Finite Partitions and Their Generators

Zeitechr. f. math. Log& und Crundlcrgen d. Malh. Bd. 20, S. 255 - 260 (1974)

FINITE PARTITIONS AND THEIR GENERATORS

by GEORQE WEAVER in Bryn Mawr, Pennsylvania (U.S.A.) 1.0. This paper presents a characterization of elementary equivalence due to

FRAISSE [6] and to EHRENFEUCHT [I]. While less sophisticated than that of KEISLER [8] or KOCHEN [9], this characterization can be extended to higher order logics (c.f. [lo] p. 135). Other proofs of our main result are found in [3] and [ll]; the novelty of the present proof yields: (1) a characterization of elementary equivalence (section 3) ; (2) a cannonical axiomatization for both finitely axiomatizable and decidable theories (section 4); (3) a “uniform” compactness theorem (section 5 ) ; and (4) two normal forms for first order sentences and a “uniform” interpolation lemma (section 6 ) .

2.0. Let KO be any finite set of non-logical constants containing no functional constants. Let c l , cq, . . ., c,, . . . be an infinite sequence of individual constants not in K O . Let K, be the result of adding c l , c2 , . . . , c, to KO ; and Lxn be the first order language with equality having K,, for non-logical constants. The formulas of LKn and the distinction between free and bound variables are defined following HILBERT and ACKERMANN (c.f. [7] p. 66). Sentences are formulas devoid of free variables. For each sentence A , let r ( A ) (the rank of A ) equal the number of distinct variables in A , and for each m let LKn[m] denote the sentences of LKn of rank no larger than m.

For each n , an interpretation of L K n is a pair i = ( u , f ) where u is a non-empty set (the domain of i) and f is a function defined on K, as usual; I , denotes the class of all interpretations for LKn . The definitions of truth on an interpretation, logical consequence, logical truth, and equivalence of interpretations for members of I,, are assumed. For S a set of sentences and A a sentence S k A indicates that A is a logical consequence of S and kA indicates that A is logically true.

v and for all k in K,, f ( k ) = g ( k ) if k is an individual constant; and f ( k ) = g ( k ) I un, if k is an n-ary predicate constant. For i = ( u , f ) in I , and al , . . . , a , any non-empty sequence of members of u , let i ( a l , . . .,a,) = ( u , f ’ ) be that member of Il,l+m where f‘ and f agree on K, and f ’ ( ~ , + ~ ) = a1 for all 1 , 1 =< 1 =< m.

For i = ( u , f) and j = ( v , g ) in I,, i is isomorphic to j provided there is a 1 - 1 function h from u onto v such that for all k in K, , h(f ( k ) ) = g ( k ) , if k is an individual constant, and for all a ] , . . ., a, , in u , ( a l , . . ., anL) E f ( k ) iff ( h ( a l ) , . . ., h(a,,)) E g ( k ) , if k is a m-ary predicate constant.

For each i = ( u , f ) in I?T,,, F ( i ) denotes the substructure of i whose domain is the image of the individual constants of K, under f . (Since K,, contains no functional contents, F (i) is defined except for KO containing no individual constants.)

For all i = ( u , f ) , j = ( v , g ) in I,, i is a substructure of j provided u

For all I 2 0 the relation w t is defined as follows: (i) i w 0 j provided F (i) is iso- j provided for all a in u there is b in v such that ; (a ) w n,-l j ( b ) morphic to F ( j ) ; (ii) i V)

and for all b in v there is a in u such that j ( b ) M,,-~ i ( a ) .

256 GEORGE WEAVER

3.0. Note that for all i in I,,, i and F ( i ) agree on all sentences in LK,[O] and that for i , j in in, if i and j agree on LK,[O], then F ( i ) is isomorphic to F ( j ) . Hence i LO,, j iff i and j agree on all sentences in LKn[O]. Here this fact is generalized to show that for all 1 2 0, i LO^ j iff i and j' agree on all sentences in LK;,[l] and hence that i is equivalent to j (in LKn) iff i w 1 j for all 1 .

Lemma 1. For all i , j in I,, and all m , if i LO ,,I j , then i and j agree on all sentences in LKn[m].

The proof proceeds by induction on m noticing that for B = 3cxA in LKn[m], if B' is the result of replacing every free occurance of cx in A by c , , + ~ , then B' is in Lh-,z+,[m - 11.

The following is a trivial consequence of the above lemma:

Lemma 2. For all i, j in I,, if for at1 m, i mrn j, then i is equivalent to j . For all i = (u, f ) in I,, and A a formula having 1 free variables, T ( A , i ) denotes

the set of &tuples of elements of u which satisfy A . T ( A , i) is called the truth set of A on i . For all i in I , and all m , where m > 0 and 1 5 1 5 m and X a finite set of formulas each having m variables ( I of them free), X is called a finite partition of u1 with respect to i and m provided { T ( A , i): A E X} forms a partition of u1 and for all A, A' in X, if A =+ A', then T ( A , i) + T(A' , i).

Let cxl, . . .,a,,, . . . be any ordering of the variables of LA-*; for each i = (u, f) in I,, and for all m > 0, let A(m) denote the set of atomic formulas whose variables are among the first m in the above ordering; let T be the class of sets of formulas where for all X, S E T provided (i) S contains every member of A ( m ) or its negation, but nothing else; and (ii) T(&X, i ) is non-empty (where & X denotes the conjunction of the members of S) . Let umli, m = (&X: S E T I ; it is easily shown that um/i, m is a finite partition of urn w.r.t. i and m.

Lemma 3. For all m , 1 where 1 5 1 5 m, m > 0 and all i = (u , f ) in I,,, if there exists a finite partition of u' w.r.t. i and m , then there i s a finite partition of ul-l w.r.t. i and m.

Proof . Let i = ( u , f) be in In. Let A , , . . ., .4, be a finite partion of u1 w.r.t. i and m. Without loss of generality we may assume that a1 , . . . , a , exhaust the variables occuring in members of this partition and that al, . . . , cxz are free. Let T be the collection of all formulas of the forms 3cx1AJ or N 3cxlAj for all j , 1 j r . Let X be any subset of T such that (i) for all j, 1 5; j r , either 3011AJ or N 3nlAJ is in X; and (ii) T ( & S , i ) is non-empty. Let P denote the set of formulas of the form &S for S in T satisfying conditions (i) and (ii) above. It can easily be verified that P is a finite partion of ul-l w.r.t. i and m. Q.E.D.

The above proof gives a functional process for constructing a finite partition of uZ-l w.r.t. i and m from a finite partition of u1 w.r.t. i and m. Thus, um-l/i, m denotes the result of applying this process to um/i, m ; similarly for all I , 1 =( 1 5 m , the finite partition of uz w.r.t. i and m constructed from umli, m by iterating this process is denoted by dli, m .

For each m , 1 and i = ( u , f) in I,, , the members of u'li, m are called the generators of u7/i , m. Note that for i = ( u , f) in I,, and all 1 , m there is a sentence in Lh-Jm]

FINITE PARTITIONS AND THEIR GENERATORS 257

which says that ulli, m is a partition of ul. Let P (uz/i, m) indicate this sentence. Finally, let P(u/ i , m) = &{P(ul/i, m): for all I, 1 4 1 m}. Note that P(u/ i , m) is a member of LK,,[m]. Further, it can be verified that for all i = ( u , f ) , j = (v, g) in I , and all m , 1 where m > 0 and 1 5 1 s m , P(uz/i , m) is true on j iff ul/i, m = vl/j, m.

Lemma 4. For m , and all i = ( u , f ) , j = (v,g) in I,, where m 2 1 and all I , 1 1 =< m , if P(ul/ i ,m) is true on j , then for all A in ul/ i ,m and a l , . . . ,al E U and b l , . . . , bl E v, then if (a, , . . . , a [ ) E T ( A , i) and (b,, . . . , b,) E T ( A , j ) , i(a1, * * * , a , ) M m - l j ( b l , . . * , bz).

Proof. Let m 2 1 . Let 1 = m and A E um/i ,m. Then A EvU’n/j,m. Let ( a 1 , . . . , a,) E T ( A , i ) and (bl , . . ., b,) E T ( A , j ) . We can easily verify that F ( i ( a l , . . .,a,)) is isomorphic to F(j(b,, . . ., b,)) and hence that i ( a l , . . .,a,) mo

-oi(b1, * * .,b,). Let 1 < m and suppose that for A E ul/i, m and a,, . . . , a, E u and b l , . . ., bl E v if

( a l , . . . , a,) E T ( A , i) and (bl , . . . , b,) E T ( A , j ) then i ( a l , . . . , al) mm-l j ( b l , . . . , bl). Let A’ E ul-i/i, m and let a l , . . ., al-l E u , bl, . . ., b Z - ] E v and assume that ( a l , . . ., ai-l) E T(A’ , i ) and ( b , , . . ., bl-l) E T(A‘ , j ) . Let a be any element of u; then there exists A in u’li, m such that ( a l , . . . , a l - l , a ) E T ( A , i), and hence 3a1A is a conjunct of A‘. Since uz/i , m = vllj, m , there exists b E v such that (b, , . . . , bz -, , b) E

e T ( A , j ) and byhypothesisj(bl, ..., b,-,,b) = j ( b , ,..., b i l l ) @ ) mm-ti(al ,..., al-,)(a) = i (a, , . . . , a l - ] , a ) . Thus, for all a E u there exists b E v such that i(al, . . . , a l - l ) (a) m(, ,L-( , - l ) ) - l j ( b l , . . ., bl...l) (b). In the same way it can be shown that for all b E v there exists a E u such that i ( a l , . . ., at-1) (a) m(,-(l-l))-l j ( b l , . . ., bt-,) (b); and hence that i(a,, . . ., al-1) mrn-(~-1)j(b~, . . ., bt-1).

It follows from this lemma that for all m 2 1 and all i = (u, f ) , j = ( v , g ) in I , , if P (uli, m) is true on j , then i m ,,, j : hence there are only finitely many non - m , interpretations for each m ; and by lemma 1, for a11 A such that r ( A ) =< m either P(u/ i , m) k A or P(u/ i , m) I= - A . Further, the following is a trivial consequence of lemmas 2 and 4.

Corol lary 1. For all i = ( u , f ) , j = (v, g ) in I,,, the following are equivalent: (1) i i s equivalent to j ; (2) for all m > 0 , i ~ , j ; (3) for all m > o and 1, 1 =< I s m , u‘li, m = dli, m; (4) for all m > 0 , P(uli, m) i s true on j.

4.0. A theory T in LK, is any set of sentences in LKa closed by logical consequence; T is axiomutizable provided every member of T is a logical consequence of a subset of T which is recursive relative to the sentences in LKn. For X any set of sentences Cn(S) denotes the result of closing S by logical consequence and M ( X ) the models of S. For T a theory and any m > 0 , let P(T , m) = v {P(u/ i , m ) : i = ( u , f ) E M ( T ) } and P(T , w ) = {P(T, m ) : m > 01.

Note. M ( S ) = M ( C n ( 8 ) ) ; if T is complete and hatj models, P ( T , m) = P(u/ i , m) for i = ( u , f ) a model of T ; P ( T , m) E T; for all sentences A , A E T provided P(T , m) / = A for all m 2 r ( A ) ; M ( P ( T , w ) ) = M ( T ) .

Thus, every theory T having models can be viewed as the union of a nested sequence, T 1 , . . ., T , , . . . of “subtheories” of T , where for all I , T 1 = T n L K J I ] ;

17 Ztschr. f. math. Logik

258 GEORGE WEAVER

Tl is closed by logical consequence in Lh;,[l]; T t is axiomatized by P(T, I). This fact is used to establish the following:

Theorem 1. For any T having models (i) T i s finitely axiomatizable iff there exists a least m such that for all m' > m , P(T , m) k P ( T , m'); and (ii) if T i s finitely axiomutizable, then there exists a least m such that (a) P ( T , m ) finitely axiomatixes T ; and (b) for all 1 < m , no sentence in LK[Z] finitely axiomatizes T .

Theorem 1 yields a characterization of E. C. classes not relying on compactness. Let Z be a class of interpretations for L,, 2 is an E . C . class provided there is a sentence A such that M ( A ) = 2. It can easily be verified that Z is an E. C. class iff there exists a least m such that 2 is closed by w m .

The following definitions prove useful in extending the above theorems to axiomatizable theories. Let S be a finite set of formulas. S is of the form u"'/ i, m provided S = {L41, . . . , A,} where for all j , 1 s j 5 r Aj = *B, & . . . & *Bk, where *B, = B, or *B, = -Bs , for all s , 1 s s 5 k , and A ( m ) = {B,, . . ., Bk}; for all 1 , 1 5 1 5 m , S i s of the form uz/ i , m provided S = { A , , . . . , A,} where for each j , 1 5 j 5 r , Aj = *3a,-lBl & 1 - * & *3atn-lBk, where * ~ L Y , , ~ - ~ B ~ = 3~x,-~B, or *3a,,,-lBs = = - ~ K , , ~ - ~ B ~ ; and { B l , . . ., Bk} is of the form u'+'/ i, m . For all m and 1 , 1 5 1 5 m , it is decidable whether or not S is of the form d / i , m.

Let A be a sentence. A i s of the form P(uz/ i , m) provided ,4 is the conjunction of the following:

(1) V a l . . . VaL(B1 v . * . v B,); (2) 30cl . , . 3alB1 & 3oL1 . . . 3611B2 & ' . . & 3011 . . . iIalB,; (3) V a l . . . Val((B1 3 - ( ( B , v . . . v B , ) ) & . . . $ ( B , ~ - ( B , v . * . v B , - , ) ) ) ,

where B,, . . . , B, is of the form u' / i , m. A is of the form P(u/ i , m) provided A = (B, & * * & Brn) where for all k, 1 5 k na, Bk is of the form P(uk/ i , m) . Fi- nally, A i s of the form P ( T , m) provided A = (B, v . e v B,) where for all k , 1 5 k s r , B,, is of the form P(u/ i , m) . For all in, I the following are decidable: whether or not a sentence is of the form P(ul/i , m ) , of the form P(u/ i , m ) , or of the form

Let T be a theory having models, then for each m there exists finitely many sentences of the form P ( T , m) in T ; and of these P ( T , m) is the shortest (i.e., all other sentences of the form P(T , m) contain the disjuncts of P(T , m ) ) .

One might expect that if T is a theory having models and T is axiomatizable, then P(T , w) is recursive. The following, however, shows that this is not the case.

Lemma 5. For all T , a theory having models, T is decidable iff P ( T , co) i s recursive. Proof. Suppose P ( T , w ) is recursive. The argument proceeds by noticing (i) that

for all A and all i = (u, f ) , either k ( P ( u / i , r ( A ) ) 3 A ) or k(P(u/ i , r ( A ) ) 3 - A ) ; (ii) that A E T provided for all i = (u, f ) E M ( T ) , (P(u / i , r ( A ) ) 3 A ) is logically true. Hence, since the logically true sentences are recursively enumerable, and P(T , w) is recursive, T is decidable.

Suppose T is decidable. The argument proceeds by noting for all A , A E P ( T , w ) provided A is the shortest sentence of the form P ( T , r ( A ) ) in T. Hence, P(T,w) is recursive.

P ( T , m).

FINITE PARTITIONS AND THEIR GENERATORS 25 9

Note that if P ( T , o) is recursively enumerable, then P ( T , w ) is recursive and the above yields the following:

Theorem 2. Por all T a theory having models, T is decidable iff P ( T , o) is recursively enumerable.

5.0. Let S be a set of sentences. The compactness theorem maintains that for every sentence A such that S k A there exists a finite subset S' of S such that S' k A . Thus, with each logical consequence of X we can associate some finite subset of 8; but we know that, in general, there is no way of corresponding the same finite subset with each logical consequence. However, the following theorem implies that this corre- spondence can be made in such a way that every logical consequence of S of the same complexity (measured by the number of distinct variables) can be associated with a finite subset of 8 which implies all of them; and further, if A and B are logical con- sequences of S and A is more complicated than B (again in terms of the number of distinct variables) then the finite subset of X associated with B is a subset of that associated with A .

Theorem 3 (Uniform Compactness Theorem). For S a set of sentences there exists S1, . . . , Sl , , . . and infinite sequence of nested finite subsets of S such that for all A if S t= A then S , t= A , for all m 2 r ( A ) .

Proof. Let X be any set of sentences. S k P(Cn(S) ,m) for all m ; and for all A if S k A , then P(Cn(S) , r ( A ) ) k A . By compactness, for each rn there exists a finite subset S;, of 8 such that S;, k P(Cn ( S ) , m) . Let S , = Si and S,, = S,,, -1 u SA, . The sequence S1, . . ., X,, . . . satisfies the theorem.

6.0. For A and B sentences, if A k B and A and B share a predicate constant in common, the interpolation lemma guarantees that there is a sentence C such that A k C, C i= B , and C contains exactly those non-logical constants common to both A and B. Here this result is strengthened to show that C can be chosen in such a way that for all B' such that A t= B', where the number of distinct variables in B' is no greater than the number of those in B and the constants common to A and B' are precisely those common to A and B, C k B'. I n particular, the choice of C is shown to be a function of A , K' and m where K' t K,, and m 2 r ( A ) .

For all A in LK*, let K,,(A) denote the members of K,, occuring in A ; and for all K' K,,, let denote the natural restriction of the interpretations of LKn to interpretations of LK,. Then for all A , all K' s K, and all m if A has models, P ( A , K', m) = v ( P ( ~ / @ ~ . ( i ) , m ) : i = ( u , f ) E M ( A ) ) ; and if A has no models P(A , K', m) is any contradiction in Lw[m]. Note that K,c(P(A, K' , m ) ) = K', r ( P ( A , K', m ) ) = m , and A k P ( A , K' , m ) . The following can easily be verified.

Lemma 6. For a12 A in LIc,, (i) t= (P(A, K,, , r(A)) = A ) ; (ii) k ( A = P ( A , K,,(A), r ( A ) ) ) . Lemma7. ForallsenteneesA, Bandallm 2 r ( B ) , i f A t= B,thenA t= P ( A , K,,(A),m)

and P ( A , K,(A), m) t= B . Proof. Suppose A t= B and let m be 2 r ( B ) . Then B is true on all models of

P ( A , K,(A), r ( A ) ) as well as those interpretations m,,* to such models, but these are exactly the models of P ( A , K,(A), m ) .

Theorem 4 (Uniform I n t e r p o l a t i o n Lemma). Por every A in LKn all K' E hrn and all m , if m 2 r ( A ) , then (i) A k P(A, K', m) ; and (ii) for all B if K,,(A) n K, ( B ) = K', r ( B ) 5 m and A t= B , then P ( A , K', m) t= B .

17*

260 OEORQE WEAVER

Proof. Let A be in L K n and let K’ E K , and m 2 r ( A ) , suppose B is such that K,,(B) n K, (A) = K’, m 2 r ( B ) and A C B , obviously A t= P ( A , K‘, m) .

Suppose A has models. Let i’ = (u’, f ’ ) be any model of P ( A , K’, m ) , then there is a i = (u, f > E M ( A ) such that GK9(i) mIn Q K r ( i ‘ ) . It suffices to show that there exists i = (v, 9) such that (a) @ K ~ ( A ) ( ~ ) m m @ K , ( B ) ( ~ ) and ( b ) @ ) K - , J B ) ( ~ ’ ) m m @ K , , < B ) ( ~ ) ; since from (a) it follows that i E M ( A ) , hence j E M (B) and from (6) it follows that i’ E M ( B ) . It suffices to show that { P ( u ’ / @ ~ , ( ~ ) ( ~ ’ ) , m), P ( u / @ ~ , ( ~ ) ( ~ ) , m ) } has a model. Suppose otherwise. Then, P ( ~ / @ ~ = ( * ) ( i ) , m) t= - P ( u ’ I @ ~ , ~ ~ ) (i’) , m) ; by the interpolation lemma there exists C such that K,(C) = K’, P ( u / @ ~ * ( ~ ) ( ~ ) , m) t= C, C k - P ( U ’ / @ ~ % ( ~ ) ( ~ ) , m ) and, by lemma 7, P ( u / @ ~ % ( ~ ) ( ~ ) , m) k P(C, K, , (C) , m) t= I. - P (u’/@K,(B)(i’), m) . Since P (C, K,, (C) , m) is a sentence in LKt [m] either P ( u ’ / @ ~ , ( ~ ’ ) , m) = P ( U / @ ~ , ( ~ ) , m) k P(C, K,,(C), m) I= - P (C, K,, (C) , m) . But this is impossible ; since if P ( u ’ / @ ~ ( i ’ ) , m) t= P (C, K,, (C) , m) then {P(u’/@jK,(i’), m ) , P ( U ’ / @ ~ , ( ~ , ( ~ ’ ) , m)) has no models; and if P ( U ‘ [ @ ~ , ( ~ ‘ ) , m) I= k -P(C , K, , (C) , m ) , then { P ( ~ / @ ~ % ( + 4 ) ( i ) , m ) , P(u/@K,(i), m ) } has no models.

Acknowledgments . It is a pleasure to acknowledge interesting and stimulating discussion of an earlier version of this paper with my colleague JOHN CORCORAN (S.U. N. Y. Buffalo) who suggested numerous improvements in the text. I n addition, I would also like to thank the members of my graduate seminar a t Bryn Mawr for various kinds of contributions.

or P ( U ’ / @ ~ ~ ~ ( ~ ’ ) , m) t=

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Finite Partitions and Their Generators