Computability and Logic

20.3. BETH’S DEFINABILITY THEOREM 265
Proof: A satisfiable extension of a complete theory is conservative, and a conservative
extension of a satisfiable theory is satisfiable. Thus if the T i satisfy the hypotheses
of Corollary 20.7, then T 3 as defined in Theorem 20.6 is a satisfiable extension of T 0 ,
and therefore T 1 ∪ T 2 is satisfiable.
20.8 Example (Failures of joint consistency). Let L 0 = L 1 = L 2 ={P, Q}, where P and
Q are one-place predicates. Let T 1 (respectively, T 2 ) be the set of consequences in L 1
(respectively, L 2 )of{∀xPx, ∀xQx} (respectively, {∀xPx, ∀x ∼ Qx}. Let T 0 be the set of
consequences in L 0 of ∀xPx. Then T 1 ∪ T 2 is not satisfiable, though each of T 1 and T 2 is
a satisfiable extension of T 0 . This is not a counterexample to Robinson’s theorem, because
T 0 is not complete. If instead we let L 0 ={P}, then again we do not get a counterexample,
because then L 0 is not the intersection of L 1 and L 2 , while L is. This shows the hypotheses
in Corollary 20.7 are needed.
We have proved Robinson’s theorem using Craig’s theorem. Robinson’s theorem
can also be proved a different way, not using Craig’s theorem, and then used to prove
Craig’s theorem. Let us indicate how a ‘double compactness’ argument yields Craig’s
theorem from Robinson’s.
Suppose A implies C. Let L 1 (L 2 ) be the language consisting of the nonlogical
symbols occurring in A (C). Let L 0 = L 1 ∩ L 2 . We want to show there is a sentence
B of L 0 implied by A and implying C. Let be the set of sentences of L 0 that are
implied by A. We first show that ∪{∼C} is unsatisfiable. Suppose that it is not and
that M is a model of ∪{∼C}. Let T 0 be the set of sentences of L 0 that are true in M.
T 0 is a complete theory whose langauge is L 0 . Let T 1 (T 2 ) be the set of sentences of
L 1 (L 2 ) that are consequences of T 0 ∪{A} (T 0 ∪{∼C}). T 2 is a satisfiable extension
of T 0 : M is a model of T 0 ∪{∼C}, and hence of T 2 . But T 1 ∪ T 2 is not satisfiable: any
model of T 1 ∪ T 2 would be a model of {A, ∼C}, and since A implies C, there is no
such model. Thus by the joint consistency theorem, T 1 is not a satisfiable extension
of T 0 , and therefore T 0 ∪{A} is unsatisfiable. By the compactness theorem, there is a
finite set of sentences in T 0 whose conjunction D, which is in L 0 , implies ∼A. Thus
A implies ∼D, ∼D is in L 0 , ∼D is in , and ∼D is therefore true in M. But this is
a contradiction, as all of the conjuncts of D are in T 0 and are therefore true in M.So
∪{∼C} is unsatisfiable, and by the compactness theorem again, there is a finite
set of members of whose conjunction B implies C. B is in L 0 , since its conjuncts
are, and as A implies each of these, A implies B.
20.3 Beth’s Definability Theorem
Beth’s definability theorem is a result about the relation between two different explications,
or ways of making precise, the notion of a theory’s giving a definition of
one concept in terms of other concepts. As one might expect, each of the explications
discusses a relation that may or may not hold between a theory, a symbol in the
language of that theory (which is supposed to ‘represent’ a certain concept), and other
symbols in the language of the theory (which ‘represent’ other concepts), rather than
directly discussing a relation that may or may not hold between a theory, a concept,