Zeitschr. f math, Logik und Grundlagen d Math Bd 36. S. 381 -383 (1 990)

EXTENDING w-CONSISTENT SETS TO MAXIMALLY CONSISTENT, IN-COMPLETE SETS

by GEORGE WEAVER in Bryn Mawr, Pennsylvania MICHAEL THAU and HUGUES LEBLANC in Philadelphia, Pennsylvania (U.S.A.) ’)

Given L, a first order language, HENKIN’S completeness proof for L proceeds by showing that every consistent set of sentences in L can be extended to a maximally consistent and w-complete set. While every consistent set in L can be extended to a maximally consistent set in L, there are consistent sets which cannot be extended to a maximally consistent and w-complete set in L. In such cases, the extension is accomplished by introducing infinitely many new individual constants into the non-logical vocabulary of L. Here we show that even when L contains infinitely many individual constants there are w-consistent sets in L which cannot be extended to maximally consistent and w-complete sets in L.

A set S of sentences in L is w-consistent iff for all formulas p(x) in L: if 3xp(x) is provable from S, then for some constant term t, - p(t) is not provable from S; S is w-complete iff for all formulas p(x) in L: if p(t) is provable from S for all constant terms t, then Vxp(x) is provable from S; S is maximally consistent iff S is consistent and for every sentence p in L: p or -p is in S. Recall that if S is maximally consistent, then S is w-complete iff for all formu- las p(x) in L: if 3xp(x> is in S, then for some constant term t, p(t) is in S.

Let L be a first order language whose non-logical vocabulary contains two one-place predi- cates PI, Pz, and infinitely many individual constants. The non-logical vocabulary may con- tain other predicates, and functional constants. When the non-logical vocabulary contains functional constants, there are constant terms which are not individual constants.

Let T be the union of the following sets of sentences:

{(Pl(t) 3 -P2(t’)): for all constant terms t, t’};

{(P,(t) =) 3xP2(x)): for every constant term t } ;

P X P l ( X ) I .

Proposit ion 1. T is w-consistent, but no maximally consistent and w-complete subset of L con-

Proof. To show that T is w-consistent it suffices to show that for each sentence of the form 3xp(x) which is provable from T there is a model of T and an individual constant t such that p(t) is true on this model.

tains T.

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l) An earlier draft of this paper contained an example of an w-consistent set in L which cannot be ex- tended to a maximally consistent, w-complete set in L. That example used a language whose non-logical vocabulary contained infinitely-many one-place predicates. The authors wish to acknowledge the assist- ance of ELLIOT MENDELSON (CUNY) who noticed that this example could be modified so as to involve only two one-place predicates.

3 82 G. WEAVER, M. THAU UND H. LEBLANC

Let U be a model of T with the following properties: (1) -P&) is true on U, for all constant terms t ; (2) -Pz( t ) is true on U, for all constant terms t ; ( 3 ) 3xPz(x) is true on 3; (4) the extensions of Pl and Pz in J are disjoint. Notice that Thas models satisfying these conditions. For example, let the domain of U be the non-negative integers; let all individual constants denote 1, let {0} be the extension of P, , let {2} be the extension of Pz, and let each n-ary functional constant have as its extension that n-ary function on the domain of U having constant value 1. Hence, all constant terms de- note 1. Let A l denote the extension of PI in U and A z the extension of Pz in U; XI denotes the complement (relative to the domain of U) of A 1 , and x2 denotes the complement (relative to the domain of U) of A z . Suppose that 3 x p ( x ) is provable from T. Let B be the set of objects from the domain of U which satisfy Q,(x) in U. By supposition, B is non-empty. Either B n xl is empty or B n XI is non-empty.

Suppose that B n xl is non-empty. Let a be any member of B n xl. Let t o be any individ- ual constant not occurring in p ( x ) . Let W be such that ill and W have the same domain, U and 8’ give the same extensions to all predicates and functional constants and give the same de- notations to all the individual constants except to, and t O denotes a in U’. Thus, U and W give the same denotations to all constant terms except t , and p(to) is true on U’. Further, since Pl(t) is false on W for all t , it is easily verified that W is a model of T. Thus, -p(to) is not provable from T.

Suppose that B n xl is empty. Thus, B is a subset of A l , and by condition (4), B is a subset of xz. Let a be any member of B and let to be any individual constant not occurring in p ( x ) . Let U’ be as above, except that a is now the denotation of to in W. U’ is a model of p(to). Since -Pz(tO) is true on W, U’ is a model of T. Thus, -p(to) is not provable from T.

Hence, in either case -p(to) is not provable from T, and, as 3 x p ( x ) was arbitrarily chosen, T is w-consistent.

Suppose that T‘ is a maximally consistent, w-complete subset of L and that Tis a subset of T’. Then 3xP, (x ) is in T’, and, as T‘ is w-complete and maximally consistent Pl ( t ) is a mem- ber of T’ for some t . But then -Pz(t’) is in T’ for all constant terms r’ and 3xPz(x ) is in T’. Therefore, as T‘ is maximally consistent, T’ is not w-complete. Contradition. Hence, T can- not be extended to an w-complete, maximally consistent set in L.

Notice that the above holds for first order logics with or without equality. Call S , a set of sentences in L, infinitely extendible iff there are infinitely many individual contants in L not occurring in any member of S . LEBLANC [l] has shown that every infinitely extendible, con- sistent set in L can be extended to a maximally consistent, w-complete set in L. Thus, analo- gous results hold for infinitely extendible, w-consistent sets. Notice that there are w-consist- ent sets which are not infinitely extendible but which can be extended to maximally consistent, w-complete sets in L (e.g., the set of logical truths).

It is easily shown that if a set S is w-consistent, then for every sentence Q, either Su {p} or S u { - p} is w-consistent. To understand this suppose that S is w-consistent, but that neither S u {p} nor S u { - -Q,} is w-consistent. Thus, there are formulas y ( x ) and y ( x ) such that (1) 3 x y ( x ) is provable from S u {q } ;

( 2 ) 3 x y ( x ) is provable from S u (-9);

EXTENLXNG LO-CONSISTENT SETS TO MAXIMALLY CONSISTENT, WCOMPLETE SETS 383

(3) - y ( t ) is provable from S u {p}, for all 1 ;

(4) - y ( t ) is provable from S u {-p}, for all t . It is then easily verified that 3x(y(x) & y ( x ) ) is provable from S, and that - ( y ( t ) & y ( t ) ) is provable from S, for all t . Thus, S is w-inconsistent. Contradiction.

Notice that it follows from proposition 1 that w-consistent sets are not closed under unions of chains, and that there are w-consistent sets which cannot be extended to maximally w-consistent sets. Finally, it follows from proposition 1 and the completeness theorem for w-logic that there are sets which are w-consistent in standard logic, but not consistent in w-logic.

Reference

[l] LEBLANC, H., A simplified strong completeness theorem for QC= . In: Akten des XIV. Intematio- nalen Kongress fur Philosophie 3 (1968), pp. 83-95.

G. Weaver Bryn Mawr College Department of Philosophy Bryn Mawr, PA 19010-2899 U.S.A. H. Leblanc and M. Thau Temple University Department of Philosophy Philadelphia, PA 19122 U.S.A.

(Eingegangen am 4. Mai 1989, uberarbeitete Fassung vom 10. September 1989)