Computability and Logic

13.5. NONENUMERABLE LANGUAGES 163
we have just remarked is that if Ɣ is a set of sentences of L and every finite subset
of Ɣ has a model, then Ɣ is in S*. It is not hard to show that S* has the satisfiability
properties (S1)–(S4) and (S6)–(S8) (imitating the proof of Lemma 13.2).
We now introduce some set-theoretic terminology. Let I be a nonempty set. A
family P of subsets of I is said to be of finite character provided that for each subset
Ɣ of I , Ɣ is in P if and only if each finite subset of Ɣ is in P. A subset Ɣ* ofI is
said to be maximal with respect to P if Ɣ* isinP, but no subset of I properly
including Ɣ*isinP.
To apply this terminology to the situation we are considering, it is not hard to show
that S* is of finite character (essentially by definition). Nor is it hard to show that
any maximal element Ɣ* ofS* will contain H (by showing that adding a Henkin
axiom to a given set in S* produces a set still in S*, so that if the given set was
maximal, the Henkin axiom must already have belonged to it). Nor is it hard to
show that any maximal element Ɣ*ofS* will have closure properties (C1)–(C4) and
(C6)–(C8) [since, for instance, if ∼∼B is in Ɣ*, then adding B to Ɣ* produces a set
still in S* by (S2)]. Nor, for that matter, is it hard to show that such a Ɣ* will also
have closure property (C5) [using the fact that whether or not ∃xF(x) isinƔ*, Ɣ*
contains the Henkin axioms ∼∃xF(x) ∨ F(c F ), and applying (C1) and (C3)]. Thus by
Lemma 13.5, whose proof made no essential use of enumerability, Ɣ* will have a
model.
Putting everything together from the preceding several paragraphs, if Ɣ is a set of
sentences of L such that every finite subset of Ɣ has a model, then Ɣ itself will have
a model, provided we can prove that for every set Ɣ in S* there is a maximal element
Ɣ* inS* that contains Ɣ.
And this does follow using a general set-theoretic fact, the maximal principle,
according to which for any nonempty set I and any set P of subsets of I that has
finite character, and any Ɣ in P, there is a maximal element Ɣ*ofP that contains Ɣ.It
is not hard to prove this principle in the case where I is enumerable (by enumerating
its elements i 0 , i 1 , i 2 ,..., and building Ɣ* as the union of sets Ɣ n in P, where Ɣ 0 = Ɣ,
and Ɣ n+1 = Ɣ n ∪{i n } if Ɣ n ∪{i n } is in P, and = Ɣ n otherwise). In fact, the maximal
principle is known to hold even for nonenumerable I , though the proof in this case
requires a formerly controversial axiom of set theory, the axiom of choice—indeed,
given the other, less controversial axioms of set theory, the maximal principle is
equivalent to the axiom of choice, a fact whose proof is given in any textbook on set
theory, but will not be given here.
Reviewing our work, one sees that using the maximal principle for a nonenumerable
set, we get a proof of the compactness theorem for nonenumerable languages.
This general version of the compactness theorem has one notable consequence.
13.9 Theorem (The upward Löwenheim–Skolem theorem). Any set of sentences that
has an infinite model has a nonenumerable model.
The proof is not hard (combining the ideas of Corollary 12.16 and Example 13.8).
But like the proofs of several of our assertions above, we relegate this one to the
problems.