Computability and Logic

20.3. BETH’S DEFINABILITY THEOREM 267
20.9 Lemma. α is implicitly definable from β 1 ,...,β n in T if and only if
(1)
∀x 0 ···∀x k (—α, x 0 ,...,x k — ↔ —α ′ , x 0 ,...,x k —)
is a consequence of T ∪ T ′ .
Proof: Here, of course, by —α ′ , x 0 , ..., x k — is meant the result of substituting α ′
for α in —α, x 0 ,...,x k —. Note that (1) will be true in a given interpretation if and
only if that interpretation assigns the same denotation to α and to α ′ .
For the left-to-right direction, suppose α is implicitly definable from the β i in T .
Suppose K is a model of T ∪ T ′ . Let M and N be the models into which K can be
decomposed as above, so that K = M + N. Then M and N have the same domain
and agree on what they assign to the β i . By the supposition of implicit definability,
they must therefore agree on what they assign to α. Therefore the biconditional (1)
is true in K. In other words, any model of T ∪ T ′ is a model of (1), which therefore
is a consequence of T ∪ T ′ .
For the right-to-left direction, suppose that (1) follows from T ∪ T ′ . Suppose M
and N are models of T that have the same domain and agree on what they assign to
the β i . Then M + N is a model of T ∪ T ′ and therefore of (1), by the supposition that
(1) is a consequence of T ∪ T ′ . It follows that M + N assigns the same denotation
to α and α ′ , and therefore that M and N assign the same denotation to α. Thus α is
implicitly definable from the β i in T .
One direction of the connection between implicit and explicit definability is now
easy.
20.10 Proposition (Padoa’s method). If α is not implicitly definable from the β i in T ,
then α is not explicitly definable from the β i in T .
Proof: Suppose α is explicitly definable from the β i in T . Then some definition
(2)
∀x 0 ···∀x k (—α, x 0 ,...,x k — ↔ B(x 0 ,...,x k ))
of α from the β i is in T . Therefore
(3)
∀x 0 ···∀x k (—α ′ , x 0 ,...,x k — ↔ B(x 0 ,...,x k ))
is in T ′ . (Recall that B involves only the β i , which are not replaced by new nonlogical
symbols.) Since (1) of Lemma 20.9 is a logical consequence of (2) and (3), it is a
consequences of T ∪ T ′ , and by that lemma, α is implicitly definable from the β i
in T .
20.11 Theorem (Beth’s definability theorem). α is implicitly definable from the β i in
T if and only if α is explicitly definable from the β i in T .
Proof: The ‘if’ direction is the preceding proposition, so it only remains to prove
the ‘only if’ direction. So suppose α is implicitly definable from the β i in T. Then
(1) of Lemma 20.9 is a consequence of T ∪ T ′ . By the compactness theorem, it is a
consequence of some finite subset of T ∪ T ′ . By adding finitely many extra sentences
to it, if necessary, we can regard this finite subset as T 0 ∪ T
0 ′,
where T 0 is a finite subset
of T , and T
0 ′ comes from T 0 on replacing each nonlogical symbol γ other than the β i