Computability and Logic

156 THE EXISTENCE OF MODELS
13.6 Lemma (Closure lemma). Let L be a language, and L + a language obtained by
adding infinitely many new constants to L. IfS* is a set of sets of sentences of L + having
the satisfaction properties, then every set Ɣ of sentences of L in S* can be extended to a set
Ɣ* of sentences of L + having the closure properties.
Sections 13.2 and 13.3 will be devoted to the proof of the term models lemma,
Lemma 13.5. As in so many other proofs, we consider first, in section 13.2, the
case where identity and function symbols are absent, so that (C7) and (C8) may be
ignored, and the only closed terms are constants, and then, in section 13.3, consider the
additional complications that arise when identity is present, as well as those created by
the presence of function symbols. The proof of the closure lemma, Lemma 13.6, will
be given in section 13.4, with an alternative proof, avoiding any dependence on the
assumption that the language is enumerable, to be outlined in the optional section 13.5.
13.2 The First Stage of the Proof
In this section we are going to prove the term models lemma, Lemma 13.5, in the case
where identity and function symbols are absent. So let there be given a set Ɣ* with
the closure properties (C1)–(C6), as in the hypothesis of the lemma to be proved. We
want to show that, as in the conclusion of that lemma, there is an interpretation in
which every element of the domain is the denotation of some constant of the language
of Ɣ*, in which every sentence in Ɣ* will be true.
To specify an interpretation M in this case, we need to do a number of things. To
begin with, we must specify the domain |M|. Also, we must specify for each constant
c of the language which element c M of the domain is to serve as its denotation.
Moreover, we must do all this in such a way that every element of the domain is the
denotation of some constant. This much is easily accomplished: simply pick for each
constant c some object c M , picking a distinct object for each distinct constant, and
let the domain consist of these objects.
To complete the specification of the interpretation, we must specify for each predicate
R of the language what relation R M on elements of the domain is to serve as
its denotation. Moreover, we must do so in such a way that it will turn out that for
every sentence B in the language we have
(1)
if B is in Ɣ* then M |= B.
What we do is to specify R M in such a way that (1) automatically becomes true for
atomic B. We define R M by the following condition:
R M( c1
M )
,...,cM n if and only if R(c 1 ,...,c n )isinƔ*.
Now the definition of truth for atomic sentences reads as follows:
M |= R(c 1 ,...,c n ) if and only if R M( c M 1 ,...,cM n
)
.
We therefore have the following:
(2) M |= R(c 1 ,...,c n ) if and only if R(c 1 ,...,c n )isinƔ*
and this implies (1) for atomic B.