Actes, Congrès intern, math., 1970. Tome 1, p. 245 à 250.
FORCING IN MODEL THEORY (')
by
ABRAHAM
ROBINSON
1. Introduction.
The forcing concept of Paul J. Cohen has had an immense effect on the development of Axiomatic Set Theory but it also possesses an obvious general significance.
It therefore was to be expected that it would have an impact also on general Model
Theory. In the present talk, I shall show that this expectation is indeed justified and
that the forcing notion provides us with a new tool in Model Theory, which leads to
a better understanding of concepts that have by now become classical in this area
and to their further development.
The work on which this talk is based [1, 3, 4] was begun in the summer of 1969.
While I am confident that it will have further consequences, enough results have become
available to make a presentation on the subject appropriate.
Experimentation shows that there are several ways in which the forcing concept
can be formalized within Model Theory. Here we shall explicate this notion as an
analogue of the satisfaction relation (in fact, strictly, as a generalization of it), i. e.,
in the first place as a binary metamathematical relation which may hold between a
structure and a sentence in the Lower Predicate Calculus [4]. Only the connectives ~|, V, A and only the existential quantifier will be regarded as basic (2).
2. Foundations.
We start with a specified class £ of (first order) structures. The following rules
provide a definition of the binary relation M H= X (M forces X) for structures M e E
and for sentences X which are defined in M, i. e. whose extralogical constants have
interpretations in M.
2 . 1 . For atomic X, M N= iff M N= X (M satisfies X); for X = Y A Z, M H= X
iffMN= Y a n d M H = Z ; f o r X = 7 V Z,M H= Xiff M N= Y or M N= Z;for X = (3y)Q(y),
M \¥ X iff M H= Q(a) for some a; and for X = "l Y, M H= X iff W does not force Y
for any M ' ë E , M ' = > M.
The following lemma is basic.
extension of its first argument.
It shows that the forcing relation persists under
(*) The research on which this paper is based was supported in part by the National Science
Foundation Grant No. GP-18728.
(2) The approach described here differs in some points from that adopted in ref. [4].
246
2.2.
A. ROBINSON
A
If M, M'e E, M'ID M, and M H= X then M'H= X.
A structure M G E is called E-generic (briefly, generic) if for any X defined in M,
M H= X iff M H= X. The class of generic structures is denoted by GE. Then
2 . 3 . If M, M'e Gs and M <=. M' then M -< M', i. e., M' is an elementary extension of M.
2.4. If { M v } is a monotonie set of elements of G2 and M = UVMV belongs to E
then MeGz.
From now on we assume
2.5. E is inductive, i. e. it is closed under unions of monotonie sets of its elements.
By 2.4, G£ is then also inductive. Then
2.6.
Every element of E is contained in some element of G 2 .
2 . 7 . I f M e G j j M ' e l j M c M ' and X is an existential or, more generally, an
V3 sentence, which is denned in M then M ' h l entails M \= X.
3. Universal classes.
Let U be a set of universal sentences which is closed under deduction (A sentence
is universal if it belongs to the smallest class containing all atomic sentences or their
negations and closed under conjunction, disjunction, and universal, i. e. ~|(3)~l,
quantification. U is closed under deduction if every universal X defined in U and
deducible from it belongs to U). We take E to be the class of all models of U. Such
a E is called a universal class. For example, if K is a (nonempty and consistent : n. e. a. c.)
set of sentences then U may be the set of universal sentences defined in K and deducible
from it, and we then write U = Ky.
3 . 1 . Let K and K! be two n. e. a. c. sets of sentences. Then K! is model-consistent relative to K (every model of K can be embedded in a model of K') iff K v c: K v .
And K and K' are mutually model-consistent iff Ky = J£ v .
U is called irreducible iff for any two universal sentences Xx and X2,X1 V X2eU
entails X1 e U or X2 e U. A set of sentences K has the joint embedding property
if any two models M1 and M2 of K can be injected into a model M of K consistently
with the interpretation of the constants of K in M1 and M 2 .
3.2.
K has the joint embedding property iff K v is irreducible.
4. Forcing in universal classes.
A universal class E is inductive.
to it.
Accordingly, the results of section 2 above apply
The following reduction lemma is fundamental in provinding a link between forcing
and the classical concepts of model theory. The existential degree of a well-formed
formula (w. f. f.) Q(x1,..., xn) is defined as'the number of its existential quantifiers
which are not in the scope of a negation.
FORCING IN MODEL THEORY
247
4 . 1 . For given U and corresponding E, let Q(xx,..., x„) be a w. f. f. of existential
degree in which is defined in (the vocabulary of) U. Then there exists a set SQ of sets
of w. f. f., {ßv(*i,..., xn, y x , . . . , yJ } which are defined in U such that for any alt. . .,an
denoting elements of an M e E, M H= ßfai »•••>«„) iff there exist elements of M, denoted
by &!,..., bm, such that for at least one { Qv(xl9..., xn, y l 5 . . . , ym)} e SQ, the sentences Qv(alt..., an, blt..., bJ all hold in M.
Let 117 | be the cardinal of U. Since the cardinality of the set of predicates definable
in U is max (| U \, co), 4.1 leads to the following result, which may be regarded as a
kind of compactness theorem or Löwenheim-Skolem theorem.
4 . 2 . If M H= X then there exists an M'a M, M e E , such that M' N= X, where
\M'\< max (| U \, co).
Similarly
4.3.
If M e E there exists an MeGL,M'^
M such that \M'\<
max (| U \, \ M \, co).
The reduction lemma also enables us to axiomatize the class GE in an infinitary
language Lßtto where ß = (2maxflxi'0)))+. Other applications of 4.1 will be given later.
5. The forcing operator.
Given K, nonempty and consistent (n. e. a. a), let U = Ky, and consider the class
of generic structures GE in the corresponding E. Let KF be the set of sentences in
the vocabulary of K which hold in all generic structures of E. KF is called the forcing
companion of K. We call K -> KF the forcing operator. Its basic properties are
given by 5.1 and 5.2.
5.1.
KF = (KV)F; (K% = K v ; KFF = KF.
5.2.
For any n. e. a. c. sets Kt and K2, KF = K2 if and only if 1C1V =
K2y.
Also
5.3.
KF is complete if and only if K has the joint embedding property (compare 3.2).
The forcing operator will now be related to the classical theory of model-completeness. To lead up to this, we first state the following result which is a consequence
of 4.1.
5.4.
Let M, M'e E, M'eGz,
and M < M'.
Then M' G G E .
We now know (2.6, 2.3, and 5.4) that G = G E , as a subclass of E, possesses the
following properties
5.5. (i) Any M G E is contained in some M ' e G; (ii) if M, M'e G, M a M', then
M< M'; (iii) if Mei:, M'e G and M<M' then MeG.
The properties 5.5 determine the class G uniquely. That is
5.6. For a given universal class E, let G = G± and G = G2 be subclasses of E
which satisfy 5.5. Then Gx = G2.
248
A. ROBINSON
For the proof, let MeG1.
A
Using 5.5 (i) we construct a chain
M = M 0 c M1 c M2 c M 3 c . ..
such that M2jeG1, M2j+1 e G2, j = 0, 1, 2 , . . . Then M 0 ^ M 2 -< M 4
. . . and
M i - < M 3 - < M 5 - < . . . by 5.5 (ii), and so, for M ' = u M2j- = u M 2j - +1 , M -< M'
and M ^ M . , Hence M -<Ml and so M G G 2 , by 5.5 (iii). Thus G ì C G2 and,
similarly, G2 c G1? Gx = G 2 .
This shows, without the use of forcing, that there is at most one G as described by
5.5. We know from the results stated previously that there is at least one such G.
Recall that a n. e. a. c. set of sentences K is called model-complete if for any two
models M, M' of K, M a M' entails M -< M'. A set K* is called a model companion
of a n. e. a. c. set K, if K* has the same vocabulary as K, is mutually model-consistent
with K, and is model-complete. This is a generalization, due to Eli Bers, of the notion
of model completion (K* is a model completion of K if, in addition to the conditions
just stated, K* ID K and K* is also model-complete relative to K, i. e. K* v D is complete for the diagram D of any model M of K). It is known that for a given X there
is, up to logical equivalence, at most one model companion (and, hence, at most one
model completion) K*, and this is also an immediate consequence of the results given
below.
Suppose that K. possesses a model companion K*. Let £ be the universal class
of models of U = K v . Then the class of models of K*, G, is a subclass of E since
K% = Ky (see 3.1). Also, G satisfies 5.5 (In particular G satisfies 5.5 (ii) since K*
is model-complete). Hence G = G% and
5.7. Suppose K has a model companion, K*. Then KF is the deductive closure
of K* (and, hence, is the largest model companion of K).
Thus, the notion of forcing companion is a generalization of the notion of model
companion.
6. Subclasses of E.
For given U and E, we introduce three more subclasses of E, which are related to G s .
Let M el, and let X be a sentence such that the relations and functions of X occur
in M (and U), but not necessarily its constants. M forces X weakly, M H= * X if no
M' E E , M ' D M, forces X. Then M W * X iff X holds in all generic structures M' => M
in which it is denned.
M G E is called pregeneric if for any sentence X which is defined in M, either M [4= * X
or M 14=* ~|X. The class of pregeneric structures will be denoted by P r .
6.1.
Suppose that MePz, M1, M2eGYi, M c Ml9 M c M 2 , and let X be a
sentence which is defined in M. Then X either holds in both M± and M2 or in neither
one of these structures.
6.2. Let E be the class of models of U = Ky where K is n. e. a. c. Suppose that
the class of models of K (which, in any case, is a subclass of E) is a subclass of P x and
FORCING IN MODEL THEORY
includes G E . Suppose that GE is the class of models of a set K*,
completion of K.
249
Then K* is a model
Let E be a universal class which is given by a set U, as before. M G E is existentially complete (within E) if for every existential sentence X which is defined in M
and for every extension M' of M in E, M' H= X entails M H= X. The class of existentially complete elements of E will be denoted by E E ,
A structure M G E is existentially universal (in E) if it satisfies the following condition. Let { Qv(Xi
x„, yl9...,
ym)} be a set of existential predicates formulated
in the vocabulary of U, n > 0, m > 0. Suppose that for some bl9..., bm (denoting
elements) of M there exists an M' el, M' ^ M such that for certain a\,..., a'n of M'
all Qv(a[,. . .,a'n,bx
fcm)
hold in M'. Then there exist al9..., a„ in M such that
all Qv(a\ » • • • *flu> &i » • • • * &m) h°ld m M. The class of existentially universal structures in E will be denoted by >4E.
For a given M, the number of distinct sets {Q v (x 1 ,..., x„, bl9..., bm)} is at most
2max(|u|,|Ml,co) r p ^ e n a b i e s u s t 0 s n ow, by a procedure of successive extension (compare the proof of 2.6 above which is given in ref. 4) that every M el can be embedded
in an M'e A^. Moreover, if U = Ky , M' may be chosen as the union of a monotonic set of models of K.
The four subclasses of E that we have introduced are related by
6.3.
P E => EE
ID
GE
ID
AE
Of the inclusion relations contained in 6.3, the first and last are consequences of the
reduction lemma 4.1, while £ E ID G E is contained in 2.7. Suitable examples (section 8,
below) show that any two of the four classes may be distinct. We have, as a consequence of 5.3 and 6.3,
6.4. If U = Ky where K possesses the joint embedding property then any two
existentially universal structures in E are elementarily equivalent in the vocabulary
of K.
7. Finite forcing; forcing of infinite sentences.
In Paul Cohen's original method, the forcing objects or conditions are finite sets
of basic sentences (atomic sentences or their negations) of Set Theory. An analogous
approach may be adopted for general Model Theory [1, 3]. This leads to a concept
of finite forcing (or fforcing) and to corresponding finitely generic structures. The
resulting theory is in some ways quite similar and in others radically different from the
theory of (infinite) forcing described in the preceding sections. A major difference
is that it is no longer true that every structure (in the class under consideration) can
be embedded in a finitely generic structure. Nevertheless, we can still define a fforcing
operator K -*• Kf such that if K possesses a model companion K* then Kf is its
deductive closure and, hence, is itself a model companion of K. The situation can
also be looked at from the point of view of Boolean valued Logic.
In another direction, a forcing theory in which the forced sentences are elements
of an infinitary language, has been developed by Carol W. Coven (unpublished).
250
A. ROBINSON
8. Examples and applications.
(i) Suppose K is a. set of axioms for commutative field theory. Then KF is the
theory of algebraically closed fields (" the " model completion of K) and so is Kf.
The class GE coincides with £ E and is the class of all algebraically closed fields. All
fields are contained in P E and so are all integral domains. Az consists of all fields
that are of infinite degree of transcendence over their prime fields (universal domains).
(ii) Let K be a set of axioms for the theory of groups. In this case E E is the class
of so called algebraically closed (or, existentially closed, see ref. 2) groups and AE is
a subclass of EE all of whose elements are elementarily equivalent (in the language
of group theory). Using forcing, A. Macintyre showed recently that the elements
of EE are not all elementarily equivalent. It is known [2] that K does not possess
a model companion. This implies that neither EE nor GE are arithmetical classes
(varieties).
(iii) Let K be the set of all sentences formulated in terms of equality, addition, and
multiplication, and true for the system of natural numbers. Then K is complete and,
hence, possesses the joint embedding property. It follows that KF also is complete,
although it can be proved that KF 4= K (This contrasts with Kf = K for the fforcing
operator in this case). Hence N £ G E , although NeE^.
It can be shown that K
does not possess a model companion, so KF cannot be model-complete. It can also
be shown that KF is not recursively enumerable. However, it contains many theorems
of elementary Arithmetic. Thus, KF and the associated classes GE and AE are appealing,
if somewhat enigmatic, mathematical objects.
REFERENCES
[1] J. BARWISE and A. ROBINSON. — Completing theories by forcing, Annals of Mathematical
Logic, vol. 2 (1970), pp. 119-142.
[2] P. EKLOF and G. SABBAGH. — Model-completions and modules, Yale University, February
1970 (mimeographed), to be published in the Annals of Mathematical Logic.
[3] A. ROBINSON. — Forcing in model theory, Yale University, October 1969 (mimeographed),
to be published in the Proceeding of the Colloquium on Model Theory, Rome, November 1969.
[4] —. — Infinite forcing in model theory, Yale University, July 1970 (mimeographed),
to be published in the Proceedings of the Second Scandinavian Symposium in Logic, Oslo,
June 1970.
Yale University
Department of Mathematics,
New Haven
Connecticut 06520
(U. S. A.)