Computability and Logic

19.2. SKOLEM NORMAL FORM 251
19.9 Theorem (Strong Löwenheim–Skolem theorem). Let A be a nonenumerable
model of an enumerable set of sentences Ɣ. Then A has an enumerable subinterpretation
that is also a model of Ɣ.
Proof: It will suffice to prove the theorem for the special case of sets of ∀-sentences.
For suppose we have proved the theorem in this special case, and consider the general
case where A is a model of some arbitrary set of sentences Ɣ. Then as in our earlier
discussion, A has an expansion A # to a model of Ɣ # , the set of Skolem forms of
the sentences in Ɣ. Since Skolem forms are ∀-sentences, by the special case of the
theorem there will an enumerable subinterpretation B # that is also a model of Ɣ # .
Then since the Skolem form of a sentence implies the original sentence, B # will also
be a model of Ɣ, and so will be its reduct B to the original language. But this B will
be an enumerable subinterpretation of A.
To prove the theorem in the special case where all sentences in Ɣ are ∀-sentences,
consider the set B of all denotations in A of closed terms of the language of Ɣ.
(We may assume there are some closed terms, since if not, we may add a constant
c to the language and the logically valid sentence c = c to Ɣ.) Since that language
is enumerable, so is the set of closed terms, and so is B. Since B is closed under
the functions that are denotations of the function symbols of the language, it is the
domain of an enumerable subinterpretation B of A. And by Proposition 19.7, every
∀-sentence true in A is true in B,soB is a model of Ɣ.
Two interpretations A and B for the same language are called elementarily equivalent
if every sentence true in the one is true in the other. Taking for Ɣ in the above
version of the Löwenheim–Skolem theorem the set of all sentences true in A, the
theorem tells us that any interpretation has an enumerable subinterpretation that is
elementarily equivalent to it. A subinterpretation B of an interpretation A is called
an elementary subinterpretation if for any formula F(x 1 , ..., x n ) and any elements
b 1 , ..., b n of |B|, the elements satisfy the formula in A if and only if they satisfy
it in B. This implies elementary equivalence, but is in general a stronger condition.
By extending the notion of Skolem normal form to formulas with free variables,
the above strong version of the Löwenheim–Skolem theorem can be sharpened to
a still stronger one telling us that any interpretation has an enumerable elementary
subinterpretation.
Applications of Skolem normal form will be given in the next section. Since
we are not going to be needing the sharper result stated in the preceding paragraph
for these applications, we do not go into the (tedious but routine) details
of its proof. Instead, before turning to applications we wish to discuss another,
more philosophical issue. At one time the Löwenheim–Skolem theorem (especially
in the strong form in which we have proved it in this section) was considered
philosophically perplexing because some of its consequences were perceived
as anomalous. The apparent anomaly, sometime called ‘Skolem’s paradox’, is that
there exist certain interpretations in which a certain sentence, which seems to say
that nonenumerably many sets of natural numbers exist, is true, even though the
domains of these interpretations contain only enumerably many sets of natural numbers,
and the predicate in the sentence that we would be inclined to translate