- Email: [email protected]
IV: Semantics of (S)1
IV SEMANTICS OF (S)
1
1. Representability of infinite sequences. I n the present Chapter we shall work with infinite sequences almost all terms of which are equal to 1. Every such sequence al, a2, . . . , a , , 1, 1, . . . can conveniently be represented by the integer m
= pY-1
.p2--1
... pEw-1
where pi is the i-th prime. Conversely, every integer m determines a sequence almost all terms of which are equal to 1. I n order to obtain this sequence we put
a,
=
WP,,m)
where W ( x ,y) is the least integer z such that y is not divisible by xz. We shall also use the shorter symbol z, instead of W(pn,m). We shall often use the auxiliary function
It folIows from this definition that W(pn, Ci(m, a ) ) = WP,,m ) if i f n W(Pi, Cdm, a ) ) = a. The function C enables us therefore to construct from a given integer m another integer representing a sequence which differs but in the i-th term from the sequence represented by m. Note the following useful property of the function C: (1)
If i # j, then Ci(Cj(m,a ) , b ) = Ci(C,(m,b ) , a).
This Chapter is based entirely on works of Tarski. Cf. his papers 1217 and [22]. 1.e. all with an exception of at most finite number.
67
VALUES OF FUNCTIONAL FORMS
2. Values of functional forms and the notion of satisfaction for matrix forms. Before we give exact definitions of these basic semantical notions we shall explain briefly their intuitive content. Let p be a f u n c t i o n a l form, e.g. p = p'
=
v,
+ (a, x a2) or p = p" = (pv,)(v, x v,
M
a,).
Speaking intuitively, every such f o r m represents an arithmetical function of as many variables as there are f r e e v a r i a b l e s and bound v a r i a b l e s with the index 2 o c c u r r i n g in p. For instance p' represents the function F(x, y, z ) = x + y . z and 9'' represents the function F ( x ) whose value is 1 if 5 is not a square of an integer and which is equal to y if x = y2. Let us ascribe arbitrary numerical values to free v a r i a b l e s occurring in p and also to the b o u n d v a r i a b l e s which occur in p and have therein the index 2. The function F represented by p takes then on a numerical value which we shall call the value of p for the given values of the variables. For instance, the value of p' is 7 if we ascribe the value 3 to v, and the values 2 to a, and a,. If we ascribe the value 2 to a,, then the value of p" is 1; if the value of a1 is 4, then the value of p" is 2. Let now @ be a m a t r i x f o r m , e.g. p' M pn-+p' M 1 and let us ascribe arbitrary values to the f r e e v a r i a b l e s o c c u r r i n g in @ as well as to the b o u n d v a r i a b l e s which occur in@ and have therein the index 2. The intuitive meaning of@is this: If the values of p' and p" are equal, then the value of 9' is 1. Hence @ represents a theorem of arithmetic and this theorem can be either true or false. I n the first case we say that the values given to the v a r i a b l e s satisfy the m a t r i x f o r m @, and in the second that these values do not satisfy@. For instance in the example considered above the values 2, 1, 2 given to the v a r i a b l e s v,, a,, a,satisfy@. It is convenient to ascribe values to all v a r i a b l e s simultaneously, independently of whether they occur in the expression which we consider or not. Every such system of values can be is the identified with an infinite sequence a,, a,, . . . in which value given to the v a r i a b l e v, and a2, the value given to the v a r i a b l e a,. Since we shall never deal simultaneously with an
58
SEMANTICS OF
(s)
infinite number of expressions, we do not need to consider wholly arbitrary sequences but can limit ourselves to sequences with almost all terms equal to 1. Every such sequence can be represented by an integer in the way explained in section 1. It follows that the value of a f u n c t i o n a l form tp is a function of cp and of an integer m which synthetizes the values ascribed to the variables. For the same reason the notion of satisfaction is a binary relation between m a t r i x -forms and integers. We shall denote by VaZ(v, m ) the value of cp for values of variables represented by the integer m and shall write Gt@f(@,m) instead of “the values of variables represented by m satisfy @”.
3. Inductive definition of Val ( 9 , m ) and GtGf ( @ , m ) . In this section we shall give an exact definition of the notions which were explained intuitively in the previous section. To obtain this definition we first define VaZ(cp,m) for the simplest f u n c t i o n a l f o r m s ah,vh, and 1 and then define it for the f u n c t i o n a l forms v y , q.~x y under the assumption that it has been defined for the f u n c t i o n a l f o r m s cp and y. Under the same assumption we define also the meaning of the formula GtGf(cp M y , m ). Next we define the meaning of the formula Gt@f(di+?P,m)under the assumption that the meaning of the formulas GtGf(@,m) and GtGf(y, m) are already defined. Finally we define Vd((,uvh)@,?n) under the assumption that the meaning of the formula G,tGf(@,m‘) is already defined for an arbitrary integer m‘. I n this way Val( v, m) and GtGf(@,m) will be defined for arbitrary cp in and @ in %Rf. We divide our inductive definition into eight parts:
+
sf
INDUCTIVE DEFINITION OF
v U l ( f p , ?n)
AND
Bt#f(@, m )
59
VaZ((,uvh)@, m ) = 1 if there is no a such that G,tsf(@, Cfi-l(m,a)), otherwise VaZ((pvh)@, m ) = the least such integer a. To illustrate how this definition works we take
(8)
fP = (PVl”V1 x
TI)
-
((a1x
ar) + 1>1
and calculate VaZ(cp,m). By (2), (3), (a), and (5) we obtain 1 whence by (6) VaZ(vl x vl,m ) =z;, VaZ((al x q) 1, m ) G,tGf(((vlx vl) Since x2
M
+
(al x q) I),
+
==:+
c1(m,a ) )3 (a2 = Gi;
+ I).
+ 1 is never a square, we infer that there is no a such that GfGf(((v1x
Vl)
= (a1 x
a1)
+ I), Cdm, a ) )
and consequently VaZ(cp,m ) = 1 according to (8). One should not be deceived by the superficial similarity of the inductive definition given in this section and the inductive defbition given in Chapter 11,section 2, p. 28. To explain the chief difference between these definitions we remark the following. In both definitions we have integers on which the induction proceeds (they are denoted by “by’ and “c” in Chapter 11, and by “9” and “@” in the present Chapter). Furthermore, we have in both definitions the parameters (“a”, “i”, (‘p” in Chapter 11, and “m” in the present Chapter). Now the parameters in the definitions of Chapter I1 are kept constant whereas in the present definition they are variable (cf. (8)).This has the effect that we can calculate the values of functions defined in Chapter I1 by tracing backwards the steps of the definition, and arriving to an end after a finite number of steps. E.g. if we have to calculate Oc(a, b ) we try to decompose b into simpler constituents; if we find for instance that b = b, -+ b,, we reduce our problem to a calculation of Oc(a, b,) and Oc(a, b,). After a finite number of such steps we arrive finally to the values Oc(a, I), Oc(a, vh), and Oc(a, ah) which are given explicitly and obtain then the value of Oc(a, b ) by repeated substitutions. The situation is entirely different in case of the definition given
60
SEMANTICS OF
(s)
in the present Chapter. Indeed, if v = (pvh)@,then the calculation of VaZ(9,m ) is reduced to that of GtSf(@,rn’) for infinitely many different values of m‘. Hence the calculation of VaE(v,m) cannot be completed in a finite number of steps and the definition given in the present Chapter has an “infinitary” character which distinguishes it essentially from the superficially similar “finitary” definitions of Chapter 11. Because of the infinitary character of the definition (1)- (8) we cannot expect that the same method which we used in Chapter I1 will allow us to replace the inductive definition by an explicit one. As a matter of fact it can be shown that an explicit definition of Val and Gt3f is possible only when we use the general notion of an arbitrary set of integers. In the next section we shall outline an explicit definition of VaZ and Gt3f. The proof that it cannot be replaced by a purely arithmetical one (i.e. such which avoids the notion of an arbitrary set) will be given in Chapter VI.
3E
4. Explicit definition of Val and Gtef 3. We shall say that a set of integers represents a function if I
(1) for every n there is in d a n z such that K,(x) = n, (2) for every x and y in Z if K,(x) = K,(y), then K&) = K,(y).
To explain this definition we remark that a function F can be identified with the set of ordered pairs (n, F ( n ) )and a pair (a, b ) can be identified with the integer J ( a , b ) (cf. Chapter I, p. 14). In this way a function F is converted into a set Z and it is easy to show that this set must satisfy the conditions (1) and (2). For every di in r3nf let us denote by a(@) the set of those m for which G,t%f(di,m).Similarly for every p in Sf let us denote by U(y) the set representing the function VaZ(v7m) treated as the function of m alone. It can be easily shown that % ( I ) is the set of all integers having the form J(m, 1): 2 E a(1)ZZ K,(S) = 1. a
Cf. Tarski [20], pp. 311-312.
EXPLICIT DEFINITION OF
Val
AND
Bt8f
61
It is also easy to prove that %(vh) Consists of all integers of the form J(m, Ea-J i.e. of the form J(m, W ( P ~ -m~) ),, and that %(a,) consists of all integers of the form J(m, Za) = J(m, W(p,, m ) ) : Kl(Z)), W(p,, K,(s)).
2 E '%(vh)
K2(x)
= w(P2h-1,
x
K,(4
=
E '%(a,)
We introduce further certain operations on sets of integers which we denote by symbols similar to those used for operations on expressions :
23 i- a 23
i
Q
5
5
W 3 b ) ( 3 C )[ ( b E B) * (c E Q)*
(Kl(b)= W C ) = &(4)* (K2(b)+ K2(c) = K2(4)1, h ( 3 b ) ( 3 C ) [ ( bE % ) . ( C E Q).
(Kl(b) = KAC)= w4) (K2(b).IC2(c) * = K2(4)1, 23 w 6 = W 3 y ) ( J ( x ,Y) E 93 * Q), 23 -+ Q 5 -B v Q,
MhB = h [ K 2 ( s )= min,(C~h-l( K I ( x ) a, ) E B or K2(z)= 1 provided that no such a exists].
It can be shown without essential difficulties that WV
+ w)
=
wcp x Y ) =
WJ) iWw), Wcp) i W Y ) ,
Wcp w
Y)=W V ) WY), %(@+!P) = a(@) I,%(!P),
W(IUVh)@) =
Y)z,W@).
Let now 9 be an arbitrary expression. It follows easily from the definition of the class of e x p r e s s i o n s that there exists a sequence of expressions
md a sequence
(i) Ql, Q2, . . ., Qs = Q (ii) il, i,,
. . ., in
62
SEMANTICS O W
(s)
whose elements are integers 1 or 2 such that for every j of the following conditions is satisfied :
< n one
(iii) Q, = 1 or Q, E 233‘6 or 52, ~ 2 3 fand i, = 1, (iv) Qj = Q k + Q, or Qj = Qk x S, and i, = ik = i, = 1 (k < j , 1 < j ) , (v) S,= Qk M Q, and ij = 2, i k = i, = 1 (k < j , 1 < j ) , (vi) Q, = Q, -+Q, and ii = i k = i, = 2 (k < j , 1 < j ) , (vii) Q, = (pvh)Qk and ij = 1, i, = 2 ( j < k). Put %, = %(Q,) for j of sets such that (viii) (ix) (x) (xi) (xii)
In In In In In
the the the the the
=
1, 2,
. . ., n. We
obtain thus a sequence
cases (iii) 2Ij is either %(I), or %(v,), or %(a,). cases (iv) 8iis either %k 8 , or 8b %,. case (v) ‘Xiis %k M 8,. case (vi) %iis 8 k 4 8,. case (vii) 8, is 4Mh%k.
4
4
Conversely, if a sequence (xiii)
a2,. . . , 8,
of sets satisfies conditions (viii) - (xii), then j = l , 2 ,..., n. Observing that if Q is a m a t r i x form, then
=
%(Qn,) for
G;tSf(S, m) = m E %(52), and if Q is a f u n c t i o n a l form, then
VaZ(Q,m ) = the x for which J(m,x ) E 8(Q) we can express the explicit definition of Val and Gt9f as follows: x = Val(Q,m ) if and only if there exist sequences (i), (ii), (xiii) satisfying conditions (iii) - (xii) and such that i, = 1 and J(m, x ) E 8%; G,tsf(Q,m) if and only if there exist sequences (i), (ii), (xiii) satisfying conditions (iii) - (xii) and such that i, = 2 and m E 8,.
63
CLASS %t
The logical form of these definitions could be simplified by an identification of sequences of integers with integers representing these sequences (cf. Chapter I , section 2, p. 15). Furthermore, we can identify finite sequences of sets of integers with single sets: Instead of the sequence (xiii) we can consider the set 3E of integers g with the properties
L(g) = n, gi
E
!Xi for i = 1, 2, . . ., L(g).
We omit the details of these simplifications since they are very easy and not essential for our further purpose.
5. Class Sr.A m a t r i x f o r m @ will be said to belong to the class Sr if every integer satisfies @: @
E
Sr 55 (@ E rnf)*(m)G,tGf(@, m).
I n case when @ is a sentence, we shall often say “@ is true” instead of “@ is in the class Sr”.We shall abstain however from using the word “true” in cases when@ contains free variables. Instead of @ E S t we shall write sometimes t+@. An example of a m a t r i x which belongs to the class Sr is @ = q M 1 where
v = (Pl)((VI
x
Vl) M
1%
x a1)
+ 1).
Indeed, we have seen in section 3, p. 59 that VaZ(v,m) = 1 for every m,and since ‘VaZ(1,m) = 1, we obtain VaZ(cp,m) = VuZ(1, m) and hence G,taf(@,m)for every m. Theorem 1. If @ andY are in Sr,then so is M p ( @ , Y ) . Proof. We can assume that @ =!P 52 since otherwise the theorem is evident. Using the formula ( 7 ) of section 3 and the assumptions that
G,t$f(Y,m ) and GtGf(Y3 SZ, m) we obtain GtSf(l2, m ) whence 52 E % . Since 52 = Mp(@,!P) the theorem is proved. Theorem 1 says that the class S r is closed under the rule (or
64
SEMANTICS OF
(s)
better the operation) of "modus ponens". We shall now prove the same for the operation of s u b s t i t u t i o n . To achieve this we need two lemmas: L e m m a 2. I f 9 is a n e x p r e s s i o n and m', m" two integers satisfying the following conditions
if a, occurs in 9, then %ii= %;j, if I n d ( v i , 9 )= 2, then Eij-l= Eii-l, then
on Q. If SZ = 1, equation (2) is evident. Proof. We use -induction -n If SZ = v,, then ?iiiih-l= rn%-l since Ind(v,, v h ) = 2, and hence the equation (2) is satisfied because its left hand side is equal to %k-l The proof in the case 9 = ah is and its right hand side to similar. Assume now that (2) holds for two f u n c t i o n a l forms rp and y and let 9 be one of the expressions p + y , rp x y , p M y . If nz' and mKsatisfy the assumptions of the lemma with respect to 9 they do so with respect to rp and y since Oc(a, rp) = 2 or Oc(a, y) = 2 implies that Oc(a, 9 )= 2 and Ind(v,, p) = 2 or Ind(v,, y ) = 2 implies that I n d ( a , Q) = 2. Hence the equation (2) holds for the f u n c t i o n a l f o r m s rp and y , and we easily infer that this equation holds also for the f u n c t i o n a l forms rp + y , rp x y and for the matrix form p m y . In a similar way we show that if (1) holds for two m a t r i x forms @ andY, i t does so for the m a t r i x f o r m @+Y. Let us finally assume that (1) holds for the m a t r i x f o r m @, and let 9 be the f u n c t i o n a l form ( , m h ) @ . Assume that m' andm" satisfy the hypothesis of the lemma with respect to Q. If ai occurs in @, it does so in 12whence %iL= If Ind(v,.,@) = 2 and j # h, then Ind(vi,Q) = 2 and hence i + i ~ i= ~-~ It follows that, if a is an arbitrary integer, then the integers m* = CZ,,-~(m', a ) and
%Ii.
c u s s 'kr
65
m** = C2h-I (mH, a ) satisfy the assumption of the lemma with respect to @. By the inductive hypothesis we obtain therefore
Giti?f(@, Cm--1 (m', a ) ) = G;tsf(@,C2h-I (m",a ) ) for every a. Since Val(J2,m') (or VaZ(J2,m f f ) )is defined as the smallest a satisfying the left (or the right) hand side of this equivalence or as 1 if no such integer a exists, we infer that Val(s2,m') = VaZ(J2,m"). Lemma 2 is thus proved. Lemma 3. If is a functional form, @ a matrix form, and if a is a functional form such that the bound variables which occur in a and have the indices 2 occur neither in qmor in@,tihen (1)
(2)
Val(S(i,a, q),m)= Val(9,&(m, Val(a,m ) ) ) , Giti?f(S(i,a,@),m) = G;ti?f(@,Czi(m, VaZ(a,m))).
Proof. We apply again the method of induction. If is 1, then (1) is evident. If y = ai with j # i, then the left hand side of (1) is and the right hand side is W(pzg,C,(m, VaZ(a,m))) If = ai, then the left hand side of (1) is TraZ(a,m) and the right side is W(pZi,Czi(m,VaZ(a,m ) ) )= VaZ(a,77%). If q~ = vi, then the left hand side of ( 1 ) is and the right hand side is W(p2i-l, CZi(m,Val(a,m ) ) ) = E2i-1.Hence (1) is satisfied for the case when p is one of the simplest f u n c t i o n a l f o r m s 1, ai, vi. Assume that ( 1) holds for two f u n c t i o n a 1 f o r m s v1 and v2and put Q, = q1 v2. Since S(i, a , 9)= X(i, a, Q ) ~ ) X(i, a, v2) we obtain by the inductive hypothesis
=zzj.
zzi
zv-l
+
+
+
V a W ( i ,a, TI, m)= Val(fJ(i,a, Q)J, m ) Val(&, a, v2),m) = = Val(% C&, Val(a,m ) ) )+ V a k % ,C2@, Val(a,m ) ) )= Val(%+ Vz, CZi(m, VaJ(a,m ) ) )= Val(%C,,(m, Val(a,m ) ) ) .
+
It follows from these equations that the lemma is true for the f u n c t i o n a l f o r m q~ = v1 v2, and we can show quite similarly that it is true also for the f u n c t i o n a l form tpl x y 2 and for the m a t r i x f o r m v1 w y2.
+
5
66
SEMANTICS OF
(s)
If we assume that ( 2 ) holds for the m a t r i x f o r m s and we can show by the same method as above that (2) is also valid for the m a t r i x f o r m Finally, let us assume that ( 2 ) holds for a m a t r i x f o r m @ and put p = (,uvA)@.Since S( i , a, p) = ( p A ) S ( ia,@), , the integer VaZ(S(i,a, q),m) is equal to the least a such that
G;tef(fJ(i,a, @I, C,-,(m, a ) )
(3)
or to 1 if no a with the property (3) exists. According to the inductive hypothesis (3) is equivalent t o (4)
Wf(@, CdC2h-ltm, 4 , V 4 a , C,h-l(m, a ) ) ) ) .
Hence, Vd(X(i, a, q),m) is the smallest a for which (4) holds or 1 if there is no such a. Observe now that the integer m’ = C,-l(m, a ) satisfies the equations E; = for all j f 2h - 1. According to the assumption of the lemma the index of vAin a is 1 (because v,, occurs in q). Hence, by lemma 2, VaZ(a,m’)= VaZ(a,m)and (4)is thus reduced to the equivalent formula
zj
GW@,~2i(C2A-l(m,a ) , ‘Val(a,W ) ) . Using the commutativity property of the function C established in the equation (1) of section 1 (p. 56) we transform the last formula into the following one
G M @ ,C,-l(Qzi(m, V a k , m ) ) , 4 ) . According to the definition given in section 2, the least a satisfying this formula is equal to (5)
If no such a exists, then (5) is equal to 1. It follows that (5) and VaZ(S(i,a, q),m) are equal and the lemma 3 is proved. T h e o r e m 4. If@ is in Sz:and a in %e, then X(i, a,@) is in St. Proof. Lct m be an arbitrary integer. @ being an element of Zt, we have G;tGf(@,C2i(m,VaZ(a,m ) ) ) whence, by lemma 3,
CLASS
53
67
G?t$f(s(i,a,@), m).Since m is arbitrary, we obtain X ( i , a,@) E ‘Zr, q.e.d. Theorem 4 shows that the class Sr is closed under the operation of s u b s t i t u t i o n . Theorem 5. If @ is an a x i o m of (S), then @ i s in ‘ZL It will be sufficient to prove this only for the axioms (1) and (3) of the group I1 since the proofs for the remaining axicms are very easy and do not require any new technical device. Let us assume that @ is a m a t r i x , that ai occurs in @ and that v h does not occur in @. We have to show that if m is an integer such that Gt$f(@,m), then Gtgf(x(i,(pvh)x(i,vh,
@),@)?
m).
By lemma 3 this is equivalent to (1)
G?tN@,C , i h V 4 ( p v d S ( i ,v,, @), m))).
First we calculate VaZ((pv,)S(i,v,,@), m). To find this integer we must look for the least a such that Gtsf(S(i,v,, @), Czn-l(m,a ) ) , or what is the same (2)
GiGf(@,C,,(C,-,(%
Since Vd(vh, C,-,(m, (3)
a ) ) = a, ( 2 ) is equivalent to a ) ,4 ) .
GtSf(@,C,i(c,,-,(m,
Put m’ = C,-,(m, a ) , m” in @ we have by lemma 2 and
a ) , Val(v,, Ca-,(m, a ) ) ) .
= Czi(m’,
a). Since v,, does not o c c u r
Gt8f(@,?n)_= G;tSf(@,m’)
GtSf(@,C,Jm, a ) ) = G?t$f(@, Czi(m’,a ) )= Gt$f(@,m”)
which proves that (3) is equivalent to (4)
Gt$f(@,C,,(m, a ) ) .
Tlierc exists at least one integer a satisfying the condition (4). Indeed, since Gt$f(@,m)in virtue of our hypothesis made at the
68
SEMANTICS OF
(s)
beginning of the proof, and since Czi(m,Z2J= m, we infer that the integer satisfies the condition (4). Let a, be the least integer satisfying (4). It follows that
z2*
(5) (6)
G;t$f(@,C2i(m, ao)), Vu&(pvN(&v,, @I, m) =
and these equations prove that the formula (1) is satisfied. Hence, the axiom I1 1 is in St (more exactly: all substitution-instances of this axiom are in Sr). We pass now to the axiom I1 3. Let us assume again that @ is in %, that a, occurs in @,and that vh, ak, a, do not occur in di. Assume further that Gtgf((pvh)#fi,v&,@)
+
ak
It follows easily from this assumption that VuZ((pvh)8(i,v,, di), m ) = VuZ(ak,m)
m).
+ VuZ(a,, m )
whence
(7)
V 4 ( p v , ) W , v,, @),
m)
# 1.
The left hand side was calculated above: it is equal to the least a for which (4) is satisfied or to 1 if no such a exists. Formula (7) proves therefore that an a satisfying (4) exists. Let us denote the least a of this kind by a,. It follows from the definition of a, that
-
(8) so > 1, GtBf(@,C,,(m,ao)),
Since
Gt3f(@,C2i(m, b ) ) for b
< a,.
G;tSf(x(i,ak,@), m ) = G;tgf(@,C2$(m,VuZ(ak,m ) ) )
- etgf(s(i,
according to lemma 3 and since VuZ(a,, m ) i.e.,
&k,@),
< ao,we obtain from (8)
m,
GtSf(h’(i, ak,@)-+1 m 1
+ 1, m).
This proves that axiom I1 3 is in St (more exactly that all substitution instances of that axiom are in St).
CLASS
%
69
From theorems 1, 4, and 5 we obtain easily T h e o r e m 6. All provable m a t r i c e s are in Proof. Let @ be a provable m a t r i x and
..
.,@%
Sr.
=@
its formal proof. We shall show by induction that every @$ is in Zr. This is evident if@iis in and in particular if i = 1 (since the first term of an arbitrary formal proof is a.lways an axiom). Let us assume that j n and that fp, E S ~for : i < j. Three cases are possible:
<
(1) is an axiom, there are k, 1 both less than j such that @6i = Hp(@k>@J), (2) (3) there are integers k, h, y such that k < j , y E !Re, and @j
= S(h, f p , @ k ) .
I n each of these cases !Dj is in Zr: in the case (1) in virtue of theorem 5, in the case (2) in virtue of theorem 1, and in the case (3) in virtue of theorem 4. It follows now by induction that @., is in Sr for every j . Putting j = n we obtain therefore the desired result. T h e o r e m 7. The set zr~'9.R is consistent. Proof. If this set were inconsistent, every m a t r i x and in particular the s e n t e n c e 1 M 1 1 would be Sr-provable. This is impossible since it would imply that 1 M 1 1 is in %r whereas we know that for every m GtGf(1 m 1 1, m). As a rather important corollary we obtain from theorem 7 the following result :
+
N
+
+
Theorem 8. The set S i s consistent. Indeed, S is a subset of 'Xr A 102 and the subset of a consistent set is itself consistent. Theorem 9. The set Sr A r331 is complete. Proof. Let @ be an arbitrary sentence. Since no ai o c c u r s in @ and every vi which o c c u r s in @ has therein the index 1, the assumptions of lemma 2 are satisfied for arbitrary m' and m". It
70
SEMANTICS OF
(s)
follows that if there exists at least one integer m such that G;t!Zf(@,m), then every integer satisfies this condition and hence @ is in %r A 9JI. If no m satisfies the condition G,tSf(@,m),then for every m
Wf(@ --f 1 M 1
+
+ 1, m)
and hence @ -+ 1 w 1 1 E S r A 9JI. Hence either @ or *@ is in Sr A mt which proves that this set is complete. To finish this Chapter we discuss still the problem of the u)consistency of the set Xr A !JX. We need the following auxiliary theorem : Theorem 10. If @ is a m a t r i x in which exactly one free variable ai occurs, and if vh is a b o u n d variable which does not occur in @, then Val((pv,JS(i, vh,@),m ) is the least p such that S(i,LIP,@) is in S r provided that such integers exist; otherwise vh,@),m) is 1. VuZ((,uvh)X(i, Proof. Let us assume that
S ( i ,D,,@) is in 23,
(1)
i.e., that G;t!Zf(S(i, L I P , @ ) , m) for an arbitrary m. According to lemma 3 this assumption is equivalent to the formula We shall show that this condition is in turn equivalent to
Gtsf(fl(i, v h , @), ca-,(nztPI)* Indeed, formula (2) is equivalent to G,tsf(@,C,,(C2h-l(m,p ) ,p ) ) , and since vh does not o c c u r in @, we may replace here C%-l(m, p ) by m without influencing the validity of the formula (cf. lemma 2). We prove further that if (2) holds for a t least one m, it does so for a.ny m. Indeed, S(i,vh,@) is an expression in which only the variable vh has the index 2 and in which no f r e e v a r i a b l e occurs. According to lemma 2 formula (2) is equivalent to (2)
G,taf(s(i,vh,@),m') where m' is an arbitrary integer such that
-#
m2h-1
=
w(pzh-19
p ) ) = P'
CLASS
%r
71
I n particular, we can take as m' the integer C,-,(n, p ) where n is wholly arbitrary since this choice of m' satisfies the above equation. Hence, if ( 2 ) holds for at least one m, it does so for every m. We have thus shown that (1) is equivalent to each of the following conditions: (2) holds for a t least one m ; ( 2 ) holds for every m. We can now prove theorem 10. Assume that there are integers satisfying (1) and let p be the least of them. Let m be arbitrary. We have then the formula (2) and for no q < p can the formula
G;tBf(W, Vh,@),
Qul-lh
a))
be satisfied. Indeed, if this formula were true X(i, D,, @) would be in Sr,and this contradicts the definition of p . Hence p is the least integer for which (2) holds which proves that v a ( p v , , ) w , %@), m ) = P. Assume now that (1) does not hold for any p . Hence ( 2 ) is false for arbitrary p and m, and we obtain according to definition of Val (section 3, equation (8), p. 59) va(pv,,)S(i,Vh,@), m ) = 1 for arbitrary m. Theorem 10 is thus proved. As an easy corollary we obtain Theorem 11. The set S r A %! is m-consistent. Proof. Let @ be a m a t r i x in which exactly one f r e e v a r i a b l e a1 occurs. Assume that X(l,D,, -A@) is in Sr for n = 1, 2, . . . . It follows that for no n S( 1, D,, @) is in S r and hence val((/tvh)fl(
vh,
m, =
which proves that (1)
(pvh)S(l,v,,, 0)M D, is in
Sr.
Using theorem 1 of Chapter 111, section 4, p. 45 we obtain that the following s e n t e n c e (pvh)fl(l, vh,@)
Dl
--f
[S(l, (pvh)S(l, vhy@),@) -tfi(l, D1,@)l
72
SEMANTICS OF
(s)
is in Zr. On account of (1) we obtain therefore that
S ( l , (Pvh)x(l, vh,@)*@)
D1?@)
is in Zr.This proves that S(1,(pvh)S(l,v,,@),@) is not in %r since otherwise S(1, Ill,@) would be in ZT which is not the case, as we have shown above. Theorem 11 is thus proved. It implies, of course, that also the set X is o-consistent. We note still the following corollaries to the theorem 10: T h e o r e m 12. If @ is a m a t r i x in which exactly one free variable 9 occurs and if v,, is a bound variable which does not occur in @, then (EVh)X(i,Vh,@) E St: (AVh)&(i,Vh, @) E St
(ZIn)S(i,Dn,@) E Sr, (n)&(i, on, @) E %r.
1
1. Representability of infinite sequences. I n the present Chapter we shall work with infinite sequences almost all terms of which are equal to 1. Every such sequence al, a2, . . . , a , , 1, 1, . . . can conveniently be represented by the integer m
= pY-1
.p2--1
... pEw-1
where pi is the i-th prime. Conversely, every integer m determines a sequence almost all terms of which are equal to 1. I n order to obtain this sequence we put
a,
=
WP,,m)
where W ( x ,y) is the least integer z such that y is not divisible by xz. We shall also use the shorter symbol z, instead of W(pn,m). We shall often use the auxiliary function
It folIows from this definition that W(pn, Ci(m, a ) ) = WP,,m ) if i f n W(Pi, Cdm, a ) ) = a. The function C enables us therefore to construct from a given integer m another integer representing a sequence which differs but in the i-th term from the sequence represented by m. Note the following useful property of the function C: (1)
If i # j, then Ci(Cj(m,a ) , b ) = Ci(C,(m,b ) , a).
This Chapter is based entirely on works of Tarski. Cf. his papers 1217 and [22]. 1.e. all with an exception of at most finite number.
67
VALUES OF FUNCTIONAL FORMS
2. Values of functional forms and the notion of satisfaction for matrix forms. Before we give exact definitions of these basic semantical notions we shall explain briefly their intuitive content. Let p be a f u n c t i o n a l form, e.g. p = p'
=
v,
+ (a, x a2) or p = p" = (pv,)(v, x v,
M
a,).
Speaking intuitively, every such f o r m represents an arithmetical function of as many variables as there are f r e e v a r i a b l e s and bound v a r i a b l e s with the index 2 o c c u r r i n g in p. For instance p' represents the function F(x, y, z ) = x + y . z and 9'' represents the function F ( x ) whose value is 1 if 5 is not a square of an integer and which is equal to y if x = y2. Let us ascribe arbitrary numerical values to free v a r i a b l e s occurring in p and also to the b o u n d v a r i a b l e s which occur in p and have therein the index 2. The function F represented by p takes then on a numerical value which we shall call the value of p for the given values of the variables. For instance, the value of p' is 7 if we ascribe the value 3 to v, and the values 2 to a, and a,. If we ascribe the value 2 to a,, then the value of p" is 1; if the value of a1 is 4, then the value of p" is 2. Let now @ be a m a t r i x f o r m , e.g. p' M pn-+p' M 1 and let us ascribe arbitrary values to the f r e e v a r i a b l e s o c c u r r i n g in @ as well as to the b o u n d v a r i a b l e s which occur in@ and have therein the index 2. The intuitive meaning of@is this: If the values of p' and p" are equal, then the value of 9' is 1. Hence @ represents a theorem of arithmetic and this theorem can be either true or false. I n the first case we say that the values given to the v a r i a b l e s satisfy the m a t r i x f o r m @, and in the second that these values do not satisfy@. For instance in the example considered above the values 2, 1, 2 given to the v a r i a b l e s v,, a,, a,satisfy@. It is convenient to ascribe values to all v a r i a b l e s simultaneously, independently of whether they occur in the expression which we consider or not. Every such system of values can be is the identified with an infinite sequence a,, a,, . . . in which value given to the v a r i a b l e v, and a2, the value given to the v a r i a b l e a,. Since we shall never deal simultaneously with an
58
SEMANTICS OF
(s)
infinite number of expressions, we do not need to consider wholly arbitrary sequences but can limit ourselves to sequences with almost all terms equal to 1. Every such sequence can be represented by an integer in the way explained in section 1. It follows that the value of a f u n c t i o n a l form tp is a function of cp and of an integer m which synthetizes the values ascribed to the variables. For the same reason the notion of satisfaction is a binary relation between m a t r i x -forms and integers. We shall denote by VaZ(v, m ) the value of cp for values of variables represented by the integer m and shall write Gt@f(@,m) instead of “the values of variables represented by m satisfy @”.
3. Inductive definition of Val ( 9 , m ) and GtGf ( @ , m ) . In this section we shall give an exact definition of the notions which were explained intuitively in the previous section. To obtain this definition we first define VaZ(cp,m) for the simplest f u n c t i o n a l f o r m s ah,vh, and 1 and then define it for the f u n c t i o n a l forms v y , q.~x y under the assumption that it has been defined for the f u n c t i o n a l f o r m s cp and y. Under the same assumption we define also the meaning of the formula GtGf(cp M y , m ). Next we define the meaning of the formula Gt@f(di+?P,m)under the assumption that the meaning of the formulas GtGf(@,m) and GtGf(y, m) are already defined. Finally we define Vd((,uvh)@,?n) under the assumption that the meaning of the formula G,tGf(@,m‘) is already defined for an arbitrary integer m‘. I n this way Val( v, m) and GtGf(@,m) will be defined for arbitrary cp in and @ in %Rf. We divide our inductive definition into eight parts:
+
sf
INDUCTIVE DEFINITION OF
v U l ( f p , ?n)
AND
Bt#f(@, m )
59
VaZ((,uvh)@, m ) = 1 if there is no a such that G,tsf(@, Cfi-l(m,a)), otherwise VaZ((pvh)@, m ) = the least such integer a. To illustrate how this definition works we take
(8)
fP = (PVl”V1 x
TI)
-
((a1x
ar) + 1>1
and calculate VaZ(cp,m). By (2), (3), (a), and (5) we obtain 1 whence by (6) VaZ(vl x vl,m ) =z;, VaZ((al x q) 1, m ) G,tGf(((vlx vl) Since x2
M
+
(al x q) I),
+
==:+
c1(m,a ) )3 (a2 = Gi;
+ I).
+ 1 is never a square, we infer that there is no a such that GfGf(((v1x
Vl)
= (a1 x
a1)
+ I), Cdm, a ) )
and consequently VaZ(cp,m ) = 1 according to (8). One should not be deceived by the superficial similarity of the inductive definition given in this section and the inductive defbition given in Chapter 11,section 2, p. 28. To explain the chief difference between these definitions we remark the following. In both definitions we have integers on which the induction proceeds (they are denoted by “by’ and “c” in Chapter 11, and by “9” and “@” in the present Chapter). Furthermore, we have in both definitions the parameters (“a”, “i”, (‘p” in Chapter 11, and “m” in the present Chapter). Now the parameters in the definitions of Chapter I1 are kept constant whereas in the present definition they are variable (cf. (8)).This has the effect that we can calculate the values of functions defined in Chapter I1 by tracing backwards the steps of the definition, and arriving to an end after a finite number of steps. E.g. if we have to calculate Oc(a, b ) we try to decompose b into simpler constituents; if we find for instance that b = b, -+ b,, we reduce our problem to a calculation of Oc(a, b,) and Oc(a, b,). After a finite number of such steps we arrive finally to the values Oc(a, I), Oc(a, vh), and Oc(a, ah) which are given explicitly and obtain then the value of Oc(a, b ) by repeated substitutions. The situation is entirely different in case of the definition given
60
SEMANTICS OF
(s)
in the present Chapter. Indeed, if v = (pvh)@,then the calculation of VaZ(9,m ) is reduced to that of GtSf(@,rn’) for infinitely many different values of m‘. Hence the calculation of VaE(v,m) cannot be completed in a finite number of steps and the definition given in the present Chapter has an “infinitary” character which distinguishes it essentially from the superficially similar “finitary” definitions of Chapter 11. Because of the infinitary character of the definition (1)- (8) we cannot expect that the same method which we used in Chapter I1 will allow us to replace the inductive definition by an explicit one. As a matter of fact it can be shown that an explicit definition of Val and Gt3f is possible only when we use the general notion of an arbitrary set of integers. In the next section we shall outline an explicit definition of VaZ and Gt3f. The proof that it cannot be replaced by a purely arithmetical one (i.e. such which avoids the notion of an arbitrary set) will be given in Chapter VI.
3E
4. Explicit definition of Val and Gtef 3. We shall say that a set of integers represents a function if I
(1) for every n there is in d a n z such that K,(x) = n, (2) for every x and y in Z if K,(x) = K,(y), then K&) = K,(y).
To explain this definition we remark that a function F can be identified with the set of ordered pairs (n, F ( n ) )and a pair (a, b ) can be identified with the integer J ( a , b ) (cf. Chapter I, p. 14). In this way a function F is converted into a set Z and it is easy to show that this set must satisfy the conditions (1) and (2). For every di in r3nf let us denote by a(@) the set of those m for which G,t%f(di,m).Similarly for every p in Sf let us denote by U(y) the set representing the function VaZ(v7m) treated as the function of m alone. It can be easily shown that % ( I ) is the set of all integers having the form J(m, 1): 2 E a(1)ZZ K,(S) = 1. a
Cf. Tarski [20], pp. 311-312.
EXPLICIT DEFINITION OF
Val
AND
Bt8f
61
It is also easy to prove that %(vh) Consists of all integers of the form J(m, Ea-J i.e. of the form J(m, W ( P ~ -m~) ),, and that %(a,) consists of all integers of the form J(m, Za) = J(m, W(p,, m ) ) : Kl(Z)), W(p,, K,(s)).
2 E '%(vh)
K2(x)
= w(P2h-1,
x
K,(4
=
E '%(a,)
We introduce further certain operations on sets of integers which we denote by symbols similar to those used for operations on expressions :
23 i- a 23
i
Q
5
5
W 3 b ) ( 3 C )[ ( b E B) * (c E Q)*
(Kl(b)= W C ) = &(4)* (K2(b)+ K2(c) = K2(4)1, h ( 3 b ) ( 3 C ) [ ( bE % ) . ( C E Q).
(Kl(b) = KAC)= w4) (K2(b).IC2(c) * = K2(4)1, 23 w 6 = W 3 y ) ( J ( x ,Y) E 93 * Q), 23 -+ Q 5 -B v Q,
MhB = h [ K 2 ( s )= min,(C~h-l( K I ( x ) a, ) E B or K2(z)= 1 provided that no such a exists].
It can be shown without essential difficulties that WV
+ w)
=
wcp x Y ) =
WJ) iWw), Wcp) i W Y ) ,
Wcp w
Y)=W V ) WY), %(@+!P) = a(@) I,%(!P),
W(IUVh)@) =
Y)z,W@).
Let now 9 be an arbitrary expression. It follows easily from the definition of the class of e x p r e s s i o n s that there exists a sequence of expressions
md a sequence
(i) Ql, Q2, . . ., Qs = Q (ii) il, i,,
. . ., in
62
SEMANTICS O W
(s)
whose elements are integers 1 or 2 such that for every j of the following conditions is satisfied :
< n one
(iii) Q, = 1 or Q, E 233‘6 or 52, ~ 2 3 fand i, = 1, (iv) Qj = Q k + Q, or Qj = Qk x S, and i, = ik = i, = 1 (k < j , 1 < j ) , (v) S,= Qk M Q, and ij = 2, i k = i, = 1 (k < j , 1 < j ) , (vi) Q, = Q, -+Q, and ii = i k = i, = 2 (k < j , 1 < j ) , (vii) Q, = (pvh)Qk and ij = 1, i, = 2 ( j < k). Put %, = %(Q,) for j of sets such that (viii) (ix) (x) (xi) (xii)
In In In In In
the the the the the
=
1, 2,
. . ., n. We
obtain thus a sequence
cases (iii) 2Ij is either %(I), or %(v,), or %(a,). cases (iv) 8iis either %k 8 , or 8b %,. case (v) ‘Xiis %k M 8,. case (vi) %iis 8 k 4 8,. case (vii) 8, is 4Mh%k.
4
4
Conversely, if a sequence (xiii)
a2,. . . , 8,
of sets satisfies conditions (viii) - (xii), then j = l , 2 ,..., n. Observing that if Q is a m a t r i x form, then
=
%(Qn,) for
G;tSf(S, m) = m E %(52), and if Q is a f u n c t i o n a l form, then
VaZ(Q,m ) = the x for which J(m,x ) E 8(Q) we can express the explicit definition of Val and Gt9f as follows: x = Val(Q,m ) if and only if there exist sequences (i), (ii), (xiii) satisfying conditions (iii) - (xii) and such that i, = 1 and J(m, x ) E 8%; G,tsf(Q,m) if and only if there exist sequences (i), (ii), (xiii) satisfying conditions (iii) - (xii) and such that i, = 2 and m E 8,.
63
CLASS %t
The logical form of these definitions could be simplified by an identification of sequences of integers with integers representing these sequences (cf. Chapter I , section 2, p. 15). Furthermore, we can identify finite sequences of sets of integers with single sets: Instead of the sequence (xiii) we can consider the set 3E of integers g with the properties
L(g) = n, gi
E
!Xi for i = 1, 2, . . ., L(g).
We omit the details of these simplifications since they are very easy and not essential for our further purpose.
5. Class Sr.A m a t r i x f o r m @ will be said to belong to the class Sr if every integer satisfies @: @
E
Sr 55 (@ E rnf)*(m)G,tGf(@, m).
I n case when @ is a sentence, we shall often say “@ is true” instead of “@ is in the class Sr”.We shall abstain however from using the word “true” in cases when@ contains free variables. Instead of @ E S t we shall write sometimes t+@. An example of a m a t r i x which belongs to the class Sr is @ = q M 1 where
v = (Pl)((VI
x
Vl) M
1%
x a1)
+ 1).
Indeed, we have seen in section 3, p. 59 that VaZ(v,m) = 1 for every m,and since ‘VaZ(1,m) = 1, we obtain VaZ(cp,m) = VuZ(1, m) and hence G,taf(@,m)for every m. Theorem 1. If @ andY are in Sr,then so is M p ( @ , Y ) . Proof. We can assume that @ =!P 52 since otherwise the theorem is evident. Using the formula ( 7 ) of section 3 and the assumptions that
G,t$f(Y,m ) and GtGf(Y3 SZ, m) we obtain GtSf(l2, m ) whence 52 E % . Since 52 = Mp(@,!P) the theorem is proved. Theorem 1 says that the class S r is closed under the rule (or
64
SEMANTICS OF
(s)
better the operation) of "modus ponens". We shall now prove the same for the operation of s u b s t i t u t i o n . To achieve this we need two lemmas: L e m m a 2. I f 9 is a n e x p r e s s i o n and m', m" two integers satisfying the following conditions
if a, occurs in 9, then %ii= %;j, if I n d ( v i , 9 )= 2, then Eij-l= Eii-l, then
on Q. If SZ = 1, equation (2) is evident. Proof. We use -induction -n If SZ = v,, then ?iiiih-l= rn%-l since Ind(v,, v h ) = 2, and hence the equation (2) is satisfied because its left hand side is equal to %k-l The proof in the case 9 = ah is and its right hand side to similar. Assume now that (2) holds for two f u n c t i o n a l forms rp and y and let 9 be one of the expressions p + y , rp x y , p M y . If nz' and mKsatisfy the assumptions of the lemma with respect to 9 they do so with respect to rp and y since Oc(a, rp) = 2 or Oc(a, y) = 2 implies that Oc(a, 9 )= 2 and Ind(v,, p) = 2 or Ind(v,, y ) = 2 implies that I n d ( a , Q) = 2. Hence the equation (2) holds for the f u n c t i o n a l f o r m s rp and y , and we easily infer that this equation holds also for the f u n c t i o n a l forms rp + y , rp x y and for the matrix form p m y . In a similar way we show that if (1) holds for two m a t r i x forms @ andY, i t does so for the m a t r i x f o r m @+Y. Let us finally assume that (1) holds for the m a t r i x f o r m @, and let 9 be the f u n c t i o n a l form ( , m h ) @ . Assume that m' andm" satisfy the hypothesis of the lemma with respect to Q. If ai occurs in @, it does so in 12whence %iL= If Ind(v,.,@) = 2 and j # h, then Ind(vi,Q) = 2 and hence i + i ~ i= ~-~ It follows that, if a is an arbitrary integer, then the integers m* = CZ,,-~(m', a ) and
%Ii.
c u s s 'kr
65
m** = C2h-I (mH, a ) satisfy the assumption of the lemma with respect to @. By the inductive hypothesis we obtain therefore
Giti?f(@, Cm--1 (m', a ) ) = G;tsf(@,C2h-I (m",a ) ) for every a. Since Val(J2,m') (or VaZ(J2,m f f ) )is defined as the smallest a satisfying the left (or the right) hand side of this equivalence or as 1 if no such integer a exists, we infer that Val(s2,m') = VaZ(J2,m"). Lemma 2 is thus proved. Lemma 3. If is a functional form, @ a matrix form, and if a is a functional form such that the bound variables which occur in a and have the indices 2 occur neither in qmor in@,tihen (1)
(2)
Val(S(i,a, q),m)= Val(9,&(m, Val(a,m ) ) ) , Giti?f(S(i,a,@),m) = G;ti?f(@,Czi(m, VaZ(a,m))).
Proof. We apply again the method of induction. If is 1, then (1) is evident. If y = ai with j # i, then the left hand side of (1) is and the right hand side is W(pzg,C,(m, VaZ(a,m))) If = ai, then the left hand side of (1) is TraZ(a,m) and the right side is W(pZi,Czi(m,VaZ(a,m ) ) )= VaZ(a,77%). If q~ = vi, then the left hand side of ( 1 ) is and the right hand side is W(p2i-l, CZi(m,Val(a,m ) ) ) = E2i-1.Hence (1) is satisfied for the case when p is one of the simplest f u n c t i o n a l f o r m s 1, ai, vi. Assume that ( 1) holds for two f u n c t i o n a 1 f o r m s v1 and v2and put Q, = q1 v2. Since S(i, a , 9)= X(i, a, Q ) ~ ) X(i, a, v2) we obtain by the inductive hypothesis
=zzj.
zzi
zv-l
+
+
+
V a W ( i ,a, TI, m)= Val(fJ(i,a, Q)J, m ) Val(&, a, v2),m) = = Val(% C&, Val(a,m ) ) )+ V a k % ,C2@, Val(a,m ) ) )= Val(%+ Vz, CZi(m, VaJ(a,m ) ) )= Val(%C,,(m, Val(a,m ) ) ) .
+
It follows from these equations that the lemma is true for the f u n c t i o n a l f o r m q~ = v1 v2, and we can show quite similarly that it is true also for the f u n c t i o n a l form tpl x y 2 and for the m a t r i x f o r m v1 w y2.
+
5
66
SEMANTICS OF
(s)
If we assume that ( 2 ) holds for the m a t r i x f o r m s and we can show by the same method as above that (2) is also valid for the m a t r i x f o r m Finally, let us assume that ( 2 ) holds for a m a t r i x f o r m @ and put p = (,uvA)@.Since S( i , a, p) = ( p A ) S ( ia,@), , the integer VaZ(S(i,a, q),m) is equal to the least a such that
G;tef(fJ(i,a, @I, C,-,(m, a ) )
(3)
or to 1 if no a with the property (3) exists. According to the inductive hypothesis (3) is equivalent t o (4)
Wf(@, CdC2h-ltm, 4 , V 4 a , C,h-l(m, a ) ) ) ) .
Hence, Vd(X(i, a, q),m) is the smallest a for which (4) holds or 1 if there is no such a. Observe now that the integer m’ = C,-l(m, a ) satisfies the equations E; = for all j f 2h - 1. According to the assumption of the lemma the index of vAin a is 1 (because v,, occurs in q). Hence, by lemma 2, VaZ(a,m’)= VaZ(a,m)and (4)is thus reduced to the equivalent formula
zj
GW@,~2i(C2A-l(m,a ) , ‘Val(a,W ) ) . Using the commutativity property of the function C established in the equation (1) of section 1 (p. 56) we transform the last formula into the following one
G M @ ,C,-l(Qzi(m, V a k , m ) ) , 4 ) . According to the definition given in section 2, the least a satisfying this formula is equal to (5)
If no such a exists, then (5) is equal to 1. It follows that (5) and VaZ(S(i,a, q),m) are equal and the lemma 3 is proved. T h e o r e m 4. If@ is in Sz:and a in %e, then X(i, a,@) is in St. Proof. Lct m be an arbitrary integer. @ being an element of Zt, we have G;tGf(@,C2i(m,VaZ(a,m ) ) ) whence, by lemma 3,
CLASS
53
67
G?t$f(s(i,a,@), m).Since m is arbitrary, we obtain X ( i , a,@) E ‘Zr, q.e.d. Theorem 4 shows that the class Sr is closed under the operation of s u b s t i t u t i o n . Theorem 5. If @ is an a x i o m of (S), then @ i s in ‘ZL It will be sufficient to prove this only for the axioms (1) and (3) of the group I1 since the proofs for the remaining axicms are very easy and do not require any new technical device. Let us assume that @ is a m a t r i x , that ai occurs in @ and that v h does not occur in @. We have to show that if m is an integer such that Gt$f(@,m), then Gtgf(x(i,(pvh)x(i,vh,
@),@)?
m).
By lemma 3 this is equivalent to (1)
G?tN@,C , i h V 4 ( p v d S ( i ,v,, @), m))).
First we calculate VaZ((pv,)S(i,v,,@), m). To find this integer we must look for the least a such that Gtsf(S(i,v,, @), Czn-l(m,a ) ) , or what is the same (2)
GiGf(@,C,,(C,-,(%
Since Vd(vh, C,-,(m, (3)
a ) ) = a, ( 2 ) is equivalent to a ) ,4 ) .
GtSf(@,C,i(c,,-,(m,
Put m’ = C,-,(m, a ) , m” in @ we have by lemma 2 and
a ) , Val(v,, Ca-,(m, a ) ) ) .
= Czi(m’,
a). Since v,, does not o c c u r
Gt8f(@,?n)_= G;tSf(@,m’)
GtSf(@,C,Jm, a ) ) = G?t$f(@, Czi(m’,a ) )= Gt$f(@,m”)
which proves that (3) is equivalent to (4)
Gt$f(@,C,,(m, a ) ) .
Tlierc exists at least one integer a satisfying the condition (4). Indeed, since Gt$f(@,m)in virtue of our hypothesis made at the
68
SEMANTICS OF
(s)
beginning of the proof, and since Czi(m,Z2J= m, we infer that the integer satisfies the condition (4). Let a, be the least integer satisfying (4). It follows that
z2*
(5) (6)
G;t$f(@,C2i(m, ao)), Vu&(pvN(&v,, @I, m) =
and these equations prove that the formula (1) is satisfied. Hence, the axiom I1 1 is in St (more exactly: all substitution-instances of this axiom are in Sr). We pass now to the axiom I1 3. Let us assume again that @ is in %, that a, occurs in @,and that vh, ak, a, do not occur in di. Assume further that Gtgf((pvh)#fi,v&,@)
+
ak
It follows easily from this assumption that VuZ((pvh)8(i,v,, di), m ) = VuZ(ak,m)
m).
+ VuZ(a,, m )
whence
(7)
V 4 ( p v , ) W , v,, @),
m)
# 1.
The left hand side was calculated above: it is equal to the least a for which (4) is satisfied or to 1 if no such a exists. Formula (7) proves therefore that an a satisfying (4) exists. Let us denote the least a of this kind by a,. It follows from the definition of a, that
-
(8) so > 1, GtBf(@,C,,(m,ao)),
Since
Gt3f(@,C2i(m, b ) ) for b
< a,.
G;tSf(x(i,ak,@), m ) = G;tgf(@,C2$(m,VuZ(ak,m ) ) )
- etgf(s(i,
according to lemma 3 and since VuZ(a,, m ) i.e.,
&k,@),
< ao,we obtain from (8)
m,
GtSf(h’(i, ak,@)-+1 m 1
+ 1, m).
This proves that axiom I1 3 is in St (more exactly that all substitution instances of that axiom are in St).
CLASS
%
69
From theorems 1, 4, and 5 we obtain easily T h e o r e m 6. All provable m a t r i c e s are in Proof. Let @ be a provable m a t r i x and
..
.,@%
Sr.
=@
its formal proof. We shall show by induction that every @$ is in Zr. This is evident if@iis in and in particular if i = 1 (since the first term of an arbitrary formal proof is a.lways an axiom). Let us assume that j n and that fp, E S ~for : i < j. Three cases are possible:
<
(1) is an axiom, there are k, 1 both less than j such that @6i = Hp(@k>@J), (2) (3) there are integers k, h, y such that k < j , y E !Re, and @j
= S(h, f p , @ k ) .
I n each of these cases !Dj is in Zr: in the case (1) in virtue of theorem 5, in the case (2) in virtue of theorem 1, and in the case (3) in virtue of theorem 4. It follows now by induction that @., is in Sr for every j . Putting j = n we obtain therefore the desired result. T h e o r e m 7. The set zr~'9.R is consistent. Proof. If this set were inconsistent, every m a t r i x and in particular the s e n t e n c e 1 M 1 1 would be Sr-provable. This is impossible since it would imply that 1 M 1 1 is in %r whereas we know that for every m GtGf(1 m 1 1, m). As a rather important corollary we obtain from theorem 7 the following result :
+
N
+
+
Theorem 8. The set S i s consistent. Indeed, S is a subset of 'Xr A 102 and the subset of a consistent set is itself consistent. Theorem 9. The set Sr A r331 is complete. Proof. Let @ be an arbitrary sentence. Since no ai o c c u r s in @ and every vi which o c c u r s in @ has therein the index 1, the assumptions of lemma 2 are satisfied for arbitrary m' and m". It
70
SEMANTICS OF
(s)
follows that if there exists at least one integer m such that G;t!Zf(@,m), then every integer satisfies this condition and hence @ is in %r A 9JI. If no m satisfies the condition G,tSf(@,m),then for every m
Wf(@ --f 1 M 1
+
+ 1, m)
and hence @ -+ 1 w 1 1 E S r A 9JI. Hence either @ or *@ is in Sr A mt which proves that this set is complete. To finish this Chapter we discuss still the problem of the u)consistency of the set Xr A !JX. We need the following auxiliary theorem : Theorem 10. If @ is a m a t r i x in which exactly one free variable ai occurs, and if vh is a b o u n d variable which does not occur in @, then Val((pv,JS(i, vh,@),m ) is the least p such that S(i,LIP,@) is in S r provided that such integers exist; otherwise vh,@),m) is 1. VuZ((,uvh)X(i, Proof. Let us assume that
S ( i ,D,,@) is in 23,
(1)
i.e., that G;t!Zf(S(i, L I P , @ ) , m) for an arbitrary m. According to lemma 3 this assumption is equivalent to the formula We shall show that this condition is in turn equivalent to
Gtsf(fl(i, v h , @), ca-,(nztPI)* Indeed, formula (2) is equivalent to G,tsf(@,C,,(C2h-l(m,p ) ,p ) ) , and since vh does not o c c u r in @, we may replace here C%-l(m, p ) by m without influencing the validity of the formula (cf. lemma 2). We prove further that if (2) holds for a t least one m, it does so for a.ny m. Indeed, S(i,vh,@) is an expression in which only the variable vh has the index 2 and in which no f r e e v a r i a b l e occurs. According to lemma 2 formula (2) is equivalent to (2)
G,taf(s(i,vh,@),m') where m' is an arbitrary integer such that
-#
m2h-1
=
w(pzh-19
p ) ) = P'
CLASS
%r
71
I n particular, we can take as m' the integer C,-,(n, p ) where n is wholly arbitrary since this choice of m' satisfies the above equation. Hence, if ( 2 ) holds for at least one m, it does so for every m. We have thus shown that (1) is equivalent to each of the following conditions: (2) holds for a t least one m ; ( 2 ) holds for every m. We can now prove theorem 10. Assume that there are integers satisfying (1) and let p be the least of them. Let m be arbitrary. We have then the formula (2) and for no q < p can the formula
G;tBf(W, Vh,@),
Qul-lh
a))
be satisfied. Indeed, if this formula were true X(i, D,, @) would be in Sr,and this contradicts the definition of p . Hence p is the least integer for which (2) holds which proves that v a ( p v , , ) w , %@), m ) = P. Assume now that (1) does not hold for any p . Hence ( 2 ) is false for arbitrary p and m, and we obtain according to definition of Val (section 3, equation (8), p. 59) va(pv,,)S(i,Vh,@), m ) = 1 for arbitrary m. Theorem 10 is thus proved. As an easy corollary we obtain Theorem 11. The set S r A %! is m-consistent. Proof. Let @ be a m a t r i x in which exactly one f r e e v a r i a b l e a1 occurs. Assume that X(l,D,, -A@) is in Sr for n = 1, 2, . . . . It follows that for no n S( 1, D,, @) is in S r and hence val((/tvh)fl(
vh,
m, =
which proves that (1)
(pvh)S(l,v,,, 0)M D, is in
Sr.
Using theorem 1 of Chapter 111, section 4, p. 45 we obtain that the following s e n t e n c e (pvh)fl(l, vh,@)
Dl
--f
[S(l, (pvh)S(l, vhy@),@) -tfi(l, D1,@)l
72
SEMANTICS OF
(s)
is in Zr. On account of (1) we obtain therefore that
S ( l , (Pvh)x(l, vh,@)*@)
D1?@)
is in Zr.This proves that S(1,(pvh)S(l,v,,@),@) is not in %r since otherwise S(1, Ill,@) would be in ZT which is not the case, as we have shown above. Theorem 11 is thus proved. It implies, of course, that also the set X is o-consistent. We note still the following corollaries to the theorem 10: T h e o r e m 12. If @ is a m a t r i x in which exactly one free variable 9 occurs and if v,, is a bound variable which does not occur in @, then (EVh)X(i,Vh,@) E St: (AVh)&(i,Vh, @) E St
(ZIn)S(i,Dn,@) E Sr, (n)&(i, on, @) E %r.