Zdtscht. f . math. Loflik und Gncndluuen d . Bid. Bd. 26, S . 111 - 113 (1980)

A NOTE ON THE COMPACTNESS THEOREM IN FIRST ORDER LOGlC

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

The compactness theorem is generally regarded (e.g. VAUGHT [S]) as the most fun- damental result in first order model theory. There are several equivalent formulations of this theorem. I n the’ following, we are concerned with two of them: (1) if evcry finite subset of a set of sentences has a model, then the set itself has a model; and ( 2 ) if the sentence A is a logical conscquence of the set of sentences S, then some finite subset of &has A among its logical consequences. Several authors (e.g. [l], [2], [5 ] , [S]) refer t o (1) ’8s the compactness theorem. However this practice is not universal; MAL’CEV [4] and ROBINSON [6] refer to (1) as the localization theorem reserving “com- pactness’’ for more topologically flavored equivalents; other authors (e.g. SHOEN- FIELD [7]) refer to ( 2 ) as the compactness theorem; yet another terminology is found in [3]. To distinguish between (1) and ( 2 ) above, we refer to the first as the satisfiability formulation. and to the second as the consequence formul a t’ ion.

Intuitively, the consequence formulation guarantees, when. A is A logical conse- quence of S, that only some of the infornietion conveyed or cxprcssed by S is relevant to the trut,h of A , and that this relevant information is expressed by some finite sub- set of S. Given this intuition, i t seems reasonable to ask what of thc infvrmation ex- pressed by &’ is relevant to t,hc truth of A and which of the finite subsets of S express this information. However, none of the arguments in the literature provide answers t,o either of these questions. Typically, these arguments follow onc of bwo patt.erns: (A) the satisfiability formulation is established, and observing that A is a logical consequence of S provided S and the contradictory of A have no models, the argument proceeds indirectly ; and (B) t.he strong completeness and strong soundness theorems are estab- lished and the argument proceeds by noting that proofs are finite. Here an argument for the consequence formulation is given which attempts to answer t,he questions raised above ; like (A) above, this argument uses the satisfiability formulation, but proceeds directly to construct the finite subset, of S .

The argument proceeds by extending notions developed in [9J. There, i t was shown for first order languages over a finite set of non-logical constants (excluding functional constants) that the appropriate subset of S can be chosen as a function, not of A , but of the number of distinct variables occurring in A ; more precisely, for each natural number n, there exists 8‘ a finite subset of S among whose logical consequences are all those consequences of S having n or fewer distinct variables. Fambarity with [9] is presumed and the major definitions are not repeated in detail.

Let K be any set of non-logical const.ants, excluding functional const,ants. Lct LK be the first order logic with equality over K . The notions of formula and sentence in LK are defined as in [9] (p. 235). For each sentence A , r(A) (the rapzk of A ) is the num- ber of distinct variables occurring in A. For each m 2 0, L,[v] ,denotes those sentences in LK of rank 4 m.

112 QEORQE WEAVER

Interpretations for Llc are ordered pairs i = (u, f ) where u is a non-empty set (the domain of i ) and f is a function defined on K as usual. The definitions of truth on a n interpretation, model, logical consequence, subinterpetations, isomorphism and equivalence of interpretations are assumed. For S a set of sentences, M ( S ) denotes the models of S; S t= A indicates that A is a logical consequence of S; and N ( A ) denotes B ( ( A } ) . Given A a sentence, K ( A ) denotes the set of members of K occurring in A. Finally, when no confusion results, we let L denote LA and L[m] denote LK[wb].

Given finite K , for each natural number 15 0 the relation w1 is defined between pairs of interpretations as in [9] (p. 255). I n [9] (p. 257) it was shown for all inter- pretations i, j that i w t j provided i and j agree on L[E]; that there is a sentence in L[2] (denoted P(u/i, 2)) such that j is a model of this sentence provided i w 1 j ; and finally, that there are only finitely many non--l interpretations for each 1 2 0.

K , i r K’ denotes the reduct of i to an interpretation for LIcj. Given S , A , K’, and n, where K’ is a finite subset of K , let P(S, K’, n) denote the disjunction of all sentences P(u/i i’ K’, n) where i is a model of S, if S has models; otherwise some sentence in L*.[n] having no models. Notice that i is a model of P(S , K ‘ , n) provided there is j a model of S such that i I’ K’ -,& j r K’. Let S, S’ be two sets of sentences; we say that S and S’ are synonymous with respect to K’ and n provided P(S’, K’, n) and P(S , K’, n) are equivalent (i.e. M ( P ( S , K , n)) = = X(P(S’, K’, n))) . It can easily be verified that when S and S’ are synonymous with respect to K‘ and n, then S and S‘ have the same consequences in Lhl[n]. Further, i t is easily seen that the relation of synonymy with respect to K‘ and n is an equivalence relation; and that the partition induced by this relation on the power set of LK has only finitely many cells. For, since there are only finitely many interpretations of L,,. which are not w I , and each of these corresponds to a sentence of the form P ( u / i r K‘, n ) , there are only finitely many non-equivalent sentences of the form P(S, K’, n).

Given any K , i an intcrpretation for Li, and K’

Lemma 1. Given K , K’, n where K’ is a finite subset of K and given S a set of sen- tences in L,, , there exists S’ a finite subset of S such that for i a model of S’ and S” any finite subset of S there is j a model of S” such that i r K‘ wI1 j r K’.

Proof, Consider the relation of synonymy with respect to K’ and n restrictcd to the finite subsets of S. Let A be the partition induccd on the finite subsets of X by this relation. By our previous remarks, A is finite. Let A* be the result of picking one representative from each cell in A . Let S‘ be the union of A*. S‘ is finite. Let i be any model of S’ and let S” be any finite subset of S; by construction there is S” ’ a finite subset of S such that S“ ‘ is synonymous with S” with respect to K’ and n and S” ‘ f S’. Thus i is a model of P(S”, K’, n) and there is j a model of S“ such that i r K’ j I’ K’.

L e m m a 2. Given R, K’, n where K‘ i s a finite subset of K and given S a subset of LIL arid 1 an interpretation {or L,, , if tvcry finate subset of S has a model j such that i r A‘ -,) j r X’ then S has a model j ’ such that i r K’ wI1 j’ P K‘.

Proof. Under the hypothesis of this lemma, every finite subset of Sw(P(u / i r I<’, )I , )}

hah a model, thus by the satisfiability formulation, there is j ‘ a model of S such that

Theorem 1. [Uniform Compactness Theorem : Consequence Formulation.] Given K , I<’, S and n, where K’ is a finite subset of K and S is a set of sentences in Lit, there

j’ r K‘ i r K ’ .

A NOTE ON TEE COMPACTNESS THEOREM M FIRST ORDER LOGIC 113

exists s" a finite subset of S which is synonymous to S with respect to K' and 11, and hence, for all A in L K , if r (A) 5 n and K ( A ) 5 K', then X i= A iff 8' k A.

Proof. Given K', n and S, construct S' by lemma 1. By lemma 2, S' is synonymous with S with respect to K' and n.

Thus, the above argument shows that when A is a logical consequence of S , only the information about K ( A ) expressible by sentences of rank r (A) or less is relevant to the truth of A .

A crucial point in the above is that A* be non-empty. When K is countable, this follov,s without the use of the axiom of choice; and thus our proof of the consequence fonnulation is no stronger than the usual proofs for countable languages. Yurther, cvcn though for uncountable K our proof requires the axiom of choice, similar remarks apply herc ; for the satisfiability formulation for uncountable languages requires the axiom of choice. Even in the countable case, our construction is not effective; for the existence of an effective construction would imply the decidability of elcrnentary arithmetic.

References

[l] I~ELL, J. L., and A. €3. SLOMSON, Models and Ultraproducts. North-Holland, Amsterdam 1969. [2] C I I ~ ~ G , C. C., and H. J. BIISLER, Model Theory. North-Holland, $msterdam 1973. [3] HERMES, H., Introduction to Mathematical Logic. Springer-Verlag, New York 1973. [4] XAL'CEV, A. I., A general method for obtaining local theorems in group theory. The Neta-

mathematics of Algebraic Systems (B. F. WELLS 111, editor and translater), North-Holland, Amsterda.m 1971.

[5 ] MOSTOWSKI, A., Thirty Years of Foundational Studies. Barnes and Noble, New York 1966. [6] R,OXINSON, A., Introduction to Model Theory and the Metamathematics of Algebra. North-

[7] SHOESFIELD, J. R., Mathematical Logic. Addison-Wesley, Reading 1967. [8] VXXHT, R. L;, Model Theory Before 1945. Proc. of the Tarski-Symposium (L. HEARIS e t al.,

[91 WEAVER, G., Finite partitions and their generators. This Zeitschr. 20 (1970), 255- 260.

Holland, Amsterdam 1963.

editors), American Mathematical Society, Providence 1974, pp. 153 - 172.

(Eingegangen am 30. M&rz 1978)

8 Zkchr. f. math. Logik