Zeitschr. f. math. Logik tcnd Grundlagen d. Math. Bd. 2s. S. 215-218 (1982)

A KOTE ON THE INTERPOLATION THEOREM I N FIRST ORDER LOGIC

by GEORGE WEAVER in Bryn Mawr, Pennsylvania (U.S.A.)

1. The purpose of this paper is to present a model theoretic argument for the inter- polation theorem. Model theoretic arguments in the literature are of three basic types : BUCHI-CRAIG, ROBINSON and HENKIN.

BUCHI-CRAIG arguments formulate the interpolation theorem in terms of pseudo- elementary classes ([3]). Here there are both algebraic and topological arguments. Algebraic arguments include those using ultrapowers ([7]) and ultralimits ([2]), and topological arguments include those which put a topology on the space of the models and use the normality of this topology to establish the interpolation theorem ([14]).

ROBIXSON arguments prove the interpolation theorem from ROBINSON’S joint consistency lemma. Arguments here are distinguished one from another by the tech- niques used to prove ROBINSON’S lemma: elementary chains ([l], [ 5 ] ) ; saturated and special models ([4]. [ 5 ] ) ; and recursively saturated models ([l], [9]).

HENKIN arguments are based on refinements of the HENKIN style proof of the compactness theorem. These include those using separable pairs of theories ([6], [5]), and thoye using consistency properties ([12], [ll], [8], [El ) .

All of the ROBINSON arguments, those HENKM arguments using separable pairs of theories, and at least one of those using consistency properties ([ 153, proceed by assuming that no interpolant exists and deriving a contradiction. The remaining arguments, while direct, neither construct the interpolant nor give conditions necessary and sufficient for a sentence to be an interpolant.

The argument presented here is “essentially” a ROBINSON argument, but differs from the arguments above in that (1) it proceeds directly; (2) its main ideas derive from the EHRENFEUCHT-FRAISSE characterization of elementary equivalence ; (3) it make.; eiqential use of finitary SCOTT sentences; and (4) it specifies for each p and K‘, a finite non-empty set of non-logical constants, the set of interpolants for p and K’, and .;hon-j (a) that they differ from one another only in the number of distinct variables they contain, and (b) that the choice of an interpolant between p and y (when their common non-logical vocabulary is contained in K ) is a function of the number of distinct rariables in y and the non-logical vocabulary of y.

2. Let K be any set of non-logical constants, excluding functional constants. LA is the first order language (with equality) over K . The notions of formula and sentence in L, are defined as in [lS] (p. 235). For p a sentence r(p) (the rank of (p) is the number of diitinct variables occurring in p. For each n >= 0, LI,[n] contains all p such that

Interpretations for LIi are ordered pairs 9l = (A, fa), A a non-empty set. and f% is a function defined on K in the usual way. For p a sentence. S a set of sentences K ( v ) denotes the non-logical constants occurring in Q;; S C ~1 indicates that p is a logical consequence of S ; Yl k S (94 C p) indicates that 8 is a model of S ( y ) . Th(9l) denote5 the complete theory of 8, and 8 EZ 23 indicates that 8 and ‘23 are equivalent. When no confusion results K is omitted from LIc and L,-[n].

T(Q;) 2 1 1 .

216 GEORGE WEAVER

Given K finite, for each natural number m 0 the relation N nL is defined between pairs of interpretations as in [16] (p. 255). The following results of [16] are used below: (1) i?l N ,lL 23 provided '21 and '23 agree on all sentences in L[m] ; (2) given '21 and n7 there is a sentence in L [ m ] (denoted P(AI'21, m)) such that for all %, B k P(A/%, m) provided '% w l r L B ; and (3) for each rn, there are only finitely many non--

Given K (finite or infinite), '21 an interpretation for LK and K' g K , '21 I K' denotes the reduct of M to K'. Given X, K' , '16 where K' is a finite subset of K , P(S, K' , n) = V{P(A/% 1 K , n) : 'iX k XI, when X has models; otherwise some contradiction in LI;,[n]. P(p , K , n) denotes P({'p}, K , n). Notice that (1) if S has no models and K' has no individual constants P(S, K', 0) is not defined; (2) 23 k P(S, K', n) provided there is '% I= S such that % I K '21 I K'; (3) if n < m, P(S, K', m) != P(8, X' , n);

3. Given K' any finite subset of K , and X, S' subsets of L K , n 2 0, S and S' are K' synonymous provided (1) for all '21 k X there is % k X' such that '21 I K' E '93 I K' and (2) for all % I= S' there is '21 k S such that % 1 K' = '21 I K ' ; and S and 8' are K', n synonymous provided (1) for all '21 k X there is % k x' such that '21 1 K' - > & % 1 K' and (2) for all 23 != X' there is '% k X such that 3 I K' -,z % I K'.

Given S, S', and y it is easily verified that (1) if X k y, S and X' are K ( y ) , r (y ) syn- onymous, then S' != y ; (2) S and P(S, K ( y ) r ( y ) ) are K ( y ) , r ( y ) synonymous; and (3) the compactness theorem guarantees that if X k y then S and some finite subset of S are K ( y ) , r ( y ) synonymous (cf. [17]).

There are basically three ideas in the following argument: When 9 I= y and K('p) A K ( y ) = K' and K' is non-empty then (1) only that information expressed by 'p

about K' is relevant to the truth of y (i.e., any set X which is K' synonymous to cp has y among its logical consequences); (2) there is a set of sentences which is K' syn- onymous to 'p; and (3) not all of the information expressed by this set is relevant to the truth of y (i.e., there is a finite subset of S which is K(y) , r ( y ) synonymous to 8). The ROBINSON lemma is used to establish (1) ; the satisfiability formulation of the compactness theorem is used to establish (2) ; and the consequence formulation of the compactness theorem is used to establish (3) (cf. [17]).

Lemma. Given K , 'p in L K , n 2 r('p). If K" i s any finite subset of K and y has models, then

interpretations.

and (4) 4 9 = P(93 K('p), r(p))).

(1) 'p k P('p3 K', n ) ;

(2) for all y, if 'p k y, r(y) 5 n and K ( y ) E K", then P(p, K", n) k y ;

(3 ) for K' any finite subset of K , all m 2 0,

(4) for all S, K', if K' i s a finite subset of K , K(p) n K" g K' and S and 'p are K' syn- onymous, then X k P ( p , K", m) for all nz;

( 5 ) { P ( ~ J , K", n): n >= 0 } and 'p are K" synonymous. Proof . (I), (2) and (3) are immediate from the above remarks. (4) Suppose S and

cp are K' synonymous, Let '@ k S. Then there is % k q~ such that '21 I K' = % 1 K'. Thus, by the ROBINSON joint consistency lemma, there is 6 k Th(% I K(p)) w Th(B 1 K"). Thus, Q k 'p, hence 6 != P('p, K", m) (by (3) above); and 23 k P(p, K", m). (5) Let T = {P('p, K", n ) : n 2 01. Let !€I I. T. Then, for each n there is I. 'p such that

'p k P('p, K', na);

A NOTE ON THE INTERPOLATION THEOREM IN FIRST ORDER LOGIC 217

B I K” N n gfL 1 K“. Thus, every finite subset of Th(8 I K”), p has a model, and (by compactness) there is % t p such that rz1 I K” f 23 1 K”. Then, by (3) above. T and p are K” synonymous.

Theorem. Given K. p a sentence in LA-, K’, K” finite non-empty subsets of K and 712 such that r(p) 5 711.

(1) There is t such that p k P(p, K’, t ) and for all y , if K ( y ) K’, r (y ) =< m, K(p) n K” ( 2 ) for all t , t satisfies (1) above provided, P(p, K‘, t ) and (P(p, K‘, n) : n ECO} are K“. nz synonymous.

Proof. (1 ) By the lemma, (P(p, K’, n ) : 71. EO} I= P(p, K“, m). Then, by compact- ness, there is t such that P(9, K‘, t ) t P(p, K”, 772) and P(p, K’, t ) k y. (2) Suppose P(p, K‘, t ) t P(p, K”. m). Let % k P(p, K’, t ) . Then there is 0. k ip such that rz1 1 K“ W m 0. I K”. But as q~ t P(p, K’, n) for all n, 0. t {P(p, K’, n ) : n E w}. Thus, P(p, K’, t ) and {P(ip, K‘, n ) : n ~ w } are K“, ?n synonymous.

Suppose P(p, K , t ) and {P(y, K’, n ) : n E C O } are K”, na synonymous. Let 8 k P(9, K’, t). Then there is ‘$3 k (P(p, K’, n) : n E CO> such that % 1 K ‘ - f,l 8 I K”. By the lemma, ‘$3 t P(p, K”, m); hence % t P(p, K”, m).

Thus, given p and K‘, all interpolants for p and K‘ have basically the same form, differing from one another only in the number of distinct variables t ; and, the value of t for any y is a function of K(y) and r(y) . It is natural to study the action of this function. In particular, consider the following questions: (1) is the value of t indepen- dent of K(y)? and ( 2 ) is t = r (y)? Notice that the second question is equivalent to the question of whether or not the interpolation theorem holds for those first order languages with only finitely many variables.

In [16] (pp. 259-260) results were announced which imply that the answer to the second question is yes. There, it was argued that if K’ is a finite non-empty subset of K and r ( y ) 5 m, then P(p, K’, n ~ ) t y for all y such that K(p) n K ( y ) 5 K ’ , p k y and r ( y ) 5 rn. However, the above is false. To see this, notice that if it were true then for all K’, K”, KO finite non-empty subsets of K such that K’ = K” n KO for all 111 2 0 and all %, ‘E if 3 I K‘ - f , l % I K‘, then there is 0. such that I K“- ,,IL 0. I K” and ’II I KO - ,), 0. I KO. Let (II, ‘E be such that % I K‘ - I ) ) ‘$3 I K’. It suffices to show that { P ( A / a 1 K“, m), P(B/% I KO, m)} has a model. Suppose otherwise; then P ( A / a I K “ , m) k-P(B/B I KO, m), and. by the above, there is ly such that K ( y ) = K’, r ( y ) = ni. P(A/% I K”, m) t y and y k -P(B/’$3 I KO, m). Thus, y is true on ‘% I K’ and % I K’. P(B/’$3 I K , m ) != y ; and P(B/’$3 I K’, nz) k -P(B/% I KO, m), contradiction.

Consider the following: Let K = {R, R‘, el, . . . , cm) where R, R‘ are binary re- lational constants, c,, , . ,, c,, are individual constants KO = {R. c l , . . ., cm), K” = (R. R‘}, and K’ = {R}. Suppose (II and 23 are such that (1) % is a model of VxyRry and some axiom of infinity in R’; and (2) (II is a model of VxyRxy and A = ( f@(c, ) : 1 5 i 5 m}. It is easily shown that 8 I K N ) , ~ ‘8 1 K , but if m 2 3 there is no 6 such that rz1 I KO - m 0.1 KO and 0.1 K” -nz B 1 K”.

The ansn-er to the first question above is also negative. For if the value of t was independent of K ( y ) , then there is t such that for all finite K”, P(p, K , t ) t P(p, K”, m). Hence, if (ZI 1 K‘ - t ‘$3 1 K’, there is 0. such that 0. I KO - 2.l I KO and 0. I K“ - f f l % I K ’ . But the example above is easily adapted to provide a counterexample here.

K’, and ip k y, then P(p, K’, t ) t y ;

218 GEORGE WEAVER

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(Eingegangen am 18. Juli 1980)