C ---> (A (PC3) (-, A -> -, B) -> (B -> A). B)) - ((C --> A) --j (C - B)), Writing the axioms with variables for arbitrary formulas means that each of (PCi)-(PC3) includes infinitely many axioms, one for each assignment of formulas to the variables occurring. [ |
This agreement is signaled by referring to each of (PC1)-(PC3) as an axiom schema.] For example, by virtue of (PC1), each of (a ((B is an axiom. Even though there are infinitely many axioms, the notion of axiom is effective, since each axiom must have one of three forms. The only rule of inference is modus ponens (see Example 4.4.6) : From (a -+ (Bi)) 631) formulas A and A -> B the formula B may be inferred. The exact form which the definition of proof (Section 5.1) takes for 9.2 I The Statement Calculus 377 the statement calculus is as follows. A (formal) proof is a finite column of formulas, each of whose lines is an axiom or may be inferred from two preceding lines by modus ponens. A (formal) theorem is a formula which occurs as the last line of some formal proof. We shall symbolize the assertion that A is a theorem by I- A. An illustration of a formal proof is given next. It is a proof of the formula a a a. It follows that F a --> a. (a --> ((a3 -- a) -- (0) - ((a -> ((B --+ (0) --j (a - a)) (1) (2) (3) (4) (5) a --+ ((a3 -> a) -> a) (a -> ((B --> a)) -> (a - a) a -' (a3 -' (t) a a Axiom schema (PC2) Axiom schema (PC1) 1, 2 modus ponens Axiom schema (PC1) 3, 4 modus ponens When a proof is given, an analysis is usually given in parallel, as above. This is not required, however, because there is an effective procedure for supplying an analysis. We observe that we can just as easily prove I- a3 -+ a3 or F- (e A a) --> (e A a) by repeating the above sequence of formulas with M or e A a in place of a. Indeed, if in the above formal proof we substitute any formula A for the statement variable a, we get a formal proof of the formula A -' A. But if, instead, we substitute the variable "A" for a (and, "B" for a |
3) we get a proof schema of the theorem schema "A - A." A theorem schema, like an axiom schema, has the merit that a theorem results when the same formula is chosen for all occurrences of any letter that appears in it. We now extend- the definition of theorem to that of deduction from assumptions. If r is a (possibly infinite) set of formulas and A is a formula, then we define D(r, A) to be the set of those finite columns X of formulas whose last line is A, such that each line of X is either an axiom or an element of r or else may be inferred from two earlier lines of X by modus ponens. If, for given I' and A, D(P, A) is nonempty, then A is said to be deducible from assumptions I', symbolized r i- A, and a member of D(r, A) is called a (formal) demonstration of A from I. Basic conditions which these definitions satisfy include the following. 378 First-order Theories I CHAP. -9 (i) If there is an effective procedure for deciding whether a given formula is a member of the set r, then, for each A, there is an effective procedure for deciding whether a column of formulas is or is not a member of D(r, A) [that is, is a demonstration of A from r I. (ii) P F- A whenever A is a member of I' or an axiom. (iii) if rF-Aand rF-A->B,then rF-B. (iv) If r F- A, then, for each set A of formulas, r u A F- A. (v) If I' F- A and r is the empty set, then F- A. (vi) If r F- A, then there exists a finite subset r, of r such that r1F-A. Condition (iii), for example, follows from the fact that if X C D(r, A) and Y C D(r, A --> B), then (X, Y, B), the column consisting of the formulas of X in order, followed by those of Y in order, followed by B, is a member of D(I', B). If in a formal axiomatic theory the notion of deducibility is analyzed into simple steps and the axioms (or, axiom sche |
mas) are few in number, then formal demonstrations and formal proofs of even quite an elementary character tend to become long. However, having once given an explicit definition of what constitutes a deduction from assumptions (and, hence, a formal proof) it is not always necessary to appeal directly to the definition. The alternative is to establish theorems, called derived rules of inference, which assert the existence of proofs under various conditions. An illustration of such a rule for the statement calculus is provided by (iii) above. A useful instance of (iii) is the derived rule If F- A and F- A --> B, then F- B. An application of this rule or of the generalization [which follows from (iii) and (iv) ], If r F- AandF- A -'B,thenrI- B, is commonly called "modus ponens" because of the similarity of each to the rule of inference, which has a like form. Another derived rule, one which plays a crucial role in the proof that the formalized statement calculus fulfills its intended role (and which appears later in an extended form), is given in THEOREM 2.1 (the deduction theorem for the statement calculus). If r is a set of formulas and A and B are formulas, then r u JAI F- B implies I' F- A --, B. 9.2 I The Statement Calculus 379 Proof. t Assume that r, A F- B and let the column X=(C1,C2,...,Q be a formal demonstration of B from r u { Al. For each i = 1, 2,, n we define by induction a column Y; as follows. Case 1. Case 2. If C; is an axiom or an element of r, let Y; be the column (A-->C1),A- Q. If C, is A and Case 1 does not hold, let Y. be the column whose lines in order are the proof, given earlier, of A --* A. Case 3. If C, is inferred from two earlier lines C, and C, -- C, of X by modus ponens, and the preceding cases do not hold, and j is the least index for which there is such a C,, let Y, be the column (Y j, Yk, (A - (Ci `i C,)) - ((A - C,) - (A - C,)), (A |
-, --' (A - C,),A--+C,). Here k is the least index for which Ch is C, -+ C,. It is left as an exercise to prove by induction that for each i = 1,, n, Y, is in D(r, A -- Q. Since C. is B, this gives the desired 2, result that r F- A -> B. COROLLARY. If A1j A2, F- At --* (A2, A. F- B, then (... (Am - B)... Repeated application of the theorem gives the corollary. The converse of this result is the next theorem. Its proof is left as an exercise. THEOREM 2.2. A2,..., A. F- B. If F- A, -+ (A2 ( (Am --> B) )), then At, In view of property (vi) of deducibility, Theorems 2.1 and 2.2 accomplish the reduction of the notion of deducibility to that of provability. A comparison of these theorems with the Corollary to Theorem 4.4.1 shows the parallel between this result and the reduction of the notion of valid consequence to the notion of validity. It follows that if we can show that a formula A is a theorem if it is a tautology, we will have demonstrated the equivalence of the informal and the formal statement calculus, both by themselves and when applied under a set of assumption formulas. We do this in the next two theorems. First, it may be noted t Hereafter we shall abbreviate "r u JAI F- B" to "r, A F- B" and "JAI, A,, f- B" to "A,. A,,.,A, F- B.", A. 1 380 First-order Theories I CHAP. 9 that in the present circumstances we understand a tautology to be a formula such that for each assignment of truth values to its constituent statement variables, it is assigned truth value T in accordance with the truth tables for -i and -'. The theorem which asserts that every tautology is a theorem is an example of a completeness theorem in the positive sense, as discussed in Section 5.4. It can be derived easily from the following lemma. LEMMA 2.1. Let A be a formula of the statement calculus in |
which, Pk. Define p occur only statement variables from the list P1j P2, to be Pi or -' Pi according as P; takes the value T or F and A' to be A or -1A according as A takes the value T or F for an assignment of truth values to P1j P2, (1), A. Then P., P2,...,P' -A' for every assignment of truth values to P1, P2, Proof. The proof is by induction on the number of symbols in A, counting each occurrence of -, or -+ as a symbol. If n = 0, then A is some Pi. Then A' is P; and (1) is immediate. Assume the lemma true for all formulas with less than n symbols and consider A with n symbols., P. Case 1. A is of the form -1 B. Then, by the induction hypothesis, PP, P2,..., P.' I- B' (2) for all assignments of truth values to P1, P2,, P. Subcase 1.1. B takes the value T. Then A takes the value F, B' is B, and A' is -, A, that is, -, -s B. Now F- B -' -, -, B (see Exercise 2.3), P,, F- B; then, by modus ponens, p, and (2) reads P;, P2, P2i,P,'F- -i B, which is (1). Subcase 1.2. B takes the value F. Then A takes the value T, B' is B, -1 B, and A' is A, that is, -, B. Then (2) gives P,, P2f which is (1)., P,' F- Case 2. A is B --j C. Then, by the induction hypothesis, (3) (4) P1, P2,...,P.F- B', P1, P2,..., P,r F- C'. Subcase 2.1. C takes the value T. Then A takes the value T, C' is C and A' is A, that is, B --> C. Hence, (4) is P,, P2,, P,' F- C and (PCI) gives F- C -1 |
(B --> C), so that, by modus ponens, P;, P2,,P,' F- B-C, which is (1). Subcase 2.2. B takes the value F. Then A takes the value T, B' is 9.2 1 The Statement Calculus 381 -1 B, and A' is A, that is, B --> C. Hence, (3) is P, Ps,, P', F- -1 B and this, with I- -- B --+ (B --> C) [see Exercise 2.3], yields (1) again by modus ponens. Subcase 2.3. B takes the value T and C takes the value F. Then A takes the value F, B' is B, C' is -i C, and A' is -1 A, that is, (B --a C)., P,', I- B and (4) Hence, (3) is P;, P2i, Pk F- -1 C. These, together with the theorem B --- (-1 C - -1(B -' C)), yield P;, P27, Pk 1- -1 (B -- C), which is (1). is P,, Pa, THEOREM 2.3 (the completeness theorem for the statement calcuIf A is a tautology, then A is a theorem; that is, if i A, then lus). F- A. Proof. Let P1, P2, ring in A and define P;, Pa, i A, A' is always A, and then, by Lemma 2.1, P,, P2, every assignment of truth values to P1, P2, (5), Pk be the distinct statement variables occur, Pk and A' as in Lemma 2.1. Since, P,t F- A for, P,. In particular, P,, P2,..., Pk_ 1, Pk F- A, P1, P2,..., Pk_ 1, -, PA: I- A (6) for every assignment of truth values to P1, P2, deduction theorem it follows that, Pk_l. From the P1, P2,... P.'_ 1 (7) Pi, P2f (8) These deductions, together with the theorem., P |
. -I F- -, Pk -> A. Pk -+ A, A), (Pk--+A) -' ((--1 Pk -+A), Pk _ 1 F- A. Thus the which the reader may prove, give P,, P2, assumption Pk is eliminated. Repeating this process k - 1 times eliminates all the assumptions, so that F- A. The converse of the completeness theorem is easily proved as we show next. If A is provable, then A is a tautology; that is, THEOREM 2.4. if F- A, then r- A. Proof. We observe first that each instance of an axiom schema is a tautology; that is, the theorem is true for the axioms. Further, by Theorem 4.3.3, if r- A and K A -+ B, then t= B. Since every theorem is either an axiom or comes from the axioms by one or more uses of modus ponens, every theorem is a tautology. 382 First-order Theories I CHAP. 9 That is, the notions of validity and provability for the statement calculus are coextensive. This result was proved first in 1921 by the American logician, Emil Post. There is more to be said about the foregoing result. We first remark that we assume it clear that the process provided in the definition (Section 4.3) for determining the truth value of a formula A for a given assignment of truth values to the statement variables in A is effective. Since any A has only a finite number of variables, and hence only a finite number of sets of values of its variables, this leads to an effective procedure for deciding whether A is a tautology or not. Hence, since H A if K A, there is an effective procedure for determining whether a formula of the statement calculus is a theorem; that is, the notion of theorem is effective. More generally, the notion of provability is effective; that is, there is an effective procedure for obtaining a proof of a theorem (which is known to be such because it has been shown to be a tautology). This follows from the fact that the procedures given in the proofs of Theorem 2.3 and Lemma 2.1 are effective. We shall substantiate this, in part, by showing that the proof of Lemma 2.1 provides an effective procedure for finding |
a proof of A' from assumptions P;, P,,, P. If A has no occurrence of --+, this is provided directly. If A has occurrences of ---, the proof provides directly an effective reduction of the problem of finding a proof of A' to the two problems of, P. The same finding proofs of B' and C' from assumptions P,, P,, reduction can then be repeated upon the latter two problems, and so on. Since the reduction process terminates after a finite number of P. repetitions, there results an effective proof of A' from PP, P,, A similar analysis can be made of the proof of Theorem 2.3. Our next theorem follows directly from the Corollary to Theorem 2.1 and Theorems 2.2-2.4. Its application to obtaining derived rules of inference for the statement calculus is illustrated in the examples which follow. THEOREM 2.5. A,, A2, (AQ A., A. 1- B iff... (Am --i B)...) is a tautology. EXAMPLES 2.1. Theorem 2.5 enables one to establish derived rules of inference with appropriate tautologies for justification. Below are listed a few such rules, with the tautology which justifies each placed opposite. 9.2 I The Statement Calculus 383 A f- A V B, A V B, A I- B, A, B I- A, -, B -' -, A I- A -' B, -,B-> -,A, A l- B, K A -i, (A V B). A V B -+ (--i A -+ B). A -- (B A). K (- B -- A) --> (A -- B). (A --+ B). (-1B- -1A) i 2.2. As an illustration of imbedding a system of logic in an axiomatic theory, an idea which was proposed at the end of Section 1, we outline how the statement calculus can be imbedded in an axiomatic theory. This may be accomplished by (i) including among the formation rules for formulas of the theory the following : If A and B are formulas, then so is (A) -* (B), If A is a formula, then so is -,(A); (ii) adding to the axioms of the theory the three axiom schemas |
we have chosen for the statement calculus (where "formula" is now taken in the extended sense of "formula of the theory"); (iii) adding modus ponens to the rules of inference. Formulas of the theory may then be regarded as formulas of a statement calculus in which the role of the statement letters is played by those formulas which are not of the form (A) -4 (B) or --, (A) (that is, formulas which cannot be decomposed into further formulas using --+ and --, in the way shown). As a result of the imbedding, every tautology will be a theorem of the theory. More important, the statement calculus is available as a theory of inference. This theory is adequate to provide the logical skeleton of various kinds of proofs that are encountered frequently. A few examples follow. (a) To establish that a formula B of a theory in which the statement calculus is imbedded is a theorem, it is sufficient to prove that -1 B -* -,A and A are theorems. This procedure is justified by the fifth instance of Theorem 2.5 in Example 2.1. Similarly the rule -, B --> -,A I- A -' B justifies a proof by contraposition. (b) Let us use "C" to denote a contradiction. In formal terms, the proof of a formula A by contradiction may be stated as If -1 AF- C,thenI- A. This rule stems from the tautology (-, A -)- C) --j A. In practice, such a proof may take the following form. One shows that -, A F- B and H -1 B and infers that -, A I-- B A -, B, and then F- A. (c) To establish that a conditional A -* B is a theorem with a proof by con- tradiction, the following rule is often used: If A, -, B I- C (a contradiction), then l- A -' B. The reader may justify this. 384 First-order Theories I CHAP. 9 (d) A "proof by cases" is not uncommon in mathematics. Such a proof of a, A,,, of V Am., A,,, -4 B are provided and it is con- formula B begins with the enumeration of a finite set A,, A2, formulas which are exhaustive in the sense that I- A, V |
A2 V Then proofs of At -+ B, A2 - B, cluded that B is a theorem. The rule at hand is If I- At V A2 V and H A. -- B, then I- B. V Am, H A, -+ B,,!- Am_, -+ B, Upon combining Theorem 2.5 and the Corollary to Theorem 4.4.1 we obtain THEOREM 2.6. A,, A2,, A. l B iff A,, A2,, A. I- B. As the reader may verify, the implication (1) IfA,,A2,,Amt= B, then A,, A2, A. F- B, which is included in the theorem is equivalent to the completeness theorem. We wish to show that (1) can be extended to (2) For any set r of formulas, if r K B, then r I- B, which is known as the strong completeness theorem for the statement calculus. We begin with some definitions. A set r of formulas of the statement calculus is called inconsistent if for some formula B we can deduce both B and -1 B (and, hence, B A -i B) from r. If r is not inconsistent, then it is called consistent. We extend a definition given in Section 4.5 by calling any set r of formulas (simultaneously) satisfiable if there exists truth-value assignments to the statement variables such that each member of r receives truth value T. In more detail, a truth-value assignment to the formulas of the statement calculus is simply a mapping v on the set of formulas onto IT, F} such that (i) for each formula A, v(-,A) is T or F according as v(A) is F or T, and (ii) v(A --- B) = F if v(A) = T and v(B) = F. Then r is simultaneously satisfiable if there exists a v satisfying (i), (ii), and (iii) for all A in r, v(A) = T.t For the case where r consists of a single formula A, satisfiability and validity are connected by (3) K A iff ( -i A } is not satisfiable, and provability and consistency are connected by t The same description of a truth-value assignment may also be used to clarify the meaning of |
the notation "t B" used in (2) above. 9.2 I The Statement Calculus 385 (4) JA) is consistent if not H -, A. Using (3) and (4), it is easily shown that the completeness theorem is equivalent to (5) Every consistent formula is satisfiable. In a similar manner we shall prove that the strong completeness theorem (2) is equivalent to (6) Every consistent set of formulas is simultaneously satisfiable. Further, we shall provide a proof of (6), and thereby (2) will be established. In order to prove the equivalence of (2) and (6), we shall need the following generalizations of (3) and (4). (7) r K A if r u { A } is not simultaneously satisfiable. (8) If r is consistent and cc r, then not r F- -, C. To prove (7), assume first that not r K A. Then there exists a truthvalue assignment v such that v(C) = T for each C in r and v(A) = F. Then it is clear that v demonstrates that r u I-, A} is simultaneously satisfiable: For the converse, assume that r K A. If r is a consistent set of formulas, then for each truth-value assignment such that every member of r takes the value T, A also takes the value T, and hence r u { -, A) is not simultaneously satisfiable. If r is inconsistent, then it is not simultaneously satisfiable (for this the reader is asked to either supply a proof or look ahead to the proof of Theorem 6.1) and, trivially, IF U { -, A } is not simultaneously satisfiable. The proof of (8) is left as an exercise. We continue by proving the equivalence of (2) and (6). Assume that (2) holds and let r be a consistent set of formulas. If c E r, then, by (8), not r I- -1 C. Hence, by (2), not r K -, C. From (7) it then follows that r u { -,, c) = r is simultaneously satisfiable. For the converse, we assume that (6) holds and that r r-: B. Then (7) implies that. r u { -, B) is not simultaneously satisfiable, so that, by (6), r u { |
B} is inconsistent, whence r F- B. Finally, to complete our objective we prove (6). This is our next theorem. THEOREM 2.7. If r is a consistent set of formulas of the state- ment calculus, then r is simultaneously satisfiable. Proof. Since the primitive symbols of our system are denumerable, and its formulas are certain strings of primitive symbols, it is possible 386 First-order Theories I CHAP. 9 to enumerate the formulas. Let some enumeration be given, so that we may speak of "the first formula," "the second formula," and so on, referring to this enumeration of the formulas. We shall use this enumeration to derive from r a maximal consistent set of formulas, that is, a set I' such that r is consistent and, if A is any formula such that r U {A} is consistent, then A E P. Given r, we define an infinite sequence Po, P,, r2, as follows: Po = P and, if the (n + 1) th formula is A, then r.+, = Pn U { A) if this is a consistent set. Otherwise r, = P,,. It follows by induction - are consistent sets, since Po is consistent. Let r be that Po, Ti, P2,.. Then P is a consistent set. For the union of the sets Po, P,, P2, the contrary assumption implies the inconsistency of some finite subset of r and hence that of some P;, contrary to what was observed above. Moreover, r is a maximal consistent set. For let A be any formula such that r U JA) is consistent. Say that A is the (n + 1) th formula. The consistency of r U {A} implies that r. u {A} is a consistent set. Hence, by the definition of r.+,, A is a member of Pn+, and hence a member of P. We list next five consequences of the maximal consistency of P. (i) AEPifTI- A. (ii) If B is any formula, then exactly one of the pair B, -, B is in P; (iii) If B E P, then A -' B C P for any formula A. (iv) If A a P, then A -* B E P for any formula B. (v) IfAEPandBFQP,thenA - BvP. To prove (i |
), let us assume first that A E P. Then P I- A since A I- A, For the converse, assume that r I- A. This means that Ti I- A for some finite subset Ti of P. Then the set r U { A } is consistent. For the contrary assumption implies that there exists a finite subset F2 of r and a formula B such that P2, A I- B A -, B. But then P,, P2 I- B A -, B, which contradicts the consistency of P. Finally, the maximal consistency of P implies that A E P. The proofs of (ii)-(v) are left as exercises. Now consider the mapping v on the set of formulas onto IT, F} such that v(A) = IT lF ifAEP, ifAvP. This qualifies as a truth-value assignment since v(-, A) is T or F according as v(A) is F or T in view of (ii) above, and v(A --' B) = F.9.3 I Predicate Calculi of First Order 387 if v(A) = T and v(B) = F in view of (iii)-(v). Thus 1, and consequently the subset r of r, is simultar.eously satisfiable. EXERCISES 2.1. Complete the proof of Theorem 2.1. 2.2. Prove Theorem 2.2. 2.3. Provide a proof of each of the following formulas of the statement cal- culus (where A and B are any formulas). (d) (A - B) --> (-1B - -,A). (a) -,A -(A -- B). (e) B-' (--C --. C)). (b) (c) A - -1 A. (f) 2.4. The theorem "If a and b are numbers such that ab = 0, then a = 0 or b = 0" is usually proved by assuming that ab = 0 and a ; 0 and deducing that b = 0. Show how to obtain a formal proof from such an informal argument. 2.5. Show that the completeness theorem is equivalent to proposition (5). 2.6. Prove proposition (8). 2.7. Referring to the proof of Theorem 2.7, show that I' has properties (B --' A) -' (( |
-, B - A) --+ A). (B (ii)-(v). 2.8. Referring again to the proof of Theorem 2.7, it should be clear that the possibility of proving by induction the existence of a maximal consistent set of formulas which includes a given consistent set rests with the assumption that the set of statement variables is denumerable. Discarding this assumptionthat is, admitting the possibility of an uncountable set of statement lettersprove the existence of a maximal consistent set which includes a given consistent set of formulas using Zorn's lemma. 3. Predicate Calculi of First Order as Formal Axiomatic Theories. Predicate logic of first order, in addition to having notations of the statement calculus, also has individual variables (and, possibly, individual constants), quantifiers, and predicate variables or predicate constants. Statement variables are not necessarily included, but there must.be a complete set of connectives for the statement calculus. Various different predicate calculi of first order are distinguished according to just which of these notations are introduced. In this section we shall present a particular formulation of each of the predicate calculi of first order. By being sufficiently ambiguous, they can be treated simulEaneously without confusion. Later, certain of these will be assigned special names. Where it is unnecessary to distinguish the va:ious predicate calculi, we speak simply of "the predicate calculus of first order." 388 First-order Theories I CHAP. 9 The axiomatization of the predicate calculus of first order which we present is taken from Church (1956). The axioms and rules of inference are essentially those in Russell (1908) but with Russell's axioms for the statement calculus replaced by (PCi)-(PC3) of Section 2. The primitive symbols are and certain sets of symbols as follows. (i) Individual symbols, some of which are classed as variables and others of which may be classified as constants. The set of variables must be infinite. (ii) Statement symbols, some of which may be classed as variables and the others as constants. (iii) For each positive integer n, a set of n-place predicate symbols, some of which may be classed as variables and the others as constants. Formula is defined inductively as follows. (I) If P is an n-place predicate symbol and x1, x2,, xn are indixn) is a formula. (Such a vidual symbols |
, then P(xi, x2, formula is called prime.) (II) If A and B are formulas, then so is (A) -+ (B). (III) If A is a formula, then so is -, (A). (IV) If A is a formula and x is an individual variable, then (x)A is a formula. (V) Only strings of primitive symbols are formulas. A string is a formula only if it is the last line of a column of strings, each either a prime formula or obtained from earlier strings by (II)-(IV). As in part (I) of the definition of formula, we shall use lower-case letters, with or without subscripts, from the latter part of the alphabet, for individual variables and, as in parts (II)-(IV) of the same definition, we shall use capital English letters from the first part of tlis alphabet for arbitrary formulas. It can be proved that the notion at, formula is effective. We make the same conventions regarding the omiss sion of parentheses when writing formulas, and introduce the same abb breviations for certain formulas as in the statement calculus. Further. 9.3 Predicate Calculi of First Order 389 we introduce (3x)A as an abbreviation for -t (x) -i A. Any occurrence of the variable x in the formula (x)A is called bound. Any occurrence of a symbol which is not a bound occurrence of an individual variable according to this convention is called free. The valuation procedure of Section 4.8, with the following modification is applicable to formulas: The statement constants arc to denote one of the truth values T or F and the statement variables are to have IT, F) as their range. An individual constant (like an individual variable with a free occurrence in a formula) is assigned a value in the domain under consideration and to a predicate constant is assigned a particular logical function. As earlier, the valuation procedure leads to the notion of a valid formula. The axioms for the predicate calculus are given by the axiom schemas (PC1)-(PC3) of the statement calculus, with "A," "B," and "C" now ranging over formulas of the new theory plus at least the following two schemas. (PC4) (x)(A -+ B) -p (A --> (x)B), where x is an individual vari- able with no free occurrences in A. (PC5 |
) (x)A -+ B, where x is any individual variable, y any individual symbol, and B is obtained by substituting y for each free occurrence of x in A, provided that no free occurrence of x is in a part of A that is a formula of the form (y)C. With the applications in mind, it is desirable to include the possibility that there is present in the predicate calculus of first order the formal analogue of the notion of equality. As it is intuitively understood, "x = y" means that x and y are the same object or that "x" and "y" are the names of the same object. For mathematical purposes, all that is required of equality is that (i) it be an equivalence relation, and (ii) it have the following substitution property: If x = y and B is the result of replacing one or more occurrences of "x" in a statement A by occurrences of '!y," then B has the same meaning as A. Now the properties of symmetry and transitivity can be derived from those of reflexivity and substitution. We take this into account in defining a predicate calculus of first order with equality. Such a predicate calculus is one of the sort described thus far with the addition of (i) the 2-place predicate.Constant "_" to the formal symbols, (ii) the clause, "if x and y are individual symbols, then (x = y) is a formula" to the definition of formula, and (iii) the following axiom schemas. 390 First-order Theories I CHAP. 9 (PC6) If x is an individual symbol, then x = x. (PC7) If x and y are individual symbols, and A is a formula, then (x = y) -+ (A - B), where B is obtained from A by replacing some free occurrence of x by a free occurrence of y. When it is necessary to distinguish between a predicate calculus with equality and one in which there is no 2-place predicate constant satis. fying (PC6) and (PC7), the latter will be called a "predicate calculus without equality." For the predicate calculus of first order there are two formal rules of inference. Modus ponens: To infer B from any pair of formulas A, A -a B. Generalization: To infer (x)A from A, where x is any individual variable. A (formal) proof is a finite column |
of formulas, each of whose lineq is either an axiom or may be inferred from two preceding lines by modus ponens or may be inferred from a single preceding line by generalizetion. As in the statement calculus, a (formal) theorem is a formula which occurs as the last line of some formal proof. Again we shall symbolize the assertion that A is a theorem by I-A. In order to extend the earlier definition of a deduction from a set Q assumption formulas to the predicate calculus, we make an auxiliary definition. A column Y of formulas is called a subcolumn of the coQ umn X of formulas if the formulas of Y appear among those of X the same order which they have in Y. Then, if r is a set of formula1 and A is a formula, we define D(r, A) to be the set of those finite col umns X of formulas whose last line is A such that each line of X is eitl 1 an axiom or an element of r, or else may be inferred from two preceding lines by modus ponens, or may be inferred from a preceding line of X by generalization on any variable-provided that B is the last A line of a subcolumn of X which is a formal proof. If, for given r and A, D(r, A) is nonempty, then A is said to be deducible from assumpp tions r, symbolized and a member of D(r, A) is called a (formal) demonstration of from r. Basic properties of deducibility include the six listed in Section. 1i (prior to Theorem 2.1) for the same notion at the statement calculul 9.3 I Predicate Calculi of First Order 391 level. Furthermore, the earlier deduction theorem can be extended to the predicate calculus. This general form, which we establish next, was first proved by J. Herbrand (1930). THEOREM 3.1 (the deduction theorem for the predicate calculus). If P is a set of formulas and A and B are formulas, then r, A I- B implies I' I- A -, B. Proof. The proof is that given for Theorem 2.1, with the following additional case inserted after Case 3. Case 4. If Ci is inferred from an earlier line C; of X by generalization on some variable and C; is the last line of a subcolumn Z of |
X which is a formal proof (and, if the preceding cases do not hold), let Yi be the column (Z, Ci, Ci -' (A -' Ci), A ---> Q. Of course the presence of this case necessitates an extension of the final step of the earlier proof-namely, the proof by induction that each Yi is a member of D(r, A -> Ci). Using Theorem 3.1 we derive next another property of deducibility. For this we first define inductively the conjunction, ATM, of any string A1, A2,, A. of formulas : / \1Ai is A1; Ai+'Ai is Ai+1 A A4AA, j = 1, 2,, m - 1. LEMMA 3.1. A1, A2,,AmI-BiffF-/\,Ai-4 B. Proof. Hints for constructing a proof are given in Exercise 3.1. We use this lemma to prove the following important result. THEOREM 3.2. If r F- A and x is an individual variable not free in any formula of r, then r F- (x)A. Proof. Assume that I' I- A and that x is a variable not free in any formula of F. By property (vi) of deducibility (Section 2), Am} of P such that there exists a finite subset Pl = {AL, A2,, A. F- A. Then Al, Ai - A is a formal theorem by Lemma 41, A2, 3.1. Let X be a proof of this theorem. Since x is not free in /\1mA;, the column X /x)(ArAi --) A) (x)(A Ai -a A) --a (MA i -+ (x) A) NnAi - (x)A 392 First-order Theories I CHAP. 9 is easily seen to be a formal proof. Hence, by Lemma 3.1, rt I- (x)A and, in turn, by property (iv) of deducibility, r I- (x)A, as required. We interrupt our discussion at this point to note that our presentation of the notion of deducibility ha§ been taken from R. Montague and L. Henkin (1956). Apparently their development was motivated by the observation |
that one of the standard definitions of D(r, A) in the literature fails to satisfy property (iv) of deducibility. In this paper the following further result is obtained. Suppose that I-i and 1-s are relations, each satisfying the conditions (ii)-(vi) plus the two conditions of Theorems 3.1 and 3.2. Then r I- t A if r 1- s A for each formula A and each set of formulas. Thereby the relation I- is characterized by this set of seven conditions. As another aspect of the notion of deducibility we note that if A(x) is a formula in which the variable x has a free occurrence, then in a demonstration which involves A(x) as an assumption formula, one is not permitted to generalize on this x. That is, x is treated as a constant. Intuitively we may say that a free occurrence of a variable in an assumption formula is employed to denote an arbitrary but fixed individual. In informal mathematics, when a variable x is employed in this way, one says that x has the conditional interpretation. In contrast, if x has a free occurrence in a formula A (x) which is an axiom of the theory under consideration, then A(x) is intended to mean the same as (x)A(x). In this circumstance one says that x has the generalityinterpretation. If A is any formula and its free variables in order of, xn, then by the closuret of A we mean first free occurrence are xt, x2, the formula (X1) (X2). (xn)A, sometimes abbreviated VA. Under the generality interpretation of free variables, A and VA are synonymous. The deduction theorem can be extended so as to give the generality interpretation to some or all of the variables having free occurrences in one or more assumption formulas. For example, there is the following result : If r, A 1- B, then r H VA -' B. For proof, assume that r, A I- B, Now VA I- A by repeated use (possibly) of (PC5) together with modus ponens. So, by the property of deducibility stated in Exercise 3.2 (b), r, VA I- B. Hence, by Theorem 3.1, r i- VA -' B. f In harmony with the definition of closure,. a formula containing no free variables |
, that is a statement according to an earlier definition, is often called a closed formula. 9.3 1 Predicate Calculi of First Order 393 The reduction of the notion of deducibility to that of provability in the case of the predicate calculus can be shown in a manner parallel to the corresponding reduction in the statement calculus, since the corollary to Theorem 2.1 and Theorem 2.2 carry over to the predicate calculus. Or, more simply, Lemma 3.1 may be called upon. It follows that if we can show that K A if F- A, we will have demonstrated the equivalence of the informal and the formal predicate calculus, both by themselves and when applied under a set of assumption formulas. As in the statement calculus, the proof is easy in one direction. THEOREM 3.3. If F A in the predicate calculus, then t-- A. Proof. As in the proof of the corresponding assertion for the statement calculus (Theorem 2.4), we observe first that the assertion is true for each instance of each axiom schema. In this connection, Theorem 4.8.1 is pertinent. Further, by virtue of Theorem 4.3.3 (extended to the predicate calculus) and Theorem 4.8.2, if C is any theorem which has been obtained from a theorem B by application of a rule of inference, and i B, then C is valid. Hence, if any formula A is a theorem, then A is valid. The converse of this result is a consequence of a theorem first proved by K. G6del (1930). Although it is not his most celebrated theorem, it is a remarkable result. We state it as the next theorem. A proof is given in Section 6. THEOREM 3.4 (G6del's completeness theorem for the predicate calculus). For each formula A in the predicate calculus, if iz A, thenF- A. We conclude this section with an assignment of names to certain predicate calculi of first order. The pure predicate calculus of first order is that in which the primitive symbols include an infinite list of statement variables and, for each positive integer n, an infinite list of n-place predicate variables, but no statement constants, no individual constants, and no predicate constants. A predicate calculus of first order in which at least one kind of constant appears is called an applied predicate calculus of first order. EXERCISES 3.1. Prove |
Lemma 3.1 by induction. In the inductive step the following tautologies are useful: 394 First-order Theories I CHAP. 9' (A'i A; -> (Al+i -+ B)) - (A{+1 A; -- B), (1V' B) -' (IV, A: -' (Ai+l - B)). 3.2. Establish the following additional properties of the relation I-. (a) If A is a formal theorem and r is any set of formulas, then r I- A. (b) If t I- A and if A I- B for every formula B in t, then A I- A. 3.3. Show that the ordering of lines in a formal proof can be avoided, by proving that the theorems of the predicate calculus constitute the smallest set of formulas containing certain formulas and closed under certain operations. 4. First-order Axiomatic Theories A first-order theory (or, a theory with standard formalization) is, a formal theory for which the predicate calculus of first order suffices as the logical basis. Those with which we shall be concerned are also axiomatic theories. An intuitive understanding of the essence of such theories is desirable before technical details are discussed. As our starting point for this we take the description in Section 5.3 of an informal theory as one whose primitive notions consist of a set X, certain of its members (individuals) and certain subsets of X" for various choices of n (primitive relations and operations in X). Now, in place of relations or operations in X, predicates may be used. For example, in place of an n-ary relation p in X we may introduce the n-place predicate P such that a (prime) is assigned the value T for an assignment of u formula P(xi, xs, in X to x;, i = 1, 2,, un) E p. In place of an n-ary operation in X (that is, a function f on Xn into X) we may introduce the (n + 1)-place predicate Q such that a formula Q(xi, x2, - -, xn+1) is assigned the value T for an assignment of u; in X to x; iff f (x,, x2,, xR) = xn+1., n, iff (u,, uz,, |
It is possible (as we shall show) to cope with n-ary operations in a more natural manner by introducing a further class of primitives, called "operation symbols" ; these are the direct formal analogues of functions whose domain is X". Now, with a specific informal theory in mind, suppose we formulate the applied predicate calculus whose individual symbols are variables and a set of constants in one-to-one correspondence with those individuals of Xwhich are primitive and whose only predicate symbols are constants which, in an interpretation having X as domain', denote the primitive relations and operations in X. (Alternatively, operation symbols may be used in place of predicates which are intended' to denote operations in X.) Finally, as axioms we take the axiom 9.4 I First-order Axiomatic Theories 395 schemas of the predicate calculus with equality together with the formalixations of those (mathematical) axioms of the informal theory. As rules of inference we take those of the predicate calculus. The result is a first-order axiomatic theory! We now turn to a precise description of a first-order theory Z. The primitive symbols are the following. (I8) An infinite sequence of individual variables, ao, a,, a2, (II.) A set of logical constants consisting of. (a) the logical symbols of the predicate calculus, parentheses, and a comma, (b) the equality symbol (III.) A set of mathematical constants consisting of (a) a set of individual constants, (b) for each positive integer n, a set of n-place predicate (or rela- tion) symbols, (c) for each positive integer n, a set of n-place operation sym- bols. The equality symbol, although regarded as a logical constant, is included in the set of 2-place predicate symbols. Statement symbols may also be included; any such may be regarded as 0-place predicate symbols. In this same spirit, individual constants may be regarded as 0-place operation symbols. The description of Z further includes the definition of a term. This is given inductively as follows. (Ii) An individual variable and an individual constant are each terms. (III) If r,, r2, then A(r1, r2, r are terms and A is an n-place operation symbol,, is a term. (III,) The only terms are those given by (Q and (III). Although some repetition is involved we give |
an inductive definition of formula. (If) If A is an n-place relation symbol and r1, r2,, r, are terms, is a formula. (Such a formula is called then A(rl, r2, prime.) In particular, if r and s are terms, then (r = s) is a prime formula., 396 First-order Theories ( CHAP P. 9 (II f) If A and B are formulas, then so are -t (A), and (A) -' (B). (111t) If A is a formula and x is a variable, t then (x)A is a formula. (lVf) Only strings of primitive symbols are formulas. A string is a formula only if it is the last line of a column of strings, each either a prime formula or obtained from earlier strings by (IIf) or (III f). We carry over to Z all of the abbreviations, conventions, and definitions employed in the predicate calculus. Further, (r = s) will be abbreviated to r = s and -t (r = s) to r s. The only part of the foregoing for which we did not prepare the reader is the notion of a term. Under the intended interpretation, a term is the name of an object of the domain, that is, an individual. In addition to variables and individual constants being terms, strings composed from variables and individual constants using operation symbols should be terms, since in the intended interpretation they denote function values. The theory Z becomes an axiomatic theory when the axioms are given and provability is defined. The axioms are of two kinds, logical and mathematical. As the logical axioms for % we take all instances of the axiom schemas for the predicate calculus of first order with equality, with the following modifications. We now permit as the "y of (PC5) any term r such that when it is substituted for (the free occurrences of) x in A, no occurrence of a variable in r becomes a bound occurrence. As the mathematical axioms we select some set of closed formulas (that is, statements) of St; these axioms are intended to provide the mathematical content of the theory. As rules of inference we take those of the predicate calculus of first order. The definitions of provability and deducibility remain unchanged from the predicate calculus but these notions are strengthened by the added mathematical axioms. EXAMPL |
ES 4.1. The formulation of group theory given in Exercise 5.4.15 leads to the following description as a first-order axiomatic theory. To the logical constants (including the equality symbol) we adjoin one individual constant e and one 2-place operation symbol - The terms of the theory are defined as follows: Each variable and each constant is a term, and if r and s are terms, then r is a term. The formulas of the theory are those as defined in a predicate calculus; plus (r = s), where r and s are terms. The mathematical axioms are f Generally we shall use letters "x," "y," as (metamathcmatical) variables which range over the variables of the theory under discussion. 9.4 1 First-order Axiomatic Theories 397 (x) (y) (z) (x 0 z) = (x y) (x) (e x = x), (x) (3,v) (y x = e). z), Alternatively, if we start with the formulation which is implicit in Exercise 5.2.7, we are led to the following description. The only mathematical constant is a binary operation symbol, and the mathematical axioms are (x) (y) (z) (x. (y z) = (x. y) (x) (y) (3z) (x = y (x)(z)(3y)(x = y z), z) - z), Each of the foregoing is a formulation of the elementary theory of groups. The word "elementary" signals that the first-order predicate calculus is the system of logic employed and that theorems of the theory are restricted to those which can be expressed by first-order formulas. Not all of group theory, as a mathematician knows this discipline, can be formalized by the elementary theory of groups. The state of affairs is that in any first-order theory one can quantify only with individual variables, and this is inadequate to formalize certain theorems. 4.2. We shall call the arithmetic of the system of natural numbers, when formalized as a first-order theory, elementary number theory, and symbolize it by N. One version (based on Peano's axioms) is the following. The mathematical constants consist of the individual constant 0, two 2-place operation symbols + and -, and the 1-place operation symbol'. The mathematical axioms consist |
of the following six axioms and one axiom schema. (x)(y)(x = y'-, x = y), (x) (x + 0 = x), 0 = 0), (x)(x (X) (X,96 0), (x) (y) (x + y' = (x + y)'), (x) (y) (x. y' = x - y + x), A(0) A (x)(A(x) -, A(x')) --> A(x), where x is any variable, A(x) is any formula, and A(0), A(x') are the results of substituting 0, x' respectively for the free occurrences of x in A(x). The intended interpretation of the mathematical constants is the obvious one. It is intended that 0 be the integer zero, that x' be the successor of x, that x + y be the sum of x and y, and that x y be their product. The axiom schema expresses the principle of mathematical induction to the extent possible in a firstOrder theory. 4.3. Since we have assumed that the equality relation is incorporated in a first-order theory, it is possible to replace an n-ary operation symbol in such a theory by an (n + 1)-ary relation symbol. The following example indicates how 398 First-order Theories I CHAP. 9 this can be done. Suppose that + is a 2-place operation symbol in a first-order theory Z. This may be replaced by the 3-place relation symbol S [where S(x, y, z) is to be read "z is the sum of x and y"] and the inclusion of the axioms (x) (Y) (3z)S(x, y, z), (x) (y) (z) (u) (S(x, y, z). A S(x, y, u) - z = u), which express the existence and uniqueness, respectively, of the sum of any two elements. Thus, for theoretical considerations, we may assume that no operation symbols are present in a first-order theory. In a similar manner, individual constants may be eliminated from a first, order theory. For example, to eliminate the individual constant c, we introduce a new unary relation symbol C and the axioms (3x)C(x), (x) (y) (C(x |
) A C(y) -'' x = y) When operation symbols and individual constants are eliminated from a theory, it is necessary to modify formulas in which they appear in an appropriate way. For example, A(c) becomes (x)(C(x) --+ A(x)). 4.4. The agreement that the mathematical axioms of a first-order theory be closed formulas (rather than simply formulas) may seem to entail some loss of generality. That this is not the case is an immediate consequence of the follow, ing result: A formula A of Z is a theorem if its closure is a theorem. Indeed, if A is a theorem, then generalization on each variable, in turn, which is free in A, yields VA as a theorem. Conversely, if VA is a theorem, then any universal quantifier in front of A can be removed, using (PC5). 4.5. The deduction theorem of the predicate calculus of first-order and its converse have the following application to first-order theories: If r is a set of, A. are formulas of Z, then a formula B is deformulas of and A,, A2,, Am) if r F- M A; --' B. In particular, B is a ducible from r u (A,, A2, theorem of T (that is, deducible from the set A of logical axioms and the set of mathematical axioms) if B is deducible from A alone or there exist mathemat,, A. such that A F- / l A; -+ B. ical axioms A,, A2, The last fact, taken together with the definition of deducibility, implies that a formula B is a theorem of Z if B is deducible from the set of mathematical axioms (as a set of assumption formulas) in the theory ', which coincides with T except that its axioms are just the logical axioms of Z. In this way the in, vestigation of various properties of first-order theories can be reduced to that of theories having a common set of axioms-to wit, those of the predicate calculus with equality. We shall take advantage of this possibility later. For first-order theories it is assumed that the mathematical constants have an interpretation in some nonempty domain D. Roughly, this means that each individual constant is interpreted as denoting a fixed.9.4 I First-order Ax |
iomatic Theories 399 member of D, that each individual variable has D as its range, that,,-place relation symbols have interpretations as subsets of D", and that p-place operation symbols have interpretations as functions on D" into D. We turn now to a detailed description of this, along with a valuation procedure (which extends that given in Section 4.8). As the starting point for the valuation procedure for a first-order theory we assume that all mathematical constants of Z can be ar, C ) ranged without repetition in an a-termed sequence (Co, C1, for some ordinal a. f Let D be a nonempty set and let (eo, et, ) be a sequence which has the same number of terms as the foregoing sequence. The nature of each e, depends on the nature of the corresponding constant C,. If C, is an m-place relation symbol, then e, is a subset of Dm; if C, is an m-place operation symbol, then e, is a function on Dm into D; if, finally, C, is an individual constant, then G, is an element of D. The sequence, is,, z _ (D) eo) Gt,... a,.... ) is called an interpretation of Z having D as its domain: this is a precise version of the definition of the same notion given in Section 5.2. If Z is an interpretation of T with domain D, we wish to define next the circumstances under which a (denumerable) sequence ) with d; C D, in brief, a D-sequence is said to (do, dt, satisfy a formula A (of T-) in D. For this we need a preliminary concept. ) and each term r we asTo each D-sequence d = (do, dt, sociate an element r(d) of D by the following recursive rule., d,,,, dR, (i) If r is the individual variable at, then r(d) = dk. (ii) If r is the individual constant C;, then r(d) = e,. (iii) If r is the term C;(ri, r2, symbol and ri, r2,, r,, are terms, then r(d) = e,(rt(d), rs(d),..., r.(3)). -, where C; is an n-place operation The element r(d |
) of D is called the value of the term r f o r the D-sequence W. Using this concept, f o r each D-sequence d = (do, dl, ) and each formula A we specify whether or not d satisfies A by the following recursive rule., d,,, (I,) If A is a prime formula C;(ri, r2, n-place relation symbol and rt, r2, satisfies A if (ri(d), r2(d),, r,,(d)) C e;., r,), where C; is an, r,, are terms, then t The reader who is not familiar with the notion of an ordinal number may assume, with little loss of generality, that the set of mathematical constants is countable. 400 First-order Theories I CHAP. 9 (II.) If A is a prime formula r = s, where r and s are terms, then d satisfies A if r(d) is the same element of D as s(d). (III.) If A is -, B, then d satisfies A if d does not satisfy B. (IV.) If A is B -> C, then d satisfies A if d satisfies C or satisfies neither B nor C. (V.) If A is (ak)B, then d satisfies A if for every d C D we have that (do,, d,-,, d, dk+,, ) satisfies B. As an illustration of the definition, we apply it to show that if C, is a unary operation symbol and C2 is a binary relation symbol, then for any D, a D-sequenced satisfies (1) (3a3)C2(as, Cias) if there exists a d C D such that (d2, eld) C C2. The reader should justify each of our steps. In unabbreviated form, (1) is (2) -1 (a3) C2(a2, C,a3). Then (a3) -, C2(as, Cia3). This means there exists d C D such that (do, d,, d2, ) satisfies (1) if it does not satisfy, d,,, (do, d,, d2j d, dg,... ) does not satisfy -,C2(a2, C,a3), and hence satisfies C2(a2, Cia3). |
Thus (d2, e,d) C e2. The proof of the following theorem is left as an exercise for the reader. THEOREM 4.1. Ifd = (do,d,, d,,, )and d' _ (do, d i,..., dn,... ) are D-scquences and if A is any formula such that for every variable ak with free occurrences in A, dk = d1ei then d satisfies A iff d' satisfies A. From this theorem it follows that if A is a statement, its satisfaction by a D-sequence does not depend on any element of the sequence; that is, a statement either is satisfied by every D-sequence or by no D-sequence. We shall call a formula true in an interpretation `.J with domain D if it is satisfied by every D-sequence. If A is a statement, then either A is true in `.D or --A is true in Z. If A is true in `D, then we shall say that Z is a model of A. A formula which contains free variables is true in `.J if its closure is true in Z. It follows that the set of all formulas of which are true in `.J is characterized by the statements which it contains. This accounts for the fact that statements occupy a central role in the study of first-order tlicories (and those theories introduced in the next section). 9.5 I Metamathematics 401 The foregoing amounts essentially to nothing more than an alternative description of the earlier valuation procedure for the predicate calculus- t Agreement with this statement will come as soon as it is recognized that an interpretation `.J with domain D of a theory X includes the equivalent of an assignment of logical functions (relative to D as domain) to the predicate symbols of Z. The circumstances under which a formula A of $ is classified is true in l is a slight extension of those under which a formula receives truth value T relative to some assignment of logical functions. Having made contact with the earlier valuation procedure, we shall take over some of the terminology introduced in Sections 4.8 and 4.9. If D is a nonempty set and A is a formula of T, then we shall say that A is valid in D if it is true in every interpretation with D as domain; A is valid, symbolized f= A, if it is valid in every D. Further, a formula A is said to be a consequence of a |
set r of formulas, symbolized F i A, if for every interpretation D and every D-sequence 7 such that I satisfies each formula of r, we also have that d satisfies A. t In the case where all formulas are statements, r t= A if A is true in every interpretation in which each member of r is true. If we understand by a model of a set r of formulas an interpretation which is a model of each member of r, then r K A if every model of r is a model of A. EXERCISES 4.1. Formulate the theory of simply ordered commutative groups (see the Exercises for Section 5.3) as a first-order theory. 4.2. Prove Theorem 4.1. 5. Metamathematics The principal reason for formulating intuitive theories as formal axiomatic theories and, in particular, as first-order theories, is that such fundamental notions as consistency and completeness can be discussed in a precise and definitive way. This is possible because the notion of t The justification for presenting two descriptions of a valuation procedure is the author's belief that each is the natural one in the setting in which it finds application. $ We note that a valid formula may be characterized as one which is a consequence of the empty set of formulas. 402 First-order Theories I C H A P. 9 proof is made explicit. Before turning to theorems related to such matters it is desirable to have some understanding of how such matters are studied and why such methods are used. In this section we shall describe the admissible methods for the study of formal theories as advocated by the school of formalists (founded by Hilbert) and then prove some theorems in accordance with these methods. A formal theory is a completely symbolic language built according to certain rules from the alphabet of specified primitive symbols. When a formal theory becomes the object of study it is called an object language. To discuss it, which includes defining its syntax, specifying its axioms and rules. of inference, and analyzing its properties, another language-the metalanguage or syntax language-is employed. Our choice of a metalanguage is the English language. In general terms the contrast between a metalanguage and the object language which is discussed in terms of this metalanguage is parallel to the contrast between the English language and the French language for one whose native tongue is English and who studies French. At the outset, vocabulary, rules of syntax, and so on, are |
communicated in English (the metalanguage). Later, one begins to write in French. That is, one forms sentences within the object language. To give a concrete example, consider the elementary theory of groups as formulated in the preceding section. The statement "The elementary theory of groups is an undecidable theory" is about group theory and written in the English language-that is, in the metalanguage. In contrast, "(a) (b) (c) (a - b = a c --+ b = c)" is a statement of group theory-that is, of the object language. A theorem about a formal theory is called a metatheorem and is to be distinguished from a theorem of the theory. It is easy to make this distinction since a theorem of the theory is written in the symbolism of the theory, whereas a metatheorem is written in English. In the preceding paragraph the statement in English regarding group theory is a metatheorem, and that written in terms of, =, and so on, is a theorem of group theory. Since the proof of a metatheorem requires a system of logic, a description of the system of logic should be available for the prospective user of the metatheorem. One possibility is to formalize the metalanguage as we have formalized the predicate calculus. But this entails the use of a metametalanguage, and the beginning of an unending regress is established. The alternative, which was proposed by 9.5 I Metamathematics 403 Hilbert, may be summarized roughly : In the metalanguage employ an informal system of logic whose principles are universally accepted. More generally, Hilbert took the position that a metatheory (that is, the study of a formal theory in the metalanguage selected) should have the following form. First of all, it should belong to intuitive and informal mathematics; thus, it is to be expressible in ordinary language with mathematical symbols. Further, its theorems (that is, the metatheorems of the formal theory) must be understood and the deductions must carry conviction. To help ensure the latter, all controversial principles of reasoning such as the axiom of choice must not be used. Also, the methods used in the metatheory should be restricted to those called finitary by the formalists. This excludes consideration of infinite sets as "completed entities" and requires that an existence proof provide an effective procedure for constructing the object which is asserted to exist. Mathematical induction is admissible as a finitary method |
of proof, since a proof by induction of the statement "For all n, P(n)" shows that any given natural number n has the property expressed by P by reasoning which uses only the numbers from 0 up to n; that is, induction does not require one to introduce the classical completed infinity of the natural numbers. Finally it is assumed that if, for example, the English language is taken as the metalanguage, then only a minimal fragment will be used. (The danger in permitting all of the English language to be used is that one can derive within it the classical paradoxes, for example, Russell's paradox.) By metamathematics or proof theory is meant the study of formal theories using methods which fit into the foregoing framework. In brief, metamathematics is the study of formal theories by methods which should be convincing to everyone qualified to engage in such activities. Before discussing some metamathematical notions and proving some metatheorems, we outline the reasons which led Hilbert to formulate metamathematics as he did. The introduction of general set theory with its abstractness. and its treatment of notions (such as the completed infinite), which are inaccessible to experience, yet with its fruitful applications to concrete problems of classical mathematics, provided the stimulus for investigations of the foundations of mathematics in the sense that this subject matter is now known. The discovery of contradictions within set theory served to strengthen and accelerate these investigations. The initial reaction to the antinomies of intuitive set theory was a reconstruction of set theory as an axiomatic theory, placing 404 First-order Theories I CHAP. 9 around the notion of set as few restrictions to exclude too large sets as appear to be required to prevent the known antinomies (see Chapter 7). Some felt that even if this venture should prove to be successful, it would not provide a complete solution to the problem because, they argued, the paradoxes raised questions about the nature of mathe. matical proofs and criteria for distinguishing between correct and incorrect proofs for which satisfactory answers had not been provided. Russell, for example, judged the cause of the paradoxes to be that each involves an impredicative procedure. t This led Russell to formulate a system of logic (his ramified theory of type, 1908) in which impredicative procedures are excluded and, with Whitehead, to attempt to develop mathematics as a branch of logic (Principia Mathematica). Both the logistic school and the advocates of the axiomatic approach |
to set theory, initiated by E. Zermelo, were in need of proofs of the consistency of their theories. It was recognized that the classical method of providing a proof-the exhibition of a model within the framework of a theory whose consistency was not in doubt-could not be applied. Further, finite models were clearly inadequate, and no conceptual framework within which an infinite model might be constructed could be regarded as "safe" in view of the antinomies. It was Hilbert who contributed the idea of making a direct attack upon the problem of consistency by proving as a theorem about each such theory that contradictions could not arise. Hilbert recognized that in order to carry out such a program, theories would have to be formalized so that the definition of proof would be entirely explicit. To this end he brought the notion of a formal axiomatic theory to its present state of perfection. To prove theorems about such theories-in particular, to attack the problem of consistency -Hilbert devised metamathematics. By restricting the methods of proof to be finitary in character, he hoped to establish the consistency of theories such as N with the same degree of impeachability as is provided by proofs of consistency via finite models when the latter technique is possible (as in group theory for instance). So much for the raison d'etre of metamathematics. We shall anticipate the results appearing in Section 10 by mentioning now the impossibility of metamathematics fulfilling the role which Hilbert intended for it. This was f A procedure is said to be impredicative if it provides a definition of a set A and a specific object a such that (i) a C A and (ii) the definition of a depends on A. For example, the procedure which leads to Cantor's paradox, is impredicative: The collection a of all subsets of the set A of all sets is both a member of A and depends upon A for its definition. 9.5 I Metamathematics 405 established as a consequence of theorems proved by Godel (1931). The specific circumstances were these. Hilbert's program slowly took form during the period 1904-1920, and in the 1920's he and his co-workers undertook its execution. Their initial goal was to prove the consistency of elementary number theory. This was a natural objective in view of the fundamental role of elementary number theory plus the possibility of the reduction of other portions of classical mathematics to that of N, via models (see Section 5 |
.4). After some partial successes, the endeavor came to a halt in 1931 with the demonstration by Godel of the impossibility of proving the consistency of any formal theory which includes the formulas of N by constructive methods, "formalizable within the theory itself." Regarding such methods, it suffices for the moment to say that so far as is known, they incorporate all methods which Hilbert was willing to permit in metamathematics. This state of affairs does not foretell the doom of metamathematics but has served to indicate its limitations. In what follows, we shall occasionally see methods of proof which lie outside the domain of metamathematics. When this is done we shall call attention to the fact. For our first example of a metamathematical notion we choose consistency. The definition in Section 5.4 (a theory is consistent iff for no formula A both A and --1 A are provable) is applicable to any formal theory having the symbol -i for negation. It is metamathematical since it refers only to the formal symbol -i and the definitions of formula and provability. A metatheorem concerning a class of theories to which the definition is applicable is proved next. THEOREM 5.1. Let Z be a formal theory which includes the statement calculus. Then T is consistent if not every formula of Z is a theorem. Proof. Suppose that T is inconsistent and that A is a formula such that both F- A and F- --1 A. Now A -+ (--I A -' B) is a theorem for any B since it is a tautology. Hence B (that is, any formula) is a theorem by two applications of modus ponens. For the converse, assume that every formula of Z is a theorem. Then if A is any formula, both A and --,A are theorems. Thus, Z is inconsistent. Henceforth it will be assumed that all formal theories include the statement calculus so that Theorem 5.1 will always hold. Our next result is a metatheorem about the statement calculus. 406 First-order Theories, CHAP. 9 THEOREM 5.2. The statement calculus is a consistent theory. Proof. Let A be a theorem. Then, in turn, A is a tautology, -, A is not a tautology, and -,A is not a theorem. The foregoing is a metamathematical proof. |
To substantiate this assertion we note first that the computation process for filling out a truth table for a given formula (regarded as a truth function) is meta.. mathematical. Hence the property of being a tautology is a metamathe.. matical property of formulas of the statement calculus. It follows that the proof of Theorem 2.4 (if A is a theorem, then A is a tautology) is metamathematical. Since the proof in question relies solely on Theorem 2.4, it also is metamathematical. A similar chain of reasoning (now using Theorem 3.3) gives a proof of the consistency of the predicate calculus as soon as a formula which is not valid is exhibited. Although the valuation procedure on which the proof of Theorem 3.3 depends is not effective in general, we apply it relative to a fixed interpretation whose domain is finite. Under these circumstances it is admissible in metamathematics, so the proof is meta, mathematical. The idea behind the proof is the fact that an n-place formula, with or without quantifiers, behaves like a statement in the sense that it assumes either the value T or F, when valuated in a domain of just one element. We begin the proof by defining for each formula A the associated statement calculus formula (a.s.c.f.) as the formula obtained from A by deleting all quantifiers, deleting all individual variables, and treating the predicate variables as statement variables. Now we observe that the a.s.c.f. of each axiom of the predicate cal. culus is a tautology and that the two rules of inference preserve the property of having a tautology as an a.s.c.f. Hence, a formula is prow; able only if it has a tautology as its a.s.c.f. Consider now the formula d1(a) A -i&'(a), where t' is a 1-place predicate variable and a is all individual variable. Its a.s.c.f. is d' A -1d' and is not a tautology, andhence the original formula is not provable. An application of Theorem 5.1 completes the proof. We state this result as THEOREM. 5.3. The predicate calculus of first order is a consistent theory. Sometimes the notion of completeness, in the sense of one or more of the definitions |
given in Section 5.4, may be treated in metamathematics. For example, Theorem 2.3, which asserts the completeness of the state- 9.5 I Metamathematics 407 ment calculus in a positive sense (as this was explained in Section 5.4), belongs to metamathematics. On the other hand, Godel's completeness theorem for the predicate calculus is outside the realm of metamathematics. The statement calculus is also complete in a sense which exhibits a negative approach to a sufficiency of theorems. The next result, which belongs to metamathematics, is of this sort. THEOREM 5.4. If A is any formula of the statement calculus, then either it is a theorem or else an incz)nsistent theory results by adding as additional axioms all formulas resulting from A by substituting arbitrary formulas for its statement variables. Proof. Let A be a formula which is not a theorem, and let us augment the axiom schemas of the statement calculus with all formulas resulting from A by substituting arbitrary formulas for its statement variables. Since A is not a theorem, it is not a tautology. Therefore, it takes the value F for some row of its truth table. Referring to one such row, we choose an instance of A as follows. Substitute a V -id for the prime formulas of A which are T, and substitute a A -1 a for those prime formulas which are F. The resulting axiom, B, will always take the value F. Then --, B is a tautology, and hence a theorem. Thus, both B and -, B are theorems. One might apply the definition of negation completeness (given in Section 5.4), with "statement" replaced by "formula," to both the statement calculus and the predicate calculus. Neither is complete in this sense. For the statement calculus this conclusion follows from the consideration of any formula A whose truth table has neither all T's not all F's; for clearly neither A nor -,A is a theorem. This is a reflection of the fact that in the statement calculus no formula corresponds to a particular statement. We may substitute any statement for a statement variable. For a similar reason the predicate calculus is not negation complete. As an example, neither al(a) nor its negation is a theorem because neither is valid. In this case Q'(a) does not |
stand for a particular statement (which one expects to be true or false) but for any statement in which a' is interpreted as a 1-place predicate and a as an individual. Actually, the metamathematical notion of negation completeness is intended for only formal axiomatic theories such as N and there its restriction to closed formulas is essential in order that it have the intended significance. For example, in N we would not want either 408 First-order Theories I CHAP. 9 (3y) (x = y - y) (which expresses, under the generality interpretation of the free variable present, "every natural number is a square") or -, (3y)(x = y y) ("every natural number is not a square") to be provable. However, one of (x) (3y) (x = y y) and -, (x) (3y) (x = y - y) should be true, and hence provable. As background for the final metamathematical notion which we shall discuss, we recall the definition of an effective procedure as given in Section 1. In brief, an effective procedure-or, as it is often called, a decision procedure-is a method which can be described in advance for providing in a finite number of steps a "yes" or "no" answer to any one of a class of questions. Such a class of questions can be identified with a predicate in the metalanguage in the obvious way. For example, the predicate "p and q are relatively prime" embraces the class of questions concerning the relative primeness of pairs of integers. Thus we may speak of a decision procedure for a predicate. (Incidentally, the Euclidean algorithm provides a decision procedure for the predicate mentioned.) The problem of discovering a decision procedure for a predicate is called the decision problem for that predicate and, if a decision procedure is found, the predicate is said to be effectively decidable; if there does not exist such a procedure the predicate is undecidable. Although we require of a formal axiomatic theory that there be a decision procedure for the notion of proof, we do not require the same for provability. In contrast to the question of whether a given sequence of formulas is a proof (which requires merely the examination of a displayed finite object), the question of whether a given formula is a theorem requires looking elsewhere than within the given object for an answer. Further, the definition of a proof sets no bounds on |
the length of a proof, and to examine all possible proofs without bound on their length is not a procedure which yields an answer to the question in a finite number of steps in the event the formula is not a theorem. This being the state of affairs, the decision problem for provability has special significance for formal theories. Accordingly, it is often called the decision problem for a theory. A theory for which the decision problem can be answered in the affirmative is said to be decidable; otherwise, it is undecidable. An example of a decidable theory is the statement calculus, for since a formula is a theorem iff it is a tautology, the method of truth tables provides a decision procedure. Some other decidable theories are described at the end of Section 9. So long as attention is restricted to results of a positive character 9.6 I Consistency and Satisfiability of Sets of Formulas 409 concerning decidability, an intuitive understanding of this concept suffices. It is up to the creator of a theorem which asserts that some theory is decidable to provide and establish a decision procedure. The situation changes radically, however, if one proposes to prove a result of a negative character, namely, that a theory is undecidable. Clearly, a precise definition of a decision procedure is indispensable in this connection. A definition which is generally agreed on is given in Section 8. 6. Consistency and Satisfiability of Sets of Formulas In this section we derive some properties of a class of theories for which the axioms are those of the pure predicate calculus (without and, later, with the axioms of equality). By being sufficiently ambiguous in the description of these theories we obtain results which have applications to both the predicate calculus and, in view of the remark made in Example 4.5, first-order theories. The applications to first-order theories consist of more definitive results concerning consistency, completeness, and categoricity than were obtained earlier for informal theories. We begin by fixing our attention on a particular theory Zo which may be the pure predicate calculus of first order, or some first-order theory with the symbol for equality deleted, or some theory in between these extremes. If Sto is not the pure predicate calculus, then it is determined by some definite choice of primitive symbols which includes individual symbols (including a denumerable set of individual variables ), possibly some predicate symbols, and possibly some opera0, at, a2, ation symbols, f but without equality. The definition of an interpretation of |
Zo may be obtained from that given for this notion in the case of a first-order theory. The definition of satisfaction of a formula by a D-sequence is obtained from the earlier one by deleting references to equality. Then it is clear that the remaining definitions given for firstorder theories apply to Zo. In order to launch our discussion of Zo, further definitions are needed. These extend some for the statement calculus given in Section 2. A formula A is satisfiable in a nonempty set D if there exists an interpretation ` of To with domain D such that A is satisfied in Z. Notice that satisfiability of A in D hinges on the possibility of making some assignments of values to the free variables in A such that there results satist Of course, we assume that the union of the set of predicate symbols (some of which may as variables and others as constants) and the set of operation symbols is nonempty. 410 First-order Theories I CHAP. 9 faction by a D-sequence which exhibits this choice of values. A formula of To is satisfiable if it is satisfiable in some D. Just as the notion of the validity of a formula may be regarded as the analogue, for to, of the notion of being a tautology in the statement calculus, so may satis. fiability be regarded as the analogue of not being a contradiction. It is clear that a formula is satisfiable (in a given domain) if its negation is not valid (in that domain), and a formula is valid (in a given domain) if its negation is not satisfiable (in that domain). A set of formulas is simultaneously satisfiable if each formula is satisfiable in some domain by some D-sequence. The definitions of an inconsistent and of a con. sistent set of formulas of To read the same as for the case of the state. ment calculus. It is left as an exercise to prove that a set of formulas is consistent if every finite subset is consistent. The main results which we shall derive in this section concern properties of a set r of formulas of to. Since, for applications to first-order theories, r will be the set of mathematical axioms, and since, as men. tioned in Example 4.4, we may take such formulas to be statements, we shall express most of our results for a set of statements. We begin by extending the result obtained in Section 2 to the effect that the notions of consistency and satisfiability of a set of formulas of the statement |
calculus are equivalent to the case of a set of statements of To. THEOREM 6.1. If the set r of statements of to is simultaneously satisfiable, then r is consistent. Proof. Assume that r is an inconsistent set of statements. Then there, Am) of r and a formula B such that exists a finite subset IA,, A2,, A. I- B A --i B. By Theorem 5.3, m > 0. Then the A,, A2, deduction theorem and the statement calculus give F- -1 AM;. The reasoning in the proof of Theorem 3.3 may be applied to this result to conclude that -1 A; A; is valid. Hence, A A; is not satisfiable,, Am) is not satisfiable in view of the which means that JAI, A2, definition of conjunction. It follows that r is not satisfiable. The reader is asked to convince himself that this proof is not admixBible in the sense of Hilbert's metamathematics. In fact, the definition of satisfiability of a set of formulas is probably not admissible. The converse of Theorem 6.1 is a much deeper result. We state it in a sharp form due to L. Henkin (1949). The proof that we shall 9.6 I Consistency and Satisfiability of Sets of Formulas 411 give-a refinement of Henkin's original proof-appears in a paper by G. Hasenjaeger (1953), who attributes the idea to Henkin. It is an elaboration of that given for Theorem 2.7. THEOREM 6.2. If r is a consistent set of statements of Zo, then r is simultaneously satisfiable in a domain whose cardinal number is equal to the cardinal number of the set of primitive symbols of Xa. Proof. We shall carry out the proof for the case where the set of primitive symbols of To is denumerable and indicate afterward the modifications needed in the general case. Let u,, u2, be symbols which do not occur among the symbols of to. Let % be the theory whose primitive symbols are those of to as individual constants. The set of foraugmented with u,, u2, mulas of T is denumerable and there is an effective procedure for listing them. This induces an effective enumeration of the statements of T and, in turn, of those statements of the form (3x |
)A(x).t Suppose, is an enumeration of all such statethat (3x)Ai(x), for i = 1, 2, ments. We shall use this last ordering to construct a consistent set of statements of T that includes P. We begin by defining a sequence of sets of statements of `.slr by induction. Let Po be P. Po, r,, F2, -, let u;, be the first constant that does not occur In the list u,, u2, in (3x)A,(x). Then take Ti to be the set whose members are (3x)A,(x) -+ and the members of Po. Assuming that P; has been defined, let u,,.,, that does not occur in be the first constant in the list u,, u2,, Ai(u;;), (3x)Ai+,(x). Then take Pi+, to be the set whose Al(u;,), members are (3x)A;+t(x) -a Ai+t(uj,..) and the members of F. Then each Pi (i = 0, 1, 2, - ) is consistent. For example, to show that Ti is consistent, assume to the contrary that Po, (3x)A,(x) -+ A,(u,,) - B A -1 B for some formula B. Then, by the deduction theorem, Po 1- ((3x)A,(x) -' Ai(u,,)) --+ B A B. In some demonstration of ((3x)A,(x) -+ A,(u,,)) -+ B A -, B, replace u,, by a new variable y which does not occur in any formula of the t The notation "A(x)" for the type of formula under consideration is a convenient one for exhibiting the result of substituting some individual symbol for the free occurrences of x. 412 First-order Theories'CHAP. 9 deduction. Since u,, does not occur in any member of To or in (3x)Al(x), we then have Tot- ((3x)Ai(x) -' AI(y)) --' B A -1 B. From this can be inferred, by the machinery of the predicate calculus, ro l- ((3x)Ai(x) - (3y)AI(y)) -+ B |
A -t B. Then a change of bound variable f gives ro F- ((3x)AI(x) -' (3x)Ai(x)) B A -1 B. But since (3x)Ai(x) -> (3x)Ai(x) is a theorem, we have To l- B A --1 B, contrary to the supposed consistency of To. Similarly, the consistency of Ti+i follows from that of Ti, and thereby the consistency of each Ti is established by induction. Let F be the union of the sets To (= T), Ti, T2,. Then I' is a consistent set. For the contrary assumption implies the inconsistency of some finite subset of r, and hence that of some Ti, contrary to what was proved above. Next we shall construct a set A of statements of T which includes r (and hence r) and which is maximal consistent in the sense explained in the proof of Theorem 2.7. For this purpose we define an infinite as follows. Let AO be the same as P. sequence of sets Ao, AI, A2, Then, if the (n + 1)th statement A of Z (in the chosen enumeration of these statements) is consistent with A, (that is, if A. U { A } is a consistent set), let A,,+i be the set whose members are A and the members of A,,; otherwise take An+i to be the same as A,,. It follows immediately by induction that each of these sets is consistent. Let A. Clearly, A includes P. Morebe the union of the sets Ao, A,, 42, over, it has the following two properties, which is all we shall use to show that A, and hence r, is simultaneously satisfiable in a denumerable domain. (i) A is a maximal consistent set of statements of Z. (ii) If a formula of the form (3x)A(x) is in A, then for some constant u;, A(u;) is in A. For the proof of (i) we note first that the consistency of A is shown by the same argument as was used above to establish the consistency f This is an application of a theorem about the predicate calculus which may be stated as follows: If y is an individual variable which is not free in a formula C and x is an individual variable which does not occur in C, if E results from D by substit |
uting x for y in C for an occurrence of C in A and if re j- D, then ro 1-- E. 9.6, Consistency and Satisfiability of Sets of Formulas 413 of I. Next, let A be any statement that is consistent with A. Suppose that A is the (n + 1)th statement of Z. Then A. U I A) is a consistent and hence in A. set. Therefore, by the definition of O"+1, A is in A,, The proof of (ii) is left as an exercise. Next we mention five further properties of A which stem from (i) and which we will need. (iii) A statement A is a member of A if A I- A. (iv) If B is any statement, then exactly one of the pair B, -, B is in A. (v) If B C A, then A -4 B E A for any statement A. (vi) If A V A, then A -+ B E A for any statement B. (vii) If AEAand BQA,then A-9BVA. These five properties of a maximal consistent set were listed in the proof of Theorem 2.7. The earlier proof of the first carries over directly to A. Proofs of the earlier statements of the remaining four (which the reader was asked to provide) also-carry over directly to A. Thus, we feel free to continue. This we do by introducing an interpretation, Z, of Z. As its domain, D, we take the set of individual constants of Z. We order all constants (individual and predicate) of Z in a sequence (Co, C,, C2,... ) and then, corresponding to this, we form a sequence (eo, e1, G2, ) as follows. If C; is an individual constant we take e1 to be C;, and if C; is an n-place predicate constant we take a to be the n-ary relation in D such that for individual constants dl, d2, -, do we have, Q. The key property of Z E e1 if A E- C;(d1, d2, (d1j d2, is the following: Each statement A of T is true in Z iff A I- A. The proof (sketched for an A containing no operation symbols) is by induction on the number m of |
symbols in A, counting each occurrence of -,, -*, and a universal quantifier as a symbol. If m = 0, then A has the form, da), where P is a predicate symbol and the d's are in D. If P(d,, d2,, A F- P(dl, d2,..., 4),, 6) is a then, clearly, every D-sequence satisfies A since (d1, d2, member of the n-ary relation assigned to P. The converse is equally obvious. Assume next that the assertion holds for all statements with fewer than m symbols and consider A with m symbols. Case 1. A is -, B. Assume that A 1- -, B. Then it is not the case 414 First-order Theories I CHAP. 9 that A I- B, by (iii) and (iv). From the induction hypothesis it follows that B is not true and, hence, - B is true in Z. The converse, that -' B is true in Z implies A I- -, B, follows by reversing this argument. Case 2. A is B -> C. This is disposed of by properties (v)-(vii) of A. The details are left as an exercise. Case 3. A is (x)B(x). If A F- (x)B(x), then by (PC5) and modus ponens, A F- B(d) where d is any individual constant. The induction hypothesis and clause (V.) of the definition of satisfaction then imply that (x) B(x) is true in Z. For the converse, assume that we do not have A I- (x)B(x). Then -, (x)B(x) and, hence, (3x) -1 B(x)-by the definition of the latter formula together with modus ponens-is in A. From (ii) it follows that there exists u; such that -1 B(u;) C A, so we do not have A I- B(u;). Hence, by the induction hypothesis, B(u;) is not true in ¶, which implies that (x) B(x) is not true in D by the definition of truth for (x)B(x). In view of the result just proved, all formulas of A are true in and so are simultaneously satisf |
iable in D. Since r is a subset of A, the theorem is proved for the case of a To whose primitive symbols are denumerable. The only modifications necessary for the proof of the general case are (i) the replacement of the ui's by symbols u where a ranges over a set with the same cardinal number as the set of primitive symbols of To, and (ii) the selection of some one well, ordering of the formulas of the new T in place of the standard enumeration used above. The depth of this result may be inferred from the fact that several profound theorems pertaining to both the pure predicate calculus and applied predicate calculi can be derived easily from it. We state as the first result in this category the completeness theorem for such theories. THEOREM 6.3 (the completeness theorem). If A is a valid for- mula of To, then I- A. Proof. Assume that A is valid and consider the closure VA of A. As observed earlier, VA is then valid and, in turn, -, VA is not satisfi, able. Hence, by Theorem 6.2, { -, VA } is inconsistent. Therefore, for some formula B, VA I- B A -, B, and then, by the deduction theorem and the statement calculus, I- VA. Then, by (PC5) and modus ponens we may clear away any universal quantifiers to obtain I- A. 9.6 I Consistency and Sati liability of Sets of Formulas 415 If To is taken to be the predicate calculus, then Theorem 6.3 becomes Godel's completeness theorem (Theorem 3.4). Godel, it may be noted, proved the completeness of the pure predicate calculus and then indicated how the method used can be extended to obtain Theorem 6.2 for the pure predicate calculus.' Incidentally, for the pure predicate calculus, Theorem 6.2 may be phrased in the following somewhat more striking form: Every consistent set of statements of the pure predicate calculus is simultaneously satisfiable in the set N of natural numbers. This version follows from the fact that the cardinality of the set of primitive symbols in this case is t4o and, since only the cardinality of a set matters when it is being considered as the domain of an interpretation, N may be used under the circumstances. THEOREM 6.4. If P is a set of statements of To which is simultaneously satisfiable, |
then r is simultaneously satisfiable in a domain whose cardinal number is equal to the cardinality of the set of primitive symbols of To. Proof. Apply Theorem 6.1 and then Theorem 6.2. From Theorems 6.1 and 6.2, when stated for the case of sets of arbitrary formulas (instead of statements) of the pure predicate calculus, follows the Skolem-Lowenheim theorem : If a set of formulas of the pure predicate calculus is simultaneously satisfiable, then it is simultaneously satisfiable in N. Lowenheim first proved this for the case of a single formula. Skolem (1929) generalized this result to the case of simultaneous satisfaction of a countable set of formulas. THEOREM 6.5. Let r be any set of statements of To such that every finite subset of r is simultaneously satisfiable. Then F is simultaneously satisfiable in a domain whose cardinality is equal to that of the set of primitive symbols of To. Proof. Assume that F is not simultaneously satisfiable. Then r is inconsistent, by Theorem 6.2. Hence, there is a formula B such that both r I- B and F H B. Since the demonstrations of B and -, B from r are finite sequences of formulas, we see that some finite subset of r is already inconsistent. This conclusion is incompatible with the hypothesis, according to Theorem 6.1. f However, Godel's proof does not extend to the case of a theory which has uncountably many primitive symbols. 416 First-order Theories I CHAP. 9 It is possible to deduce from Theorem 6.2 an extended version of Theorem 6.3, which is known as the strong completeness theorem: For any set r of statements of To, if r K B, then r h- B. Conversely, the strong completeness theorem implies Theorem 6.2 and thereby follows the equivalence of these two results. This equivalence extends one obtained in Section 2 for the statement calculus. We shall establish the strong completeness theorem next and leave the proof of the converse as an exercise. THEOREM 6.6. For any set r of statements of To, if r B, then r F- B. Proof. Assume that r K B. Then the set r u { -, B} is not simultaneously satisfiable. To prove this we note first that if r is inconsistent, then (Theorem 6.1) it is not |
simultaneously satisfiable, and hence r u {-,B} is certainly not satisfiable. If r is consistent, then (Theorem 6.2) it is simultaneously satisfiable and any model of IF is a B } is not satisfiable. From the nonmodel of B, so again IF U { satisfiability of r U { -, B} follows the inconsistency of this set. Hence, by the deduction theorem and the statement calculus, r I- B. Theorem 6.2 holds for a theory Z, like To but with equality, if we replace "a domain whose cardinal number is equal to" by "a domain whose cardinal number is less than or equal to." Before we prove this we note that the definition of "simultaneous satisfaction in a domain D" now includes clause (II.) of the definition of satisfaction of a formula by a D-sequence. That is, the symbol " _" must denote the relation of equality between individuals of D. (It is because of this inflexible interpretation of the relation of equality that it is classified as a logical constant.) To begin the proof, let E, and E2 be the set of the closures of all instances of the axiom schemas (PC6) and (PC7), respectively, for equality. Given a set r of statements of T-1, we consider the set r U E, U E2 of statements of the theory To obtained from Z, by dropping the axiom schemas (PC6) and (PC7). Since Theorem 6.2 is applicable to this To, there exists, an interpretation of To with domain D in which r U E, U E2 is simultaneously satisfied, provided that it is consistent in To (which is the case if IF is consistent in a,). To " =" there is assigned by this interpretation some binary relation e k in D. Since from E, and E2 one can deduce in To that (x) (y) (x = y --),y = x) and (x) (y) (z) (x = y A y = z --)' x = z), C k is an equivalence relation on D. The relation ek has the additional property that for any n-ary relation 9.7 I Consistency, Completeness, and Categoricity 417, dnekdn imply that, dn) C ei. This is guaranteed because, dn) C ei if ( ', d di |
ei of the interpretation of to, dlekd,, d2ekd2, (d,, d, in To we can deduce the formula (1) x,=x1'Ax2-x$^...A\xn=x.-(Ci(xl, x2,...,..., xn) HCi x, x2, 1 X. ) ) from assumptions E, U E2. Now let D' be the set of equivalence classes modulo ek. Then for each subset of D" the canonical mapping on D onto D' determines a subset of (D')n. Consequently, to each constant C, of %, may be assigned a relation in D' in the natural way. By this route we are led to an interpretation of Ti with domain Y. If c is an individual constant to which is assigned d in D, then to c is assigned the equivalence class (element of D') determined by d. Hence, the relation of equality,. Further, of individuals of D' is assigned to the equality symbol in. r is simultaneously satisfied in D' since r U E, U E2 is satisfied in D and because of the property (noted above) of ek, which stems from formula (1). From the foregoing modification of Theorem 6.2 may be inferred Theorem 6.3 in the form: If A is a valid formula of Z1i then 1- A. This result includes Godel's completeness theorem for the pure predicate calculus with equality. Theorems 6.4 and 6.5 also hold for T, when "equal to" is replaced by "less than or equal to." EXERCISES 6.1. Prove that a set of formulas of To is consistent iff every finite subset is consistent. 6.2. Referring to the proof of Theorem 6.2, prove that A has property (ii). 6.3. Referring to the proof of Theorem 6.2, the reader should agree that the proof-outline of the key property of `tJ (that a statement A of Z is true in D if A I- A) is lacking in precision. He can correct matters by proving by induction on the length of A the following ) and every formula A of LEMMA: For every D-sequence a = (do, d1, d2, Z, A(a) C A if a satisfies A, where |
A(a) is the result of substituting dk for all free occurrences of ak in A, k = 0, 1, 2, 6.4. Write out an expanded version of the proof given of Theorem 6.3, sup-. plying all missing details. 7. Consistency, Completeness, and Categoricity of First-order Theories In this section we shall discuss the notions mentioned in the section heading for an arbitrary first-order theory Z having a set r of statements 418 First-order Theories I CHAP. 9 as its mathematical axioms. The results of Section 6 become available for use in our discussion simply by changing the status of r from that of a set of axioms to a set of assumption formulas. To explain this in detail, let Zi be the theory which coincides with Z except that the axioms of Zi are just the logical axioms of T. Then, as shown in Example 4.5, the theorems of T are precisely those formulas of T1 which are deducible (in Zi) from r as a set of assumption formulas. The transition from Z to Ti amounts to nothing more than the change in status of r mentioned above. When Z is regarded as %I it qualifies as a theory of the type considered in the latter part of Section 6, so the results obtained there may be applied to Z. When discussing $ in this way, the definition of a model of Z coincides with that of a model of r. For, by definition, a model of Z is a model of the set of axioms of Z, but this amounts to an interpretation which is a model of r, since the remaining (logical) axioms of Z are true in every interpretation. Likewise, when our earlier definition of the consistency of a theory is applied to Z, it is seen to coincide with the more recent definition of consistency for r. The definitions given earlier in this chapter of negation completeness and of categoricity of an axiomatic theory may also be applied to r instead of Z. In summary, the notions which are uppermost in our mind now may be formulated at one's or its set of mathematical axioms. Sometimes pleasure for either there are psychological reasons for having a preference. Our first concern is the extension of Godel's completeness theorem and its converse to Z, thereby establishing the correctness and adequacy of the deductive apparatus which is available for Z. THE |
OREM 7.1. A model of Z is a model of the set of theorems of X. Proof. Let B be a theorem of %. Then r I- B in Z1, which means, A. of r. In turn, that Al, A2, A, A; -+ B is a theorem of Z1. If Z is a model of Z, then Z is a model Further, as a theorem of Zi, of (A,, A2, /\;A; --* B [which may be reformulated as (-, /\7,A;) V BI is a valid formula and hence has Z as a model. Thus, Z is a model of B., A. I- B f o r members A,, A2,, A.) and hence of - An alternative version of Theorem 7.1 is : If B is a theorem of T, then B is true in every model of T. We continue by proving the converse statement. Assume that B is true in every model of Z. Then VB is true in every model of X. Thus r v (-1VBJ has no model, and consequently 9.7 1 Consistency, Completeness, and Categoricity 419 is inconsistent by Theorem 6.2. Thus, for some formula C we have r, -1 YB F- C A --i C, and then, by the deduction theorem and the statement calculus, r F- bB and, in turn, r F- B, which completes the proof. Taken together, these two results mean that the theory of inference at hand (that is, that of the predicate calculus) enables one to establish only and all those formulas which are valid conas theorems of sequences of the mathematical axioms of T. We summarize this conclusion in our next theorem. THEOREM 7.2. A formula B is a theorem of T if B is true in every model of Z. We take up next the question of the consistency of (the set of mathematical axioms of) T. For such a theory, with its formal definition of deduction, consistency becomes amenable to exact discussion. Indeed, according to Theorems 6.1 and 6.2, Z is consistent if it has a model, thereby establishing that consistency and satisfiability are entirely equivalent notions. We recall that in Section 5.3 we gave a heuristic argument that in informal theories satisfiability implies consistency |
. Now we have an exact form for both that argument and the meaning of the concepts involved. A further gain that is achieved by formalization is the converse, which is a striking result when stated as, "If the set of axioms of a theory is not satisfiable, then a contradiction can be derived." Certainly this is by no means clear when operating at the intuitive level. Unfortunately, we feel obliged to detract from these lofty observations by mentioning that although in principle a model exists for every consistent first-order theory, finding or describing a model may be difficult, and it is a fact of life that many mathematical axiomatic theories are not of first order. We consider next the question of completeness of a first-order theory %. Theorem 7.2 gives an affirmative answer in the sense that validity implies provability, so we turn to the concept of negation completeness. As with consistency, this has a characterization in terms of models. THEOREM 7.3. Z is negation complete if every statement of T which is true in one model of Z is true in every model of T. If Z is inconsistent, then the left-hand side of the biconditional Proof. is trivially true and the right-hand side is vacuously true. So assume that Z has a model. Breaking the biconditional of the theorem into two conditionals, we shall prove both parts by contraposition. Suppose there is a statement A and models Za and tz of Z such 420 First-order Theories I CHAP. 9 that A is true in Zt and not true in S D2. Then neither A nor -,A is a theorem, by Theorem 7.2, and Z is not negation complete. Conversely, assume that Z is not negation complete; let A be a statement such that neither A not -,A is a theorem. Adjoin A to the set I' of axioms of Z. The set r u {A} is consistent, for otherwise we could deduce from this set a formula of the form B A -, B, and then, by the deduction theorem and the statement calculus, r i- -,A, which is contrary to assumption. Similarly, r U { -, A } is consistent. Hence, there exist models Zt and X12 of r u { A } and r u { -, A 1, respectively, by Theorem 6.2. These are also models of r ( |
that is, of Z) and A is true in )i and not true in Z2. We assume that our earlier discussion of isomorphism is adequate for gathering the meaning of this notion for the case of models of firstorder theories. It is left to the reader to prove that if a statement is true in one model of Z, then it is true in any isomorphic model. We can now prove the following THEOREM 7.4. If Z is categorical, then Z is negation complete. Proof. Assume that Z is categorical. Then a statement which is true in one model of Z is true in every model. Hence, Z is negation complete, by Theorem 7.3. Parenthetically we note at this point that from each of Theorems 7.3 and 7.4 we may infer the completeness of a theory which has no modelthat is, a theory which is inconsistent. (Of course, the completeness of an inconsistent theory is also an immediate consequence of inconsistency.) This triviality having been uncovered, henceforth we shall consider completeness only for consistent theories. From the next two theorems we may infer that the range of applicability of Theorem 7.4 is rather limited, since together they imply that practically no first-order theory is categorical. THEOREM 7.5. If has an infinite model, then for every infinite cardinal number c which is greater than or equal to the cardinality of the set of formulas of Z, Z has a model of cardinality c. t Proof. Let Z be an infinite model of Z and let A be a set of cardinone new individual constant a for each element ality c. Adjoin to t When speaking of the cardinal number of a model we have in mind the cardinal numb* of its domain. 9.7 I Consistency, Completeness, and Categoricity 421 of A and adjoin to the set r of mathematical axioms of 2 all formulas of the form a 76 fg for distinct a and P. Let St' denote this extension of St and let r' denote the set of its mathematical axioms (thus, r' is the union of r and the set of all axioms of the form a 0 p9). Since the cardinal number of the set of primitive symbols of St cannot exceed c, the cardinal number of the set of primitive symbols of St' is equal to c. Further, S is a model of any finite |
subset of r', since (i) it is a model of r, and (ii) being infinite, we can assign to any finite number of distinct a's distinct elements of the domain of Z. It follows from Theorem 6.5 (taking into account the presence of the equality relation) that r' has a model V whose cardinalitycall it c'-is less than or equal to c. But since to the equality symbol is assigned the relation of equality of individuals in the domain of `S', c' > c. Hence, `V' is a model of T, having cardinality c. THEOREM 7.6. If St has models of arbitrarily large, finite car- dinality, then it has an infinite model. Proof. For any positive, finite cardinal number n, the domain of any model of the formula Cn: (3ao) (3ai)... (3an-i) (ao 5,64 ai A ao 0 a2 A... A ao, Cn, s an_, A a, 5,6 a2 A... A an-2 0 an-1) has at least n elements. Let us adjoin to the set r all statements of and call the augmented set of axioms the sequence C1, C2, F'. Then, if St has models of arbitrarily large, finite cardinality, each finite subset of F' has a model, and hence r' has a model D by Theorem 6.5. Since to the equality symbol is assigned the relation of equality of individuals in the domain of `aJ, this domain must be infinite if every C is to be true in Z. Since every Cn is true in Z, it is an infinite model. The two preceding theorems imply that unless a finite upper bound on the cardinality of models of T can be exhibited, then St has models of any preassigned infinite cardinality. Such a theory cannot be categorical, for since isomorphic models always have the same cardinality, the existence of models of St having different cardinalities excludes the possibility of every pair of its models being isomorphic. Even when a finite upper bound on the cardinality of models of T can be found, if models of different cardinalities exist, then St is not categorical for the same reason as above. Hence, a necessary condition for categoricity of 422 First-order Theories I CHAP. 9 a first-order theory Z is that every |
model have the same finite cardinality. But even this condition is not sufficient. To prove this we note first that it is possible to augment the set r of axioms of % with an axiom that restricts the domain of any model to have a preassigned cardinal number. For example, the conjunction of the formula C (used in the proof of Theorem 7.6) and the formula -, suffices to ensure that the domain of every model has exactly n elements. Suppose now that we adjoin to the mathematical axioms of elementary group theory, as formulated in Example 4.1, the axiom which expresses the fact that there exist exactly four objects. Then every model of this theory has cardinal number 4. After the reader has studied a bit of the theory of groups presented in Chapter 8, he will be able to construct two nonisomorphic models of the theory just defined. So the condition that every model of a theory have the same finite cardinality, which is necessary for categoricity, is not sufficient. Although categoricity has essentially no applications to questions of completeness, the following generalization does lead to significant results in this area. If c is a cardinal number, a first-order theory is called categorical in power c if any two models of cardinality c are isomorphic. The following result concerning such theories was obtained independently by R. L. Vaught (1953) and J. Loi (1954). THEOREM 7.7. t If all models of Z are infinite and if, for some infinite cardinal c greater than or equal to the number of formulas of Z, Z is categorical in power c, then T is negation complete. If T is inconsistent, the theorem is true in a trivial way, so Proof. assume that X is consistent. We shall prove that for any given statement of X either it or its negation is a theorem of S, by assuming the contrary and deriving a contradiction. So let S be a statement of Z such that neither S nor -,S is a theorem. Let T' be the theory " be the which results from Z by adjoining S as an axiom and let theory which results from Z by adjoining -i S as an axiom. Since -1 S is not a theorem of Z, V is consistent, and, since S is not a theorem of Z, V" is consistent. Hence, V has a model `Y' and V" has a model `s", according to |
Theorem 6.2. Since ¶' and Z" are models of 2 as well, both are infinite by assumption. Let c be an infinite cardinal such that any two models of Z of cardinality c are isomorphic. Then, f (Added in proof.) In M. D. Morley (1962) there is announced the following theorem which meshes very nicely with the above result: If a first-order theory is categorical in one uncountable power, then it is categorical in every uncountable power. 9.7 1 Consistency, Completeness, and Categoricity 423 by Theorem 7.5, V has a model 11' and V" has a model a", both of cardinal number c. Again, (9' and (&-" are also models of T, and consequently they are isomorphic. However, this is impossible, since S is true in t' while -1 S is true in (Y". This theorem can be used to establish the completeness of a variety of theories. Some examples follow. EXAMPLES 7.1. The elementary theory of densely ordered sets is a first-order theory in which the 2-place predicate < is the only mathematical constant and whose mathematical axioms are the following. 01. 02. 03. 04. 06. 06. (x) --i (x < X). (x)(y)(x 96 y -'x < y Vy <x). (x) (y) (z) (x < y A y < z -' x < z). (x) (y) (3z) (x < y -- x < z A z < y). (3x) (3y) (x < y) (x) (dY) (3z) (y < x A x < z). The models of this theory are precisely all simply ordered dense sets of at least two different elements and have neither a least nor a greatest element. Since 01 and B, each with its natural ordering, are models, the theory is not categorical. However, according to the result stated at the beginning of Exercise 2.6.11, any two denumerable models are isomorphic (since each is isomorphic to Q with its natural ordering). It follows that Theorem 7.7 is satisfied with c = No, and thus the theory is negation complete. 7.2. The elementary theory of atomless Boolean algebras is the first- |
order theory described in Chapter 6 with the axioms given there supplemented by one which implies that each model (that is, each Boolean algebra) has no atoms. All such algebras are infinite, and it can be proved that any two denumerable atomtess algebras are isomorphic. Hence, the theory is negation complete. 7.3. The elementary theory of infinite commutative groups in which every element different from the identity has a given prime order p is the theory defined in Example 4.1, with the necessary additional axioms to ensure that every model has the distinguishing features stated. For example, among these mentioned in the proof of axioms will appear the formulas C1, Cs, Theorem 7.6. It can be shown that any two models of this theory which have the same cardinal number are isomorphic. That is, the second condition of Theorem 7.7 is satisfied for an arbitrary infinite cardinal c. Hence, for each p, the theory is negation complete., C,,, 7.4. The elementary theory of algebraically closed fields of given characteristic p may be described as follows. First, the theory of fields as defined in Chapter 8 is formalized as a first-order theory. Then, if p > 0, the formula X, which, 424 First-order Theories I CHAP. 9 translated into everyday language, states that the successive addition of any element to itself p times yields the zero element, is added as an axiom. If p = 0, we add instead the sequence of formulas -1X2, -iXs, -iXs, Finally, to restrict models to fields which are algebraically closed (which means that every polynomial equation with coefficients in the field has a root, in the field), we add another infinite sequence of axioms, A2, A3, where A. expresses the fact that every polynomial of degree n has at least one root., -,Xp,, A,,, There are pairs of denumerable algebraically closed fields of any given characteristic which are not isomorphic. However, it is known that any two uncountable algebraically closed fields of the same cardinality and the same characteristic are isomorphic. So, again, the conditions of Theorem 7.7 are satisfied for every c > No and the theory, for each choice of p, is complete. The above proofs of negation completeness are all due to V |
aught; however, the results themselves are known earlier, having been obtained by other methods. We remark further that each of these theories is decidable. This matter is discussed in Section 9. We conclude the section with the unraveling of a paradox that can be derived from two of our earlier theorems. On the one hand, Theorem 2.1.8 seems to imply that the arithmetic of the natural numbers is a categorical theory (since it asserts that any two models are isomorphic), while on the other hand Theorem 7.5 implies that it cannot be categorical. To bring this conflict into sharp focus we prove a version of Theorem 7.5 which is tailored specially- for the matter at hand: The theory N is not categorical. To prove this we introduce the first-order theory N' which coincides with N except that it has a further individual constant, u, and the following additional mathematical axioms: U 9 & 0 u00+1 u 54 0 + 1+ I + + 1 (with n occurrences of "1") Now let A be any finite subset of the set r' of mathematical axioms of N' and consider the following interpretation of A. As the domain of the interpretation we choose N, and to -}-,, 0, and 1 we assign the familiar meaning, and, if "u Fx- 0 + 1 + + I" (with m occurrence's of "1") is the last member of the above sequence of axioms that occurs in A, then to u we assign m'. Clearly this interpretation is a model of A, Hence, by Theorem 6.5, I", that is, N', has a model. This model is not 9.7 1 Consistency, Completeness, and Categoricity 425 isomorphic to (N, +, -, ', 0, 1) (for what would be the image of u under a proposed isomorphism?). This completes the proof. The fact that this theorem is not in conflict with Theorem 2.1.8 begins to take form when one attempts to formalize the proof of Theorem 2.1.8. It is found that the first-order predicate calculus is inadequate to carry out this proof because use is made of bound occurrences of predicate variables. That is, the formalization requires the so-called predicate calculus of second order, which, unlike that of first order, admits quantification of both individual and predicate variables. At this point one |
might conclude that the state of affairs might be summarized by the assertion that when the arithmetic of the natural numbers is formalized as a first-order theory it is not categorical but when formalized as a "second-order theory" it is. Matters are even'more complicated than this, however, since the latter part of the assertion must be qualified before it becomes correct. The following is an indication of the precise state of affairs. Suppose that arithmetic is formalized as a second-order theory N". In rough terms this means that we start with the first-order pure predicate calculus with equality, adjoin the constants introduced for N, and alter the definition of formula to admit as a formula (x)A for any formula A and any individual or predicate variable x. Finally, adjoin as the mathematical axioms those introduced for N except that the axiom schema for induction is replaced by a single axiom prefixed with the quantifier "(A)." The definition of an interpretation is as before. However, a description of the valuation procedure relative to an interpretation with domain D must specify the range of each n-place. We select as this range some nonpredicate variable for n = 1, 2, empty collection 61n of sets of n-tuples of elements of D. If every formula of N" is to be meaningful in an interpretation, the sets 6'ri cannot be chosen in an arbitrary mahner. For example, if A is a 1-place predicate variable and A(x) is interpreted as meaning that x is in the set S, then -,A(x) means that x is in the complement of S; hence the range for 1-place predicate variables should be closed under complementation. In general, each method of compounding formulas has associated with it some operation on the sets 6',,, with respect to which these sets must be closed.t We shall assume that the satisfy all such closure conditions. The earlier definition of a model is then applicable to N". If t It is not really necessary to postulate these closure conditions, as is explained in Henkin (1953). 426 First-order Theories I CHAP. 9 an interpretation of N" such that, for each n, <P, is the collection of all sets of n-tuples of D (that is, each n-place predicate variable ranges over all subsets of D") is a model, it is called a standard model. All other models are called nonstandard models. The existence of non |
standard models-indeed, ones in which all of the domains D, Pt, Q2, are denumerable-can be proved. Finally, we are in a position to describe precisely the meaning of Theorem 2.1.8. It is the assertion that any two standard models of N" are isomorphic ; that is, if only standard models of N" are admitted as models, then N" is categorical. Thus the formulation of the arithmetic of natural numbers as a second-order theory is stronger than the formulation as a first-order theory. But the existence of nonstandard models of N" means that even this theory is not categorical. This was discovered by Henkin (1950). EXERCISES 7.1. Formalize the theory of partially ordered sets, using a 2-place relation symbol as the only mathematical constant. Augment the axioms with one that means that there exist exactly three distinct objects, and then show that this theory is not categorical. 7.2. Given any finite set of positive integers, devise a statement such that, when it is adjoined as an axiom to elementary group theory, the cardinal number of any model of the resulting theory is one of the members of the set (and vice versa). 8. Turing Machines and Recursive Functions f Of the metamathematical notions which we have promised to discuss for first-order theories, there remains that of decidability. As we have already pointed out in Section 5, a precise definition of a decision procedure is necessary if one hopes to prove that some theory is undecidable. In this section we develop a tool for coping with decision problems in general. Then, in the next section, questions of decidability and undecidability are discussed. We begin with a sketch of how the type of metamathematical problem at hand can be recast in arithmetical form. The objects of a formal theory are various symbols, various finite sequences of symbols (the formulas of the theory), and various finite sequences of formulas (such as deductions). Since the set of symbols of those theories with which we are cont In the remainder of the chapter we dd not maintain the level of rigor and degree of selfcontainment exhibited up to this point. Results from without are introduced and some arguments arc purely intuitive in nature. 9.8 1 Turing Machines and Recursive Functions 427 cerned is denumerable, so is (Theorem 2.4.5) the set |
of all objects. Now suppose that we provide a particular enumeration of the set of all objects of such a theory. If we let a metamathematical statement of the theory refer to the indices in the enumeration instead of to the objects enumerated, a statement of number theory results. More generally, a predicate of the metalanguage of a first-order theory can be transformed into a number-theoretic predicate, that is, a function on the set N" of all n-tuples of natural numbers into IT, F). Now with each number-theoretic predicate may be correlated a function on N" into N, the so-called characteristic function of the predicate, which takes the value 0 or I according as the predicate is true or false. If by the computation problem for a number-theoretic function f is understood the problem of discovering a procedure describable in advance for computing the value off for any given argument in a finite number of steps, each determined by the preassigned recipe, then the decision problem for a predicate in the metalanguage is transformed into the computation problem for some number-theoretic function. Thus, in particular, by way of an arithmetization of the metalanguage of a theory, the decision problem for that theory reduces to the computation problem for a number-theoretic function. The process of the arithmetization of the metalanguage of a theory, which was devised by Godel for the purpose of establishing the theorems which are discussed in Section 10, is analogous to the arithmetization of Euclidean geometry via the introduction of a coordinate system. A typical example is afforded by the following arithmetization of the metalanguage of N. The starting point is a correlation of certain natural numbers with the formal symbols of N; for example, the following might be adopted : 3 --I 5 -a 7 9 11 13 15 17 19 21 and, to the ith individual variable, the ith prime greater than 22. Having assigned numbers to symbols, we next assign numbers to for, nk be the numbers of the symbols of a mulas as follows. Let n,, n2, formula A in the order in which they occur in A. Let p,(= 2), P2,. -, PA; be the first k primes in order of increasing magnitude. Then the number assigned to A is p' p;' pk'. For example, the numbers of the symbols of the formula -i |
Subsets and Splits
No community queries yet
The top public SQL queries from the community will appear here once available.