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