Note on a Problem of L. HenkinAuthor(s): Abraham RobinsonSource: The Journal of Symbolic Logic, Vol. 21, No. 1 (Mar., 1956), pp. 33-35Published by: Association for Symbolic LogicStable URL: http://www.jstor.org/stable/2268483 .

Accessed: 10/06/2014 09:00

Your use of the JSTOR archive indicates your acceptance of the Terms & Conditions of Use, available at .http://www.jstor.org/page/info/about/policies/terms.jsp

.JSTOR is a not-for-profit service that helps scholars, researchers, and students discover, use, and build upon a wide range ofcontent in a trusted digital archive. We use information technology and tools to increase productivity and facilitate new formsof scholarship. For more information about JSTOR, please contact support@jstor.org.

.

Association for Symbolic Logic is collaborating with JSTOR to digitize, preserve and extend access to TheJournal of Symbolic Logic.

http://www.jstor.org

This content downloaded from 195.34.78.42 on Tue, 10 Jun 2014 09:00:39 AMAll use subject to JSTOR Terms and Conditions

Tima JOURNAL OP SYMBOLIC LoC-C

Volume 21, Number 1, March 1956

NOTE ON A PROBLEM OF L. HENKIN

ABRAHAM ROBINSON

1. A statement X in the lower predicate calculus is said to be persistent with respect to the set of statements K ([2], [3]), if whenever X holds in a model M of K then X holds also in all extensions of M which are models of K. If X is persistent with respect to the empty set, then it may be said to be absolutely persistent.'

A statement X is called existential, if it is in prenex normal form and does not contain any universal quantifiers. This includes the possibility that X does not contain any quantifiers at all.

Let E be the class of all existential statements. Then it is not difficult to see that E is quasi-disjunctive. That is. to say, given statements Y,, Y2 in E, there exists a statement Y in E such that

(1) Y1 VY2 Y

is provable. L. Henkin [1] has raised the question how to characterize the statements

X which are persistent with respect to a given set K (e.g. a set of axioms for a field or a group) by a syntactical condition. He has shown that, in order that a statement X be absolutely persistent, it is necessary and sufficient that there exist a statement Y E E such that

(2) X Y

is provable. In the present note we shall prove the more general:

THEOREM. In order that the statement X be persistent with respect to the set K it is necessary and sufficient that (2) be deducible from K, for some existential statement Y.

This theorem includes Henkin's result, which was obtained by him by the use of some related work of Tarski [4] on universal classes. The theorem of the present paper can also be reformulated as a result on universal statements.

It may be pointed out that a syntactical characterization of persistence relative to a set K must by necessity include a reference to K, since every statement X which is deducible from K is persistent with respect to K without any restriction on its syntactical form.

Received April 4, 1955. 1 For the sake of convenience, we have modified the definition of a persistent

statement given in [3] by dropping the condition that X must be consistent with K. It is not difficult to restate the main theorem if the original definition is retained in its entirety.

33

This content downloaded from 195.34.78.42 on Tue, 10 Jun 2014 09:00:39 AMAll use subject to JSTOR Terms and Conditions

34 ABRAHAM ROBINSON

2. It is not difficult to see that the condition of the theorem is sufficient. To prove that it is also necessary, let X be persistent with respect to K. Put H = K u {X}. If H is contradictory, then OX must be deducible from K (where K may be either consistent or contradictory). In that case, let Y - (3x)[R(x) A R(x)] where R(x) is an arbitrary relation of one variable. Then Ye E, and X - Y is deducible from K- since Y is con- tradictory- and so the condition of the theorem is satisfied.

Suppose next that H is consistent. A statement Z is defined in H, if all the relations and individual constants of Z occur in statements of H. Let F be the set of existential statements Z which are defined in H such that

(3) ZDX

is deducible from K. Then F is a subset of E. Also, whenever the statements Z1 D X and Z2 2 X are deducible from K, Z1 V Z2 2 X is deducible from K as well. From these facts we conclude without difficulty that F is quasi- disjunctive.

Let G be the set of negations of elements of F, -Z e G whenever Z E F, and consider the set of statements J = H u G. Suppose first that J is consistent. Then J possesses a model M. Let N be the (complete) diagram of M, that is to say, the set of all atomic statements R(a,, . . ., aJ) whenever the relation in question holds in M, or of ,R(al, ..., aJ) whenever the atomic R(a,, ..., a") is defined in M but does not hold in it. Since X is persistent, it holds in all extensions of M which are models of K. That is to say, X holds in all models of K u N, and hence is deducible from K u N. Now if X is deducible from K alone, then we may satisfy (2) by choosing as Y any provable existential statement, e.g. (3x)[R(x) V R(x)]. Suppose then that X is deducible from K u N although it is not deducible from K. Then there exist statements Z1, .. ., Zn in N, n > 1, such that

(4) Z1A ... AZDX

is deducible from K. We put

Z, A...A Z, =Z(al, ...,ak)

where we distinguish the constants a1, ..., ak which do not occur in the statements of H. The application of a familiar rule. of the predicate calculus to (4) shows that

[(3x1) ... (3Xk)Z(Xl, . X. Xk)] D X

is deducible from K. Thus, the statement

V = (3x1) ... (3Xk)Z(Xl, ..., Xk)

belongs to F, TV belongs to G, V holds in M, which is a model of G. On the other hand, the statements Z1, .. *, Z. belong to N, and hence hold in M. This entails that Z(al, ..., ak), and with it V, also holds in M. Thus

This content downloaded from 195.34.78.42 on Tue, 10 Jun 2014 09:00:39 AMAll use subject to JSTOR Terms and Conditions

NOTE ON A PROBLEM OF L. HENKIN 35

the assumption that J is consistent leads to a contradiction; J is contra- dictory. Since H is consistent, it follows that there exist statements Y1* * ., Yn e F (i.e. Al.'YE, . ., AYE G), n > 1, such that the set

K u { .YE . . * L zyn X}

is contradictory. This entails that the statement

(5) [,Y, A ... A -Y. VX]

is deducible from K. Applying one of de Morgan's rules to (5), we conclude that

XZD Y1V ... V Yn

is deducible from K. Now F is quasi-disjunctive, so there exists a statement Y e F such that

Y, A ... A Y,, Y is provable. Hence the statement

XD Y

is deducible from K. But Y D X is deducible from K by the defining property of F, and so (2) is deducible from K. This proves the theorem.

In the above proof, it was taken for granted that the formal language involved is sufficiently rich to ensure the existence of a model of J provided that set is consistent. This was permissible, only because, for the definition of F, we considered only statements which are defined in H. Without some such restriction, it is quite possible that a class of statements within a given language be consistent without, however, possessing a model within the language.

REFERENCES

[1] L. HENKIN, Two concepts from the theory of models, this JOURNAL, Vol. 21 (1956), pp. 28-32.

[2] A. ROBINSON, On the metamathematics of algebra, Studies in logic and the foundations of mathematics, Amsterdam, North Holland Pub. Co., 1951, IX + 195 pp.

[3] A. ROBINSON, On axiomatic systems which possess finite models, Methodos, 1951, pp. 140-149.

[4] A. TARSKI, Contributions to the theory of models I, II, Indagationes Mathe- maticae, vol. 16 (1954), pp. 572-581, 582-588.

UNIVERSITY OF TORONTO

This content downloaded from 195.34.78.42 on Tue, 10 Jun 2014 09:00:39 AMAll use subject to JSTOR Terms and Conditions