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VI Syntactical Transforms
VI
SYNTACTICAL TRANSFORMS 6.1.
Introduction By a syntactical transform T we mean a correspondence which associates with every element X of a certain class C of statements in the LPC another statement X'=T(X) such that X' is obtained from X by a definite formal rule. C will be called the range of T. The above definition is very wide, and in this form, somewhat vague. It includes operations such as the passage from Y(a) to (z)Y(s),or from Y(a,b ) to Y(b,a)-a combinator-or the passage from a statement X t o its Skolem form. We note that the latter transforms provable statements into provable statements. Another transform, and one which is related to the subset of this chapter more closely, is given by the operation of relativisation (compare ref. 26, and sections 3.4 and 3.7 above). Consider for example the axioms of order
{
6.1.1.
(z)(Y)(~)[&(z, Y) A Q(Y, 2) 2 &(x, z)I (x)(y)[&(x,Y) v &(Yt 4 1
where &(z, y) stands for "x is smaller than or equal to y". Relativising by means of the predicate R(x), we obtain (Z)[R(X) 2
6.1.2.
[
[(Y"(Y)
3 [ ( z ) R ( z2 ) [ a x , y) A A
&(Y, 2) 3 &(x, 41 ...I
(x)[R(43 [(Y)[R(Y)3 Q ( x , y) v &(y, 4111.
We observe that while
108
SYNTACTICAL TRANSBOR&fS
is not provable. Thus, provability is not invariant under relativisation. On the other hand, it can be shown that if X is a provable statement without constants then its transform by relativisation, by means of R ( x ) , is deducible from
(3x)R(4-
6.1.3.
For a given transform T on a range of statements C, we shall call the set of statements 2 a supporting set of T if, for every provable X EC, the statement T(X)is deducible from 2. I n ordinary language, the operation of relativisation corresponds to the passage from a phrase
“for all x...” to
(‘for all x which belong to R...”
and from (<
there exists an x.. .” to “there exists an x which belongs to R...”
Thus, relativisation is expressed here by the insertion of the clause “which belong(s) to R...”, the imprecision of our informal language having blurred the distinction between the operations required for the two types of quantifiers. We now make the fundamental observation that, in Mathematics, assertions on the continuity or boundedness of the solution of certain problems can be obtained from the assertion of the mere existence of a solution by this kind of modification. Thus, from the statement “For every x there exists a y such that y3=x” we may pass on to “For every x1 and x2 such that ~ x 2 - - z is 1 ~small, there exist yl and y2 such that lyl-y21 is small and such that y:=xl, and yi=x2. Here again we have used inexact terms in order to emphasise the general nature of the transformation rather than the technical details. A formal theory of the continuity transform which achieves the passage from the former assertion to the latter is beyond the framework of this book and wiU be expounded elsewhere. In the following sections we shall consider instead the syntactical transform which is concerned with boundedness. We shall find that the results are applicable to some of the systems which were considered earlier in this book. Y,
109
BOUNDMU TRANSEOBM
6.2. The Bounding Transform Let C be the class of statements of the LPC which are of the form 6.2.1.
x= [(34 (ZXl)(Yl) - a *
1.-
(YJ(33)
*..(ZZJ
Z ( 5 , . * * , z,91,..*,Ym, 3,
* - a ,
z,)l
where the matrix Z does not contain any quantifiers. We include the possibilities that I = 0, m > 0, so that the statement begins with a universal quantifier, and similarly that m=O, or n=07 etc., all of which may occur separately or simultaneously. I n particular it is possible that X does not include any quantsers at all. But we exclude the appearance of two universal quantifiers which are separated by one or more existential quantifiers. For any X E C, we now d e h e the bounding transform of X , X' = B ( X ) by
I
-.-(3zt)(?h) (YmW%) .-. A W(%J, 2) A [W(yi, y) A ... A 6.2.2. A ~ ( Y , w?/) 3 [w(3, 2) A .-.A w(Zn, 2) A z(z1, 21, Y1, ym, 3, *.., %)I I. It is understood that if I = 0 then (3%) and [W(zl, s)A ...A W ( z , ,z)A X ' = [(3Z)(Y)(3Z)W%)
... (ZZ,,)[W(Zi,
2)A
0 . .
.*.
***>
are to be omitted with a similar instruction for n = 0. If m =0 then we omit (y) and [W(yl, y) A ... A W(ym,y) 3. Let Z be the union of the set of statements 6.1.2 and of
6.2.3.
(4(3Y) wz7 Y) Y) 3 R(Y)I (Z)(Y)(Z)[W(% Y) A &(Y, 4 I l W(S,741. (4(Y)EJW7
It is not difficult to show that 2 is consistent. Suppose that the statements of 2 hold in a structure M . Then the set of constants of M which satisfy the relation R(z) is not empty, by virtue of the first two statements of 6.2.3. It will be denoted by R. R is ordered by the relation Q(x, y), for which we shall write x
110
SYNTACTICAL TRANSFORMS
W ( b ,a ) holds in M . At least one R, is not empty. If a, a' E R and a ~ a then ' R, C R,, in view of the third statement of 6.2.3. The crucial result of the theory is
' is a supporting set for the bounding trans6.2.4. THEOREM. L form B on the range C. Instead of proving 6.2.4 directly, we consider the negations of X and X ' . When written in prenex normal form, these are 6.2.5.
Y = [(4, ... (~JWYJ(~Y~)(zA ... (2,) [--(X,,
*..I
Xz, Y1, - - * 7
yrn,
21,
...,Z J l I
and
Y'= [(~)(~Y)(z)(~ ..- (ZMYJ ... (3~,)(%) (2,)
6.2.6.
[
... A ~ ( Z Z2), 3 [w('ZJ,, Y) A ... A A W Y m 7 Y) A [W(;;, 4 A . * * A Wk,, 2) 3 [- Z(Z1,.*., xz: Yl, Ym, %, -.-, z,)l--.l. [~(ZI X),A
*.a,
Let C' be the class of statements Y which are obtained in this way from statements X EC. We write Y ' = B { Y } in agreement with our earlier notation for statements which belong to C as well. 6.2.4 states explicitly that for any provable X E C, X' is deducible from Z, i.e. X' holds in every model of 2 in which it is defined. It follows that 6.2.4 is equivalent to 6.2.7. THEOREM.Given any Y EC' such that Y ' = B { Y } holds in some model of 2, there exists a model for Y . For the proof, we shall assume I > 1, 772 >, 1, n > 1. It is easy to show that any other case within the scope of 6.2.7 may be reduced to one in which this assumption is satisfied. For example, if
y = (gYi)(Zi)[-' Z(Yi,Z1)l we consider instead the statement
where V(x) is an arbitrary relation of one variable. It is not difficult to see that if B { Y ) holds in some model of Z then B{Y*)also holds
111
BOUNDING TRANSFORM
in some model of Z. Then 1 = m = n = 1 for Y * . Assuming that we have already proved 6.2.7 for this case, we can find a structure M in which Y * holds. Since Y* 3 Y it then follows that Y holds in the same structure. The other cases for which either I = 0 or m = 0 or n=O can be dealt with in a similar way. However, it may be mentioned that there also exist simple direct proofs of 6.2.7 for these cases. Suppose then that I > 1, m > 1, n > 1. We write 2’ for 2. Let M be a model of Y as given by 6.2.5. The semantic interpretation of Y in terms of M shows that there exist, within M , functions i = 1, ..., m gi=fa(zl,..., x,),
-
in the ordinary mathematical sense such that for any I + n-uple of constants of M , a,, ..., a,, b,, ..., b,, the statement 6.2.8.
Z’(a1, ..., a,, fi(a1, . * - ) a,),
-.a
fm((ll,
-.-,a,), b,, ..*,b,)
holds in M . It is understood that when writing out 6.2.8 in detail, for given a,, ..., a,, we have to substitute the constants which are the functional values of fi(al, ..., a,), since the terms fi(a,, ..,, a,) are not included in the LPC as detailed in Chapter I. 6.2.8 will be said to be in Herbrand form. Conversely, if 6.2.8 holds in a structure M for all Z+n-uples a,, ..., a, b,, ..., b, then the structure satisfies 6.2.5. Similarly, 6.2.6 is satisfied by a structure M’ if and only if there exist in M’ functions g(z) and hi(%,z, x,, ..., z,), i = 1, ..., m such that the statement
6.2.9.
I
W(a1, a ) A -..A W(az, a ) 2 [W(h,(a,b, a,, uz), g(a))A A ... A v ( h , ( U , b, a,, ..., a $ ) g(U)) , A [W(b,, b ) A ... A A W(b,, b ) 5, Z’(a,, ..., a,, h,(a, b, a,, ..., a t ) , ... ..., h,(a, b, a,, a,), 4, --.bJll -.a,
holds in M’ for all a, b, a,, ..., a,, b,, ..., b,. We may now formulate our problem as follows. Given a model M’ of 1: which satisfies 6.2.9, we have to find a structure M which satisfies 6.2.8.
112
SYNTACTICAL TRANSFORMS
I n order to specify the constants of M , we introduce m new functional symbols 6.2.10.
41(%
*a',
d, A&l, ... 4. ..-9
7
Let c,, cl, ..., c, be the set of constants which are included in Y . If Y does not include any constants, let c, be a constant picked a t random from M'. We now define the class of constants of M , say, as follows. 6.2.11. c,, ..., c, are elements of r, of order 0. 6.2.12. Having defined the elements of '1 of order less than k, where k is a positive integer, we obtain the elements of order k, by filling the argument places of any +;, i = 1, ..., m, with elements of of order less than k, provided that a t least one argument is exactly of order k- 1 in each case. For example, dl(c0, ..., co) is of order 1, +2(co, ..., q5,(c0, ..., c,)) is of order 2, etc. The number of elements of given order is finite. Thus we obtain a simply infinite sequence by taking first the elements of order 0, as given, followed by the elements of successive orders, when ranged in arbitrary but definite finite sequences. We shall denote the elements of the scquence by
r
r
6.2.13.
co, c1,
...> c,, c1+1, C,+Z, ...
in agreement with 6.2.11. If cjl, ..., cjl are elements of of order less than k, then the order of C$~(C~,, ..., c&, i = 1 , ...,m, does not exceed k. We define the functions fi(zl,..., x J , i = 1 , ..., m, on I' by
r
6.2.14.
fi(cj,>*..> cjl)=+i(cjl>. * * > cil).
It must be understood that this is not a functional identity, but a rule which associates with the argument values cil, ..., cjl as functional value the purely formal expression 4i(cjl, ..., cjl) which is an element of I'. The relations of M now have to be specified in such a way that the statements
113
BOUNDINQ “RANSFORM
hold in M , for jl,..., it,kl, ..., k,, = 0, 1 , 2, 3 , .... Let H be the set of all these statements and let Hvflbe the W t e subset of H whose elements are restricted by the conditions ji
i=l,
..., 1
k,gp,
i=l,
..., n,
for v,p=O, 1, 2, .... For any relation S(xl, ..., zk)which occurs in Y and for any n-uple c,,, ..., c,~, we have to specify whether or not X(c,, ..., cjk) holds in M , in such a way that according to the usual truth-table procedure of the propositional calculus, the elements of H all become “true”. However a familiar argument (compare refs. 4, 10, 11) shows that it is sufficient to establish the possibility of such a procedure for all the sets HYp. This has the advantage that the sets of statements to be considered are finite. According to the first statement of 6.2.3, there exist constants do, ..., d, such that W(ci,di) holds in M‘, i = O , 1, ..., T . The constants do, ..., d, belong to R. We propose to show that the formulae 6.2.16. 6.2.17.
I(ci)=di 4+i(cjl,
-.*>
i=o, 1,
...,r
c$l))=g (max. (cjJ) 14k$l
constitute an inductive definition of a mapping of r into the set of constants of M‘, more particularly into the set R. Indeed 6.2.16 defines 1for the elements of r which are of order 0, in such a way that these elements are mapped into R. Let q be any positive integer and suppose that we have verified already that any element of whose order is smaller than q is mapped by rZ into an element of R.Consider any element c of I’ which is of order q. c is of the form
r
c=+i(cj,,
cji)
where ci,,..., cil are of order smaller than q. Then I(c,,), ..., A(c,,) are elements of R and are ordered by the relation Q . Thus the definition c’ = max. (cj ) l
determines a unique element of R which has the properties
for k = 1 , ..., I I(cI,)=c’ for at least one k .
1(c,J
114
SYNTACTICAL TRANSFORMS
Suppose for example that c'=A(cjt). Then 6.2.17 yields wi(cj,,
*.*)
ciJ)=g(~(%,))
6.2.9 implies that
holds in M' for all constants a, b, a,, ..., a,. We make the additional inductive assumption that for any A(cj), where cj is of order less than n, there exists a constant d such that W ( d ,il(c,)) holds in M'. This is certainly true for all ci of order 0. Putting i = 1, a=A(cj,) in 6.2.18 and a,= ... =a,=d where d is such that W(d ,l ( c j , ) ) holds, we obtain for arbitrary b,
W d , &,))
A
- - a
A
W(d,4 C j J 3 W(~l(A(C&b, d,
*.*,
4, gMc,,)).
It follows that W(hlMCj,), b, d,
*.*,
4,g(A(x,,))
holds in M', and this shows, by virtue of the second statement of 6.2.3 that g(A(c,)) belongs to R and, moreover, that there exists a constant d' such that W ( d',g(il(c,,))) holds in M'. We conclude that the definition of il is unambiguous and that A(cj) E R for all cj E I'. Having fixed v and p, let p=max. il(ci) and define another 3 . a
mapping of 6.2.19.
Then
(
'I into the set of constants of M' by n(ci)=ci, i = o , 1, ..., r
A(y5Jcf1, ..., ciJ) = hi(max. A(c,J, p , A(cj,), ..., A(ciz)), ladl
i = 1, ..., m.
6.2.20.
W(A(C,),A(Ci)),
j=o, 1) 2,
...
holds in M . Indeed for j = O , 1, ..., r , 6.2.20 reduces to W(cj, dj). For a given positive integer q, suppose that 6.2.20 has already been proved for all ci of order less than q . Consider any element cj of 'I which is of order q. ci is of the form fii(ci,, ..., cil) where the arguments are of smaller order. Put a=max. A(c,), l
b=p,
~ , = A ( c , ..., ~ ) ,a,=d(c,,)
115
BOUNDING TRANSPORM
in 6.2.18,
SO
-
6.2.21
Now
W(A(c,),&*)),
i= 1,
*.*9
holds in M’ by the assumption of induction and, in view of the third statement of 6.2.3 the same applies to
i= 1, ..., 1.
W ( A ( c . ) max. (A(cjJ)
6.2.22.
’ l
The conjunction of the statements of 6.2.22 is the implicans of 6.2.21 and so we may conclude that the implicate of 6.2.21 also holds in Af‘. Reference to 6.2.17 and 6.2.19 shows that this implicate is identical with W(A(c,),A(c7)),which proves our assertion. Next, we propose to show that for all j,, ..., j l and for k,, .,.,k,
{ Z’(A(c,), ..*,
4c,z), 4 M c i , ,
*..Y
A(+m(cjl,
c,$,
me-9
A(ck,)l
holds in X’. Substituting max. A(ci,), p , A(c,J, ..., A(c& A(c,l), ..., A(c,)
6.2.24.
1Qk
for a,b, a,, ..., at, b,, ..., b, respectively in 6.2.9, we obtain an implication whose implicans is the conjunction of the statements 6.2.22 which have already been shown to hold in M‘. It follows that the implicate which is the result of substituting 6.2.24 in 6.2.85.
A W(&(n,b,a,, .--,az),da)) A [W(h,(%b,ap --.,ad,g(a)>A A [W(b,,b) A ... A W(b,,6)3Z’(al,...,a,,h,(a,b,a, ,..., a z ) ..., ,
i
h,(a, b, a,, ...,aZ),4, ...,b J l l
also holds in N’, and the same applies in particular to the conjunct 6.2.26.
( W(b,, 6 )
A
... A W(b,, b ) 3 Z’(a,, ..., a,, h,(a, 6, a,, ..., a t ) , .-.,hm(a, 6, a,, at), 4, ...,ha). ..a,
116
SYNTACTICAL TRANSFORMS
of 6.2.25. Now the implicans of 6.2.26 in terms of 6.2.24 is the conjunction of the statements
Jww,,), PI,
6.2.27.
i = 1,
***,
n
and these hold in M’ by 6.2.20 and the definition of p . We conclude that the result of substituting 6.2.24 in 6.2.28.
Z’(a,,...,a,,h,(a,b,a,) ,...,h,(a,b,a, ,...,a),b,,...,b,)
holds in M’ and reference to 6.2.19 shows that this is precisely 6.2.23.
Now let S ( C , ~ .-., , cjk) be any atomic statement which occurs in , c,J holds one of the elements of HVp.Then we specify that S ( C , ~..., or does not hold according as the statement S(A(c,,),..., A(c,)) does or does not hold in MI. With this specification, all the elements of H1, become “true”. Indeed, consider any Zl, .,., ._,. *, E Hvp. The truth value of this statement depends only on the truth values of the atomic statements contained in it, and these are the same as the truth values of the corresponding atomic statements of 6.2.23, which yield a truth value “true” (or “holds”) for the latter statement. Hence Zz, ,... ..,.ks becomes “true” for the specified truth values of the atomic statements contained in it. This proves 6.2.7.
It may be mentioned that if Y does not contain any constants then C may be weakened by replacing the first axiom of 6.2.3 by
(34(3Y) WG Y). For the proof, we select c, and do so that W(c,, do)holds, and proceed 6.2.29.
as before.
6.3. A Counter-Example We shall now show that the main result established in the preceding section (6.2.4) is, in a sense, the best possible. For this purpose, we shall construct a provable statement 6.3.1.
x= ( ~ l W Y l ) ( M ( YI? ~ l Z~I),
where Z does not contain any further quantifiers, such that
117
CO7JNTER-EXAMPLE
X' = B ( X ) is not deducible from 2. More particularly, we put
c-
Z(X, y, z)=
41
A(& y) v
where A is a relation with two arguments. Then
x= ( ~ l ) ( ~ Y l ) ( ~ l ) [4 - %Yl) v A(% 4 1 .
6.3.2.
Now, for any constant a, the statement
-
[(Yl)A(% Y1)l
" [(!Ill 4%
Y1)l
is provable. Changing the variable y in the second disjunct to z, and replacing the first disjunct by (3yl)[- A(a, yl)], we obtain
[(ZYi)" A(% Yd11 V [(%I A(% %)I
6.3.3.
as another provable statement. We may change 6.3.3 to
-
(3~d-
M )v
[(d4, %)I1
and since A ( a ,yl)does not contain %, we may shift the universal quantifier across it, so (@?h)(%)" A(a,Yi) V A(a,21)l.
Finally, replacing a by the variable xl and prefixing (q), we obtain 6.3.4.
(%)WYl)(%)[-
4%Yl) v 4%2113.
This is X . Since the steps leading from 6.3.3 to 6.3.4 all transform provable statements into provable statements, we have established that X is provable. Now consider X ' = B ( X ) 6.3.5.
{
x ' = ( ~ ) ( 3 y ) ( 2 ) ( ~ i ) ( 3 y i ) ( z , ) [ w2 ( ~[v(!/i, i, y) A A Ew(%2) 3 [- 4 x 1 , Yl) v 4%21)1*..1.
A prenex normal form of -XI 6.3.6.
{
is given by
3
-
[w(?h [w(%, 2) A [A(xi:,, y1) A A(z17 %)I***].
Y'=( 3 ~ ) ( Y ) ( 3 2 ) ( 3 ~ i ) ( Y i ) ( 3 2 i f [ w (2) ~ iA,
We shall construct a model M' of 2 which satisfies 6.3.7.
( y"
= ( 3 ~ ) ( Y ) ( ~ 2 ) ( 3 ~ i ) ( 3 z ,[)w((yX1i ,) 2) A [ 3 2 ) A [A(x1, y1) A
[w(%
w(?hy) 3 A(%
%)I*-.]-
118
SYNTACTICAL TRANSFORMS
Since Y" 3 Y' is provable, M f will a fortiori satisfy Y', Y" can be replaced by the following statement in Herbrand form.
W s ( a ) ,4 A r w a , , a ) A
6.3.8.
[WW), / ( a ) )A A(g(a),a,) A A
--4s(a),
Na))I...l*
Thus Y" holds in a structure M if and only if M contains a constant d such that for all constants a,, a of M , there are defined function f, g, h for which 6.3.8 is satisfied. We define the structure M in the following way. The set of constants of M' consists of the elements of the doubly infinite sequence {c;], i, k = 0, 1, 2, ..., and the relations of M' are R ( 4 , &(x, Y), W(X,Y), A ( z ,Y). R ( c ~holds ) for i = O , all k, and not otherwise; Q(ci ci) holds for i = j = O , k ~ l and , not otherwise; k: W(c;,cf) holds for j = 0, i < 1, and not otherwise; A(c;, ct) holds for i = O , j ~ k and , not otherwise. Thus A(c',,cf) holds in M' if and only if W(ci,ci) holds in that structure. It is not difficult to verify directly that M' is a model of 2. To prove that Y" holds in M' we define
'.
d = co", g(cL)=cP,
f (ck)= c:+1
i, k = O , 1, 2, ...
i - k3-1 h(c,)-c, ,
Then W(g(a),d ) becomes, for a=cR, W(cg,c:) and this holds in M' by the above definition. It remains to be shown that 6.3.9.
W%,4 3 W W ) ,f ( a ) )A 4&), a,) A
-
A m ) , W))l
holds in M'. We may confine ourselves to
a,=c;,
l>i
a=cp,
since otherwise the implicans of 6.3.9 does not hold. Then the implicate becomes
-
Ci) A A(C;,C;") W(C;+',C!+l) A and we may verify again that this holds in M'. Thus M' constitutes the required counter-example. I n the preceding section, the bounding transform was defined
BOUNDING TRANSFORMS AND DEDUCIBILITY
119
only for special classes of statements in prenex normal form. Consider now an arbitrary statement in prenex normal form, e.g.
where Z is the matrix of the statement. We define the general bounding transform of X , X ' = B * { X } by
x'= (z')(X2)(3y1)(z3)-.- ( ~ n ) ( 3 y ~ ) ( ~ i ) ( z 2 ) ( 3 y..i ) ( ~ 3 ) ( z ~ ) ( ~ y ~ ) [ ~zl) ( ~ l ,[w(%s2) y') x3)3 ..' .-.[W(za,,zn)I [w(ym,y") z]...], *
[w(?h7
A [w(x3>
A
from which the law of formation of the transform wiU be apparent. B * { X ) is not, in general, identical with B { X } for X EC, but it is not difficult to see that for such statements B { X } = B * { X ) is deducible from 2. Now conside the sequence of quantifiers for any statement in prenex normal form which does not belong to the class C. The sequence must include three quantifiers such ;ts (Zz),(y), ( 3 z ) , which appear in that order, although they may be separated by further quantifiers. Using the above counter-example, we may then show that there exists a matrix Z such that if we prefix to it the given sequence of quantifiers, there results a provable statement X whose general bounding transform B* { X } is not deducible from 2. Thus, although B* has a wider range of definition than B, its introduction does not lead to any new results which are stronger than 6.2.4. For this reason we have given preference to B , which has the advantage of greater simplicity.
6.4. Bounding Transforms and Deducibility Let K be a finite or infinite set of axioms whose elements belong to the class C' (see 6.2.5). The set of the bounding transforms of the elements of K will be denoted by B { K } . We propose to prove 6.4.1. THEOREM.If the statement X E C is deducible from the set of axioms K C C', then B ( X }is deducible from the set 2 u B{K}. The assumption of the theorem is that there exist statements Yl, ..., Y , E K such that 6.4.2.
y1 A
... A
17, 3
x
120
SYNTACTICAL !l!RANSFOBMS
is provable. We shall show that
B(Y1)A B(Y,}A ... A B(Y,} 3 B(X}
6.4.3.
is deducible from Z. We write Xifor the normal form of
B{Y,) =
-
Yi,then
N
B(XJ
is a provable statement, i = 1 ..., p . We also write X = X9+1, so that XiEC, 1 < i < p + 1. Then 6.4.2 and 6.4.3 can be replaced by
x, v ... v x,v x,,
6.4.4.
and
... v B(X,}v B{xp+l}.
B(Xl} v
6.4.5.
We suppose (if necessary after a preliminary change of variables) that no variable occurs in two different components of 6.4.4. We may then shift all quantifiers in 6.4.4 to the beginning of the formula and arrange them at will, so long as we do not interfere with the mutual order of the variables which belong to the same component. More particularly, if 6.4.6.
xi= ( 3 x i ) ... (xxi4)( ~ i..). (&,)
1... "i, 7
. y'1 .
... yi,,
3
7
(32;)
...)(&!%i(
..., 2;)
zc(Zi,... i= 1, ..., p +
1
then we arrange the quantifiers in 6.4.4 so as to yield the result
v,= ( 3 x 3 ... (2h;J ( 3 x 3 ... ( 3 x 9 ... (34 ... ( 3 x i ) (Y:) *.. (Yk)(Y3 ( Y a (?I (?.a ;) **-
6.4.7.
*.-
- a -
. ... ( 3 2 ; ) ... ( 3 2 ; ) ... ( 3 2 ; ) 1 ... xi1,y1, ... ykl, z:, ... 2 . t ) v ... v z,(x;, ... 5, y;, ... y k 7z;, ... 2;)
(32;) .. ( 3 2 : ) (32:) Z1(Z!,
for r = p + 1. More generally we define the syntactical transform VT=X{Xl v ... v X,] by 6.4.7 for r = l , 2, ..., p+1. I n particular Vl=X,. Then V,, E C , and so .B{T'9+l) is deducible from C,in view of 6.4.4. We shall establish presently that 6.4.8.
W T + l }
2
W ?v ) B{XT+l}
BOUNDING TRANSFORMS AND DEDUCIBLTZW
121
is deducible from Z for r = 1, 2, ..., p . Taking this relation for granted, we obtain the following chain of statements which are deducible from Z.
'
W,+lI W J v B{X,+l} W,+d3 W a - 1 1 v B{XJ v BP,+l} W,+J3 w , - 2 1
vBPP-1) v
W
P
I
v B{X*l}
and finally 6.4.9.
B{V,,l} 3 W
l )
vB mv
*.*
v
W,+d
where we have taken into account that V,=X,. Since B(V,+,} is deducible from 2, it then follows that 6.4.5 also must be deducible from Z. We still have to show that 6.4.8 is deducible from 2. Now
so that 6.4.8 may also be written as 6.4.10.
v
'{S{J'r
Xr+l>Z3 B(Vr1 v B{Xr+1}-
This is of the same form for all r , and we may therefore confhe ourselves to r = 1, 8,= X,. Then the statement becomes 6.4.1 1.
V
P
l
v
&)I
3 B{Xd " w
2 1 .
The implicans of 6.4.11 is, explicitly,
B{X{X, v X,}} = [(3x)(y)(zZ)(32:)... (&%;J(Zg)... (&!a$ (y:) ... (&l)(Y:) ... (gg2)(3zi)... ( ~ T Z : ~ ) ( ~ .. Z ?. ( ) g.Z$,)[w(2) ZlA,. .. A w ( Z i , 2) A w ( X ; , 2) A ... A w(Xi,5)A [w(?/i,y) A ... A w(&, g) A w(y:,
,Y) A
... A w(&,,, y) r> [w(Z:,2 ) A ... A w(Z&, 2) A w ( Z : , Z) A ... ..., Xi, y:, ..., yk1, Z:, ..., X A l ) V z,(X:,..., zk)] ..- 3 .
w(Z&, 2) A [zl(X:,
We modify this expression by replacing (3x)(y)(3z) by ( 3 ~ ~ ) (Z22)(y1)(y2)(3x1)(i71~2), and by replacing 2, y, z in the relations W by xl, yl,3, or 5,y,, x,, according as these variables appear next to a symbol with upper suffix 1 or with upper suffix 2. There results a statement 2: such that B{X(Xlv X,}} E 2: is deducible from 2 (compare the end of the preceding section).
122
SYNTACTICAL TRANSFORMS
I n this expression the quantifiers of the second disjunct, B{X,), may be shifted across the matrix of the first disjunct, and arranged so that the sequence of the quantifiers is 6.4.12.
(3xi1) ( (34)...(3$)(y:) ...(y&)(y:). ..(&,) (g.2:). .. ( 3 ~ 1 ) ( 3 ~ 2 ) ( y l ) ( Y 2 ) ( 3 2 1 ) ( 3 2 2 ) ( -3- *~ ~ )
(32;)
(gz:)...(3zgO).
The resulting implicate 2; is such that B { X l ) v B{X2}= 2: is provable, and so we only have to show that 6.4.13.
is deducible from Z. Moreover, both 2: and 2: are in prenex normal form and the sequence of the quantifiers is the same for both cases. Now put Al=W(U:,Ul)A
...A W ( U ~Ul), , k!,=W(U:,
U,)A
... h W ( U : , U , )
B1=W(b:,b,) A ... A W(bkl, b,), B, = w(b:, b,) A ... A w(b;*, b,) C, = W(c:, c,) A ... A W(cil, cl), C , = W(c:, c,) A .. . A W(cg,,c,) D,=Z,(a:, ..., u:, b:, ..., bkL,c:, ..., cAJ, D2=Z2(4,..., a:, b?, ..., b:,, c:, ..., c:~) - where it is assumed that the constants uk,b,, ck, ui,bi, ci, are not already contained in 2: 3 2;. One verifies without difficulty that the statement
6.4.15.
{ A1
A
A , A [Bl A B2 3 [c,A (72 A [olv 0 2 1 1 1 3 3 [A1A iB13 c, A Dlll v [ A , A [ B , 3 c, A Qll
s provable when A,, A,, ..., D, are considered as propositional
BOUNDING TRANSFORMS AND DEDUCIBILITY
123
variables. It follows that it is provable also with the interpretation 6.4.14. We now make use of the rule 6.4.16. Let W,(x), W,(s) be two predicates such that
is deducible from a set of statements H which does not include a.
Then the statements
[(x)W1(413 r
( ~ ) ~ 2 ( 4 1
and
“34 %(.,I
2 [(gx) W,(s)I
are deducible from H . The truth of 6.4.16 can be established along the same lines 2.3.3 and the proof will be omitted here. We apply 6.4.16 to 6.4.15 with
it9
w1= [A, A A, A [B, A B23 [c,A c, A [Dl D,]]] w,=“4* [ B l I Cl A 0111 v [A, A [B,A [C, A Dzlll V
by prefixing the quantifiers of 6.4.12 successively to W, and W,, the order of the procedure being opposite to the order to appearance of the quantifiers in 6.4.12. Thus we first prefix (zz,) to W, and W, and replace czz by zz, in both statements. Next we prefix (zz,-,) 2 and replace c;,-, by znS-,, etc. As our last step, we prefix (3x3 and replace a: by xi. The final result is that W, is transformed into 8: and W , into Zl, so that 6.4.15 is turned into 6.4.13. This completes the proof of 6.4.1. A survey of the preceding chapters shows that the following sets of axioms belong to the class C‘ or more precisely, to the intersection of C and c‘: 1.2.1, 1.4.2, 1.4.3, 3.1.2, 3.2.1, 3.3.1, 3.3.3, 3.4.3-3.4.6, 3.4.38, 3.5.1-3.5.5, 3.5.14, 3.6.2-3.6.7.
The same applies to 3.1.3, 3.1.4, 4.3.6, 4.3.7, 4.3.15, 4.3.18, as appears when we write out Xn(x, y) according to its definition 3.1.1; to 3.2.5, 3.3.2, and 3.5.7, by 3.2.3 and 3.2.4; and to all sets of statements which do not include any quantifiers, including all
124
SYNTACTICAL TRANSBORMS
diagrams. Thus, all systems of axioms which are considered in the present book are subsets of C n C'.
Application The following simple example is typical of the applications which can be made of theorem 6.4.1. Let M be a completely divisible ordered abelian group which contains at least two different elements (see section 3.1). We denote the order relation in M by &'(x, y) in order to distinguish it from the relation &(x, y) of the present chapter. The remaining relations of M are E(x, y) and X(z, y, 2 ) . We define three additional relations R(x), &(x, y), W ( z ,y) for the constants of M as follows. 6.5.1. R(a) holds for all constants a of M such that apO, and not otherwise. 6.5.2. &(a,b ) holds if and only if O G a G b in M . 6.5.3. W(a,b ) holds if and only if b>O and -b
3.1.4).
Considering first the axioms 1.2.1, we observe that if a statement X in prenex normal form contains only universal quantifiers, then X 3 B { X }is provable. It follows that if X holds in a particular structure then the same applies to B{X),provided B { X ) is defined in the structure. Thus we need consider only the axioms
Y, 4 (.)(Y)(gz) X (.> 2 , Y). ( X) ( Y) ( Z Z ) S(Z,
The bounding transforms of these two statements hold in M by virtue of the inequality
I.+Yl~I~l+lYl which holds in every ordered abelian group. The absolute value 1x1 is defined rn x or - x according as x > O or x t O . The repeated
A.PPLICA!L'ION
126
application of the same inequality proves that the bounding transform of 4.1.4 is satisfied by M * . The same is true of 3.1.2, trivially so, while 3.1.3 includes only universal quantifiers when written out in prenex normal form. Applying 6.4.1, we now obtain 6.5.4. THEOREM. Let X be a statement of the class C which is formulated in terms of the relations E(x, y) and S(z, y, z ) and which holds in all completely divisible torsion-free abelian groups which contain at least two elements. Then B { X } holds in all ordered groups of the same type, provided W ( z ,y) is interpreted by means of 6.5.3. It will be seen that 6.5.4 represents a new type of transfer principle (compare section 5.2). The theorem affirms that if a statement X of a specified kind holds in one particular class of structures, then a certain syntactical transform of X holds for all structures of some other class. Taking into account Theorem 4.3.1, we may replace 6.5.4 by 6.5.5. THEOREM. Let X be a statement of the class C, formulated in terms of the relations E(x, y) and S(x, y, z), and without tonstants, which holds for the additive group of rational numbers. Then B { X ) holds in all completely divisible ordered abelian groups which contain at least two elements.
SYNTACTICAL TRANSFORMS 6.1.
Introduction By a syntactical transform T we mean a correspondence which associates with every element X of a certain class C of statements in the LPC another statement X'=T(X) such that X' is obtained from X by a definite formal rule. C will be called the range of T. The above definition is very wide, and in this form, somewhat vague. It includes operations such as the passage from Y(a) to (z)Y(s),or from Y(a,b ) to Y(b,a)-a combinator-or the passage from a statement X t o its Skolem form. We note that the latter transforms provable statements into provable statements. Another transform, and one which is related to the subset of this chapter more closely, is given by the operation of relativisation (compare ref. 26, and sections 3.4 and 3.7 above). Consider for example the axioms of order
{
6.1.1.
(z)(Y)(~)[&(z, Y) A Q(Y, 2) 2 &(x, z)I (x)(y)[&(x,Y) v &(Yt 4 1
where &(z, y) stands for "x is smaller than or equal to y". Relativising by means of the predicate R(x), we obtain (Z)[R(X) 2
6.1.2.
[
[(Y"(Y)
3 [ ( z ) R ( z2 ) [ a x , y) A A
&(Y, 2) 3 &(x, 41 ...I
(x)[R(43 [(Y)[R(Y)3 Q ( x , y) v &(y, 4111.
We observe that while
108
SYNTACTICAL TRANSBOR&fS
is not provable. Thus, provability is not invariant under relativisation. On the other hand, it can be shown that if X is a provable statement without constants then its transform by relativisation, by means of R ( x ) , is deducible from
(3x)R(4-
6.1.3.
For a given transform T on a range of statements C, we shall call the set of statements 2 a supporting set of T if, for every provable X EC, the statement T(X)is deducible from 2. I n ordinary language, the operation of relativisation corresponds to the passage from a phrase
“for all x...” to
(‘for all x which belong to R...”
and from (<
there exists an x.. .” to “there exists an x which belongs to R...”
Thus, relativisation is expressed here by the insertion of the clause “which belong(s) to R...”, the imprecision of our informal language having blurred the distinction between the operations required for the two types of quantifiers. We now make the fundamental observation that, in Mathematics, assertions on the continuity or boundedness of the solution of certain problems can be obtained from the assertion of the mere existence of a solution by this kind of modification. Thus, from the statement “For every x there exists a y such that y3=x” we may pass on to “For every x1 and x2 such that ~ x 2 - - z is 1 ~small, there exist yl and y2 such that lyl-y21 is small and such that y:=xl, and yi=x2. Here again we have used inexact terms in order to emphasise the general nature of the transformation rather than the technical details. A formal theory of the continuity transform which achieves the passage from the former assertion to the latter is beyond the framework of this book and wiU be expounded elsewhere. In the following sections we shall consider instead the syntactical transform which is concerned with boundedness. We shall find that the results are applicable to some of the systems which were considered earlier in this book. Y,
109
BOUNDMU TRANSEOBM
6.2. The Bounding Transform Let C be the class of statements of the LPC which are of the form 6.2.1.
x= [(34 (ZXl)(Yl) - a *
1.-
(YJ(33)
*..(ZZJ
Z ( 5 , . * * , z,91,..*,Ym, 3,
* - a ,
z,)l
where the matrix Z does not contain any quantifiers. We include the possibilities that I = 0, m > 0, so that the statement begins with a universal quantifier, and similarly that m=O, or n=07 etc., all of which may occur separately or simultaneously. I n particular it is possible that X does not include any quantsers at all. But we exclude the appearance of two universal quantifiers which are separated by one or more existential quantifiers. For any X E C, we now d e h e the bounding transform of X , X' = B ( X ) by
I
-.-(3zt)(?h) (YmW%) .-. A W(%J, 2) A [W(yi, y) A ... A 6.2.2. A ~ ( Y , w?/) 3 [w(3, 2) A .-.A w(Zn, 2) A z(z1, 21, Y1, ym, 3, *.., %)I I. It is understood that if I = 0 then (3%) and [W(zl, s)A ...A W ( z , ,z)A X ' = [(3Z)(Y)(3Z)W%)
... (ZZ,,)[W(Zi,
2)A
0 . .
.*.
***>
are to be omitted with a similar instruction for n = 0. If m =0 then we omit (y) and [W(yl, y) A ... A W(ym,y) 3. Let Z be the union of the set of statements 6.1.2 and of
6.2.3.
(4(3Y) wz7 Y) Y) 3 R(Y)I (Z)(Y)(Z)[W(% Y) A &(Y, 4 I l W(S,741. (4(Y)EJW7
It is not difficult to show that 2 is consistent. Suppose that the statements of 2 hold in a structure M . Then the set of constants of M which satisfy the relation R(z) is not empty, by virtue of the first two statements of 6.2.3. It will be denoted by R. R is ordered by the relation Q(x, y), for which we shall write x
110
SYNTACTICAL TRANSFORMS
W ( b ,a ) holds in M . At least one R, is not empty. If a, a' E R and a ~ a then ' R, C R,, in view of the third statement of 6.2.3. The crucial result of the theory is
' is a supporting set for the bounding trans6.2.4. THEOREM. L form B on the range C. Instead of proving 6.2.4 directly, we consider the negations of X and X ' . When written in prenex normal form, these are 6.2.5.
Y = [(4, ... (~JWYJ(~Y~)(zA ... (2,) [--(X,,
*..I
Xz, Y1, - - * 7
yrn,
21,
...,Z J l I
and
Y'= [(~)(~Y)(z)(~ ..- (ZMYJ ... (3~,)(%) (2,)
6.2.6.
[
... A ~ ( Z Z2), 3 [w('ZJ,, Y) A ... A A W Y m 7 Y) A [W(;;, 4 A . * * A Wk,, 2) 3 [- Z(Z1,.*., xz: Yl, Ym, %, -.-, z,)l--.l. [~(ZI X),A
*.a,
Let C' be the class of statements Y which are obtained in this way from statements X EC. We write Y ' = B { Y } in agreement with our earlier notation for statements which belong to C as well. 6.2.4 states explicitly that for any provable X E C, X' is deducible from Z, i.e. X' holds in every model of 2 in which it is defined. It follows that 6.2.4 is equivalent to 6.2.7. THEOREM.Given any Y EC' such that Y ' = B { Y } holds in some model of 2, there exists a model for Y . For the proof, we shall assume I > 1, 772 >, 1, n > 1. It is easy to show that any other case within the scope of 6.2.7 may be reduced to one in which this assumption is satisfied. For example, if
y = (gYi)(Zi)[-' Z(Yi,Z1)l we consider instead the statement
where V(x) is an arbitrary relation of one variable. It is not difficult to see that if B { Y ) holds in some model of Z then B{Y*)also holds
111
BOUNDING TRANSFORM
in some model of Z. Then 1 = m = n = 1 for Y * . Assuming that we have already proved 6.2.7 for this case, we can find a structure M in which Y * holds. Since Y* 3 Y it then follows that Y holds in the same structure. The other cases for which either I = 0 or m = 0 or n=O can be dealt with in a similar way. However, it may be mentioned that there also exist simple direct proofs of 6.2.7 for these cases. Suppose then that I > 1, m > 1, n > 1. We write 2’ for 2. Let M be a model of Y as given by 6.2.5. The semantic interpretation of Y in terms of M shows that there exist, within M , functions i = 1, ..., m gi=fa(zl,..., x,),
-
in the ordinary mathematical sense such that for any I + n-uple of constants of M , a,, ..., a,, b,, ..., b,, the statement 6.2.8.
Z’(a1, ..., a,, fi(a1, . * - ) a,),
-.a
fm((ll,
-.-,a,), b,, ..*,b,)
holds in M . It is understood that when writing out 6.2.8 in detail, for given a,, ..., a,, we have to substitute the constants which are the functional values of fi(al, ..., a,), since the terms fi(a,, ..,, a,) are not included in the LPC as detailed in Chapter I. 6.2.8 will be said to be in Herbrand form. Conversely, if 6.2.8 holds in a structure M for all Z+n-uples a,, ..., a, b,, ..., b, then the structure satisfies 6.2.5. Similarly, 6.2.6 is satisfied by a structure M’ if and only if there exist in M’ functions g(z) and hi(%,z, x,, ..., z,), i = 1, ..., m such that the statement
6.2.9.
I
W(a1, a ) A -..A W(az, a ) 2 [W(h,(a,b, a,, uz), g(a))A A ... A v ( h , ( U , b, a,, ..., a $ ) g(U)) , A [W(b,, b ) A ... A A W(b,, b ) 5, Z’(a,, ..., a,, h,(a, b, a,, ..., a t ) , ... ..., h,(a, b, a,, a,), 4, --.bJll -.a,
holds in M’ for all a, b, a,, ..., a,, b,, ..., b,. We may now formulate our problem as follows. Given a model M’ of 1: which satisfies 6.2.9, we have to find a structure M which satisfies 6.2.8.
112
SYNTACTICAL TRANSFORMS
I n order to specify the constants of M , we introduce m new functional symbols 6.2.10.
41(%
*a',
d, A&l, ... 4. ..-9
7
Let c,, cl, ..., c, be the set of constants which are included in Y . If Y does not include any constants, let c, be a constant picked a t random from M'. We now define the class of constants of M , say, as follows. 6.2.11. c,, ..., c, are elements of r, of order 0. 6.2.12. Having defined the elements of '1 of order less than k, where k is a positive integer, we obtain the elements of order k, by filling the argument places of any +;, i = 1, ..., m, with elements of of order less than k, provided that a t least one argument is exactly of order k- 1 in each case. For example, dl(c0, ..., co) is of order 1, +2(co, ..., q5,(c0, ..., c,)) is of order 2, etc. The number of elements of given order is finite. Thus we obtain a simply infinite sequence by taking first the elements of order 0, as given, followed by the elements of successive orders, when ranged in arbitrary but definite finite sequences. We shall denote the elements of the scquence by
r
r
6.2.13.
co, c1,
...> c,, c1+1, C,+Z, ...
in agreement with 6.2.11. If cjl, ..., cjl are elements of of order less than k, then the order of C$~(C~,, ..., c&, i = 1 , ...,m, does not exceed k. We define the functions fi(zl,..., x J , i = 1 , ..., m, on I' by
r
6.2.14.
fi(cj,>*..> cjl)=+i(cjl>. * * > cil).
It must be understood that this is not a functional identity, but a rule which associates with the argument values cil, ..., cjl as functional value the purely formal expression 4i(cjl, ..., cjl) which is an element of I'. The relations of M now have to be specified in such a way that the statements
113
BOUNDINQ “RANSFORM
hold in M , for jl,..., it,kl, ..., k,, = 0, 1 , 2, 3 , .... Let H be the set of all these statements and let Hvflbe the W t e subset of H whose elements are restricted by the conditions ji
i=l,
..., 1
k,gp,
i=l,
..., n,
for v,p=O, 1, 2, .... For any relation S(xl, ..., zk)which occurs in Y and for any n-uple c,,, ..., c,~, we have to specify whether or not X(c,, ..., cjk) holds in M , in such a way that according to the usual truth-table procedure of the propositional calculus, the elements of H all become “true”. However a familiar argument (compare refs. 4, 10, 11) shows that it is sufficient to establish the possibility of such a procedure for all the sets HYp. This has the advantage that the sets of statements to be considered are finite. According to the first statement of 6.2.3, there exist constants do, ..., d, such that W(ci,di) holds in M‘, i = O , 1, ..., T . The constants do, ..., d, belong to R. We propose to show that the formulae 6.2.16. 6.2.17.
I(ci)=di 4+i(cjl,
-.*>
i=o, 1,
...,r
c$l))=g (max. (cjJ) 14k$l
constitute an inductive definition of a mapping of r into the set of constants of M‘, more particularly into the set R. Indeed 6.2.16 defines 1for the elements of r which are of order 0, in such a way that these elements are mapped into R. Let q be any positive integer and suppose that we have verified already that any element of whose order is smaller than q is mapped by rZ into an element of R.Consider any element c of I’ which is of order q. c is of the form
r
c=+i(cj,,
cji)
where ci,,..., cil are of order smaller than q. Then I(c,,), ..., A(c,,) are elements of R and are ordered by the relation Q . Thus the definition c’ = max. (cj ) l
determines a unique element of R which has the properties
for k = 1 , ..., I I(cI,)=c’ for at least one k .
1(c,J
114
SYNTACTICAL TRANSFORMS
Suppose for example that c'=A(cjt). Then 6.2.17 yields wi(cj,,
*.*)
ciJ)=g(~(%,))
6.2.9 implies that
holds in M' for all constants a, b, a,, ..., a,. We make the additional inductive assumption that for any A(cj), where cj is of order less than n, there exists a constant d such that W ( d ,il(c,)) holds in M'. This is certainly true for all ci of order 0. Putting i = 1, a=A(cj,) in 6.2.18 and a,= ... =a,=d where d is such that W(d ,l ( c j , ) ) holds, we obtain for arbitrary b,
W d , &,))
A
- - a
A
W(d,4 C j J 3 W(~l(A(C&b, d,
*.*,
4, gMc,,)).
It follows that W(hlMCj,), b, d,
*.*,
4,g(A(x,,))
holds in M', and this shows, by virtue of the second statement of 6.2.3 that g(A(c,)) belongs to R and, moreover, that there exists a constant d' such that W ( d',g(il(c,,))) holds in M'. We conclude that the definition of il is unambiguous and that A(cj) E R for all cj E I'. Having fixed v and p, let p=max. il(ci) and define another 3 . a
mapping of 6.2.19.
Then
(
'I into the set of constants of M' by n(ci)=ci, i = o , 1, ..., r
A(y5Jcf1, ..., ciJ) = hi(max. A(c,J, p , A(cj,), ..., A(ciz)), ladl
i = 1, ..., m.
6.2.20.
W(A(C,),A(Ci)),
j=o, 1) 2,
...
holds in M . Indeed for j = O , 1, ..., r , 6.2.20 reduces to W(cj, dj). For a given positive integer q, suppose that 6.2.20 has already been proved for all ci of order less than q . Consider any element cj of 'I which is of order q. ci is of the form fii(ci,, ..., cil) where the arguments are of smaller order. Put a=max. A(c,), l
b=p,
~ , = A ( c , ..., ~ ) ,a,=d(c,,)
115
BOUNDING TRANSPORM
in 6.2.18,
SO
-
6.2.21
Now
W(A(c,),&*)),
i= 1,
*.*9
holds in M’ by the assumption of induction and, in view of the third statement of 6.2.3 the same applies to
i= 1, ..., 1.
W ( A ( c . ) max. (A(cjJ)
6.2.22.
’ l
The conjunction of the statements of 6.2.22 is the implicans of 6.2.21 and so we may conclude that the implicate of 6.2.21 also holds in Af‘. Reference to 6.2.17 and 6.2.19 shows that this implicate is identical with W(A(c,),A(c7)),which proves our assertion. Next, we propose to show that for all j,, ..., j l and for k,, .,.,k,
{ Z’(A(c,), ..*,
4c,z), 4 M c i , ,
*..Y
A(+m(cjl,
c,$,
me-9
A(ck,)l
holds in X’. Substituting max. A(ci,), p , A(c,J, ..., A(c& A(c,l), ..., A(c,)
6.2.24.
1Qk
for a,b, a,, ..., at, b,, ..., b, respectively in 6.2.9, we obtain an implication whose implicans is the conjunction of the statements 6.2.22 which have already been shown to hold in M‘. It follows that the implicate which is the result of substituting 6.2.24 in 6.2.85.
A W(&(n,b,a,, .--,az),da)) A [W(h,(%b,ap --.,ad,g(a)>A A [W(b,,b) A ... A W(b,,6)3Z’(al,...,a,,h,(a,b,a, ,..., a z ) ..., ,
i
h,(a, b, a,, ...,aZ),4, ...,b J l l
also holds in N’, and the same applies in particular to the conjunct 6.2.26.
( W(b,, 6 )
A
... A W(b,, b ) 3 Z’(a,, ..., a,, h,(a, 6, a,, ..., a t ) , .-.,hm(a, 6, a,, at), 4, ...,ha). ..a,
116
SYNTACTICAL TRANSFORMS
of 6.2.25. Now the implicans of 6.2.26 in terms of 6.2.24 is the conjunction of the statements
Jww,,), PI,
6.2.27.
i = 1,
***,
n
and these hold in M’ by 6.2.20 and the definition of p . We conclude that the result of substituting 6.2.24 in 6.2.28.
Z’(a,,...,a,,h,(a,b,a,) ,...,h,(a,b,a, ,...,a),b,,...,b,)
holds in M’ and reference to 6.2.19 shows that this is precisely 6.2.23.
Now let S ( C , ~ .-., , cjk) be any atomic statement which occurs in , c,J holds one of the elements of HVp.Then we specify that S ( C , ~..., or does not hold according as the statement S(A(c,,),..., A(c,)) does or does not hold in MI. With this specification, all the elements of H1, become “true”. Indeed, consider any Zl, .,., ._,. *, E Hvp. The truth value of this statement depends only on the truth values of the atomic statements contained in it, and these are the same as the truth values of the corresponding atomic statements of 6.2.23, which yield a truth value “true” (or “holds”) for the latter statement. Hence Zz, ,... ..,.ks becomes “true” for the specified truth values of the atomic statements contained in it. This proves 6.2.7.
It may be mentioned that if Y does not contain any constants then C may be weakened by replacing the first axiom of 6.2.3 by
(34(3Y) WG Y). For the proof, we select c, and do so that W(c,, do)holds, and proceed 6.2.29.
as before.
6.3. A Counter-Example We shall now show that the main result established in the preceding section (6.2.4) is, in a sense, the best possible. For this purpose, we shall construct a provable statement 6.3.1.
x= ( ~ l W Y l ) ( M ( YI? ~ l Z~I),
where Z does not contain any further quantifiers, such that
117
CO7JNTER-EXAMPLE
X' = B ( X ) is not deducible from 2. More particularly, we put
c-
Z(X, y, z)=
41
A(& y) v
where A is a relation with two arguments. Then
x= ( ~ l ) ( ~ Y l ) ( ~ l ) [4 - %Yl) v A(% 4 1 .
6.3.2.
Now, for any constant a, the statement
-
[(Yl)A(% Y1)l
" [(!Ill 4%
Y1)l
is provable. Changing the variable y in the second disjunct to z, and replacing the first disjunct by (3yl)[- A(a, yl)], we obtain
[(ZYi)" A(% Yd11 V [(%I A(% %)I
6.3.3.
as another provable statement. We may change 6.3.3 to
-
(3~d-
M )v
[(d4, %)I1
and since A ( a ,yl)does not contain %, we may shift the universal quantifier across it, so (@?h)(%)" A(a,Yi) V A(a,21)l.
Finally, replacing a by the variable xl and prefixing (q), we obtain 6.3.4.
(%)WYl)(%)[-
4%Yl) v 4%2113.
This is X . Since the steps leading from 6.3.3 to 6.3.4 all transform provable statements into provable statements, we have established that X is provable. Now consider X ' = B ( X ) 6.3.5.
{
x ' = ( ~ ) ( 3 y ) ( 2 ) ( ~ i ) ( 3 y i ) ( z , ) [ w2 ( ~[v(!/i, i, y) A A Ew(%2) 3 [- 4 x 1 , Yl) v 4%21)1*..1.
A prenex normal form of -XI 6.3.6.
{
is given by
3
-
[w(?h [w(%, 2) A [A(xi:,, y1) A A(z17 %)I***].
Y'=( 3 ~ ) ( Y ) ( 3 2 ) ( 3 ~ i ) ( Y i ) ( 3 2 i f [ w (2) ~ iA,
We shall construct a model M' of 2 which satisfies 6.3.7.
( y"
= ( 3 ~ ) ( Y ) ( ~ 2 ) ( 3 ~ i ) ( 3 z ,[)w((yX1i ,) 2) A [ 3 2 ) A [A(x1, y1) A
[w(%
w(?hy) 3 A(%
%)I*-.]-
118
SYNTACTICAL TRANSFORMS
Since Y" 3 Y' is provable, M f will a fortiori satisfy Y', Y" can be replaced by the following statement in Herbrand form.
W s ( a ) ,4 A r w a , , a ) A
6.3.8.
[WW), / ( a ) )A A(g(a),a,) A A
--4s(a),
Na))I...l*
Thus Y" holds in a structure M if and only if M contains a constant d such that for all constants a,, a of M , there are defined function f, g, h for which 6.3.8 is satisfied. We define the structure M in the following way. The set of constants of M' consists of the elements of the doubly infinite sequence {c;], i, k = 0, 1, 2, ..., and the relations of M' are R ( 4 , &(x, Y), W(X,Y), A ( z ,Y). R ( c ~holds ) for i = O , all k, and not otherwise; Q(ci ci) holds for i = j = O , k ~ l and , not otherwise; k: W(c;,cf) holds for j = 0, i < 1, and not otherwise; A(c;, ct) holds for i = O , j ~ k and , not otherwise. Thus A(c',,cf) holds in M' if and only if W(ci,ci) holds in that structure. It is not difficult to verify directly that M' is a model of 2. To prove that Y" holds in M' we define
'.
d = co", g(cL)=cP,
f (ck)= c:+1
i, k = O , 1, 2, ...
i - k3-1 h(c,)-c, ,
Then W(g(a),d ) becomes, for a=cR, W(cg,c:) and this holds in M' by the above definition. It remains to be shown that 6.3.9.
W%,4 3 W W ) ,f ( a ) )A 4&), a,) A
-
A m ) , W))l
holds in M'. We may confine ourselves to
a,=c;,
l>i
a=cp,
since otherwise the implicans of 6.3.9 does not hold. Then the implicate becomes
-
Ci) A A(C;,C;") W(C;+',C!+l) A and we may verify again that this holds in M'. Thus M' constitutes the required counter-example. I n the preceding section, the bounding transform was defined
BOUNDING TRANSFORMS AND DEDUCIBILITY
119
only for special classes of statements in prenex normal form. Consider now an arbitrary statement in prenex normal form, e.g.
where Z is the matrix of the statement. We define the general bounding transform of X , X ' = B * { X } by
x'= (z')(X2)(3y1)(z3)-.- ( ~ n ) ( 3 y ~ ) ( ~ i ) ( z 2 ) ( 3 y..i ) ( ~ 3 ) ( z ~ ) ( ~ y ~ ) [ ~zl) ( ~ l ,[w(%s2) y') x3)3 ..' .-.[W(za,,zn)I [w(ym,y") z]...], *
[w(?h7
A [w(x3>
A
from which the law of formation of the transform wiU be apparent. B * { X ) is not, in general, identical with B { X } for X EC, but it is not difficult to see that for such statements B { X } = B * { X ) is deducible from 2. Now conside the sequence of quantifiers for any statement in prenex normal form which does not belong to the class C. The sequence must include three quantifiers such ;ts (Zz),(y), ( 3 z ) , which appear in that order, although they may be separated by further quantifiers. Using the above counter-example, we may then show that there exists a matrix Z such that if we prefix to it the given sequence of quantifiers, there results a provable statement X whose general bounding transform B* { X } is not deducible from 2. Thus, although B* has a wider range of definition than B, its introduction does not lead to any new results which are stronger than 6.2.4. For this reason we have given preference to B , which has the advantage of greater simplicity.
6.4. Bounding Transforms and Deducibility Let K be a finite or infinite set of axioms whose elements belong to the class C' (see 6.2.5). The set of the bounding transforms of the elements of K will be denoted by B { K } . We propose to prove 6.4.1. THEOREM.If the statement X E C is deducible from the set of axioms K C C', then B ( X }is deducible from the set 2 u B{K}. The assumption of the theorem is that there exist statements Yl, ..., Y , E K such that 6.4.2.
y1 A
... A
17, 3
x
120
SYNTACTICAL !l!RANSFOBMS
is provable. We shall show that
B(Y1)A B(Y,}A ... A B(Y,} 3 B(X}
6.4.3.
is deducible from Z. We write Xifor the normal form of
B{Y,) =
-
Yi,then
N
B(XJ
is a provable statement, i = 1 ..., p . We also write X = X9+1, so that XiEC, 1 < i < p + 1. Then 6.4.2 and 6.4.3 can be replaced by
x, v ... v x,v x,,
6.4.4.
and
... v B(X,}v B{xp+l}.
B(Xl} v
6.4.5.
We suppose (if necessary after a preliminary change of variables) that no variable occurs in two different components of 6.4.4. We may then shift all quantifiers in 6.4.4 to the beginning of the formula and arrange them at will, so long as we do not interfere with the mutual order of the variables which belong to the same component. More particularly, if 6.4.6.
xi= ( 3 x i ) ... (xxi4)( ~ i..). (&,)
1... "i, 7
. y'1 .
... yi,,
3
7
(32;)
...)(&!%i(
..., 2;)
zc(Zi,... i= 1, ..., p +
1
then we arrange the quantifiers in 6.4.4 so as to yield the result
v,= ( 3 x 3 ... (2h;J ( 3 x 3 ... ( 3 x 9 ... (34 ... ( 3 x i ) (Y:) *.. (Yk)(Y3 ( Y a (?I (?.a ;) **-
6.4.7.
*.-
- a -
. ... ( 3 2 ; ) ... ( 3 2 ; ) ... ( 3 2 ; ) 1 ... xi1,y1, ... ykl, z:, ... 2 . t ) v ... v z,(x;, ... 5, y;, ... y k 7z;, ... 2;)
(32;) .. ( 3 2 : ) (32:) Z1(Z!,
for r = p + 1. More generally we define the syntactical transform VT=X{Xl v ... v X,] by 6.4.7 for r = l , 2, ..., p+1. I n particular Vl=X,. Then V,, E C , and so .B{T'9+l) is deducible from C,in view of 6.4.4. We shall establish presently that 6.4.8.
W T + l }
2
W ?v ) B{XT+l}
BOUNDING TRANSFORMS AND DEDUCIBLTZW
121
is deducible from Z for r = 1, 2, ..., p . Taking this relation for granted, we obtain the following chain of statements which are deducible from Z.
'
W,+lI W J v B{X,+l} W,+d3 W a - 1 1 v B{XJ v BP,+l} W,+J3 w , - 2 1
vBPP-1) v
W
P
I
v B{X*l}
and finally 6.4.9.
B{V,,l} 3 W
l )
vB mv
*.*
v
W,+d
where we have taken into account that V,=X,. Since B(V,+,} is deducible from 2, it then follows that 6.4.5 also must be deducible from Z. We still have to show that 6.4.8 is deducible from 2. Now
so that 6.4.8 may also be written as 6.4.10.
v
'{S{J'r
Xr+l>Z3 B(Vr1 v B{Xr+1}-
This is of the same form for all r , and we may therefore confhe ourselves to r = 1, 8,= X,. Then the statement becomes 6.4.1 1.
V
P
l
v
&)I
3 B{Xd " w
2 1 .
The implicans of 6.4.11 is, explicitly,
B{X{X, v X,}} = [(3x)(y)(zZ)(32:)... (&%;J(Zg)... (&!a$ (y:) ... (&l)(Y:) ... (gg2)(3zi)... ( ~ T Z : ~ ) ( ~ .. Z ?. ( ) g.Z$,)[w(2) ZlA,. .. A w ( Z i , 2) A w ( X ; , 2) A ... A w(Xi,5)A [w(?/i,y) A ... A w(&, g) A w(y:,
,Y) A
... A w(&,,, y) r> [w(Z:,2 ) A ... A w(Z&, 2) A w ( Z : , Z) A ... ..., Xi, y:, ..., yk1, Z:, ..., X A l ) V z,(X:,..., zk)] ..- 3 .
w(Z&, 2) A [zl(X:,
We modify this expression by replacing (3x)(y)(3z) by ( 3 ~ ~ ) (Z22)(y1)(y2)(3x1)(i71~2), and by replacing 2, y, z in the relations W by xl, yl,3, or 5,y,, x,, according as these variables appear next to a symbol with upper suffix 1 or with upper suffix 2. There results a statement 2: such that B{X(Xlv X,}} E 2: is deducible from 2 (compare the end of the preceding section).
122
SYNTACTICAL TRANSFORMS
I n this expression the quantifiers of the second disjunct, B{X,), may be shifted across the matrix of the first disjunct, and arranged so that the sequence of the quantifiers is 6.4.12.
(3xi1) ( (34)...(3$)(y:) ...(y&)(y:). ..(&,) (g.2:). .. ( 3 ~ 1 ) ( 3 ~ 2 ) ( y l ) ( Y 2 ) ( 3 2 1 ) ( 3 2 2 ) ( -3- *~ ~ )
(32;)
(gz:)...(3zgO).
The resulting implicate 2; is such that B { X l ) v B{X2}= 2: is provable, and so we only have to show that 6.4.13.
is deducible from Z. Moreover, both 2: and 2: are in prenex normal form and the sequence of the quantifiers is the same for both cases. Now put Al=W(U:,Ul)A
...A W ( U ~Ul), , k!,=W(U:,
U,)A
... h W ( U : , U , )
B1=W(b:,b,) A ... A W(bkl, b,), B, = w(b:, b,) A ... A w(b;*, b,) C, = W(c:, c,) A ... A W(cil, cl), C , = W(c:, c,) A .. . A W(cg,,c,) D,=Z,(a:, ..., u:, b:, ..., bkL,c:, ..., cAJ, D2=Z2(4,..., a:, b?, ..., b:,, c:, ..., c:~) - where it is assumed that the constants uk,b,, ck, ui,bi, ci, are not already contained in 2: 3 2;. One verifies without difficulty that the statement
6.4.15.
{ A1
A
A , A [Bl A B2 3 [c,A (72 A [olv 0 2 1 1 1 3 3 [A1A iB13 c, A Dlll v [ A , A [ B , 3 c, A Qll
s provable when A,, A,, ..., D, are considered as propositional
BOUNDING TRANSFORMS AND DEDUCIBILITY
123
variables. It follows that it is provable also with the interpretation 6.4.14. We now make use of the rule 6.4.16. Let W,(x), W,(s) be two predicates such that
is deducible from a set of statements H which does not include a.
Then the statements
[(x)W1(413 r
( ~ ) ~ 2 ( 4 1
and
“34 %(.,I
2 [(gx) W,(s)I
are deducible from H . The truth of 6.4.16 can be established along the same lines 2.3.3 and the proof will be omitted here. We apply 6.4.16 to 6.4.15 with
it9
w1= [A, A A, A [B, A B23 [c,A c, A [Dl D,]]] w,=“4* [ B l I Cl A 0111 v [A, A [B,A [C, A Dzlll V
by prefixing the quantifiers of 6.4.12 successively to W, and W,, the order of the procedure being opposite to the order to appearance of the quantifiers in 6.4.12. Thus we first prefix (zz,) to W, and W, and replace czz by zz, in both statements. Next we prefix (zz,-,) 2 and replace c;,-, by znS-,, etc. As our last step, we prefix (3x3 and replace a: by xi. The final result is that W, is transformed into 8: and W , into Zl, so that 6.4.15 is turned into 6.4.13. This completes the proof of 6.4.1. A survey of the preceding chapters shows that the following sets of axioms belong to the class C‘ or more precisely, to the intersection of C and c‘: 1.2.1, 1.4.2, 1.4.3, 3.1.2, 3.2.1, 3.3.1, 3.3.3, 3.4.3-3.4.6, 3.4.38, 3.5.1-3.5.5, 3.5.14, 3.6.2-3.6.7.
The same applies to 3.1.3, 3.1.4, 4.3.6, 4.3.7, 4.3.15, 4.3.18, as appears when we write out Xn(x, y) according to its definition 3.1.1; to 3.2.5, 3.3.2, and 3.5.7, by 3.2.3 and 3.2.4; and to all sets of statements which do not include any quantifiers, including all
124
SYNTACTICAL TRANSBORMS
diagrams. Thus, all systems of axioms which are considered in the present book are subsets of C n C'.
Application The following simple example is typical of the applications which can be made of theorem 6.4.1. Let M be a completely divisible ordered abelian group which contains at least two different elements (see section 3.1). We denote the order relation in M by &'(x, y) in order to distinguish it from the relation &(x, y) of the present chapter. The remaining relations of M are E(x, y) and X(z, y, 2 ) . We define three additional relations R(x), &(x, y), W ( z ,y) for the constants of M as follows. 6.5.1. R(a) holds for all constants a of M such that apO, and not otherwise. 6.5.2. &(a,b ) holds if and only if O G a G b in M . 6.5.3. W(a,b ) holds if and only if b>O and -b
3.1.4).
Considering first the axioms 1.2.1, we observe that if a statement X in prenex normal form contains only universal quantifiers, then X 3 B { X }is provable. It follows that if X holds in a particular structure then the same applies to B{X),provided B { X ) is defined in the structure. Thus we need consider only the axioms
Y, 4 (.)(Y)(gz) X (.> 2 , Y). ( X) ( Y) ( Z Z ) S(Z,
The bounding transforms of these two statements hold in M by virtue of the inequality
I.+Yl~I~l+lYl which holds in every ordered abelian group. The absolute value 1x1 is defined rn x or - x according as x > O or x t O . The repeated
A.PPLICA!L'ION
126
application of the same inequality proves that the bounding transform of 4.1.4 is satisfied by M * . The same is true of 3.1.2, trivially so, while 3.1.3 includes only universal quantifiers when written out in prenex normal form. Applying 6.4.1, we now obtain 6.5.4. THEOREM. Let X be a statement of the class C which is formulated in terms of the relations E(x, y) and S(z, y, z ) and which holds in all completely divisible torsion-free abelian groups which contain at least two elements. Then B { X } holds in all ordered groups of the same type, provided W ( z ,y) is interpreted by means of 6.5.3. It will be seen that 6.5.4 represents a new type of transfer principle (compare section 5.2). The theorem affirms that if a statement X of a specified kind holds in one particular class of structures, then a certain syntactical transform of X holds for all structures of some other class. Taking into account Theorem 4.3.1, we may replace 6.5.4 by 6.5.5. THEOREM. Let X be a statement of the class C, formulated in terms of the relations E(x, y) and S(x, y, z), and without tonstants, which holds for the additive group of rational numbers. Then B { X ) holds in all completely divisible ordered abelian groups which contain at least two elements.