Computability and Logic

234 THE UNPROVABILITY OF CONSISTENCY
18.3 Theorem (Gödel’s second incompleteness theorem, abstract form). Let T be a
consistent, axiomatizable extension of P, and let B(x) be a formula having properties
(P1)–(P3) above. Then not ⊢ T ∼B( 0 = 1 ).
The proof will occupy the remainder of this chapter. Throughout, let T be an
extension (not necessarily consistent) of Q. A formula B(x) with properties (P1)–
(P3) of Lemma 18.2 we call a provability predicate for T .Webeginwithafew
words about this notion. The formula Prv T (x) considered so far we call the traditional
provability predicate for T , though, as we have indicated, we are not going to give the
proof of Lemma 18.2, and so are not going to be giving the proof that the ‘traditional
provability predicate’ is a ‘provability predicate’ in the sense of our official definition
of the latter term.
If T is ω-consistent, taking the traditional Prv T (x) for B(x), we have also the
following property, the converse of (P1):
(P4)
If ⊢ T B( A ) then ⊢ T A.
[For if we had ⊢ T Prv T ( A ), or in other words ⊢ T ∃y Prf T ( A , y), but did not have
⊢ T A, then for each b, ∼Prf T ( A , b) would be correct and hence provable in Q and
hence in T , and we would have an ω-inconsistency in T .] We do not, however, include
ω-consistency in our assumptions on T , or (P4) in our definition of the technical
term ‘provability predicate’. Without the assumption of (P4), which is not part of our
official definition, a ‘provability predicate’ need not have much to do with provability.
In fact, the formula x = x is easily seen to be a ‘provability predicate’ in the sense
of our definition.
On the other hand, a formula may arithmetically define the set of Gödel numbers
of theorems of T without being a provability predicate for T .IfT is consistent and
Prv T (x) is the traditional provability predicate for T , then not only does Prv T (x)
arithmetically define the set of Gödel numbers of theorems of T , but so does the
formula Prv* T (x), which is the conjunction of Prv T (x) with ∼Prv T ( 0 = 1 ), since the
second conjunct is true. But notice that, in contrast to Theorem 18.1, ∼Prv* T ( 0 = 1 )
is provable in T . For it is simply
∼(Prv T ( 0 = 1 )&∼Prv T ( 0 = 1 ))
which is a valid sentence and hence a theorem of any theory. The formula Prv* T (x),
however, lacks property (P1) in the definition of provability predicate. That is, it
is not the case that if ⊢ T A then ⊢ T Prv* T ( A ). Indeed, it is never the case that
⊢ T Prv* T ( A ), since it is not the case that ⊢ T ∼Prv T ( 0 = 1 ), by Theorem 18.1.
The traditional provability predicate Prv T (x) has the further important, if nonmathematical,
property beyond (P0)–(P4), that intuitively speaking Prv(x) can plausibly
be regarded as meaning or saying (on the standard interpretation) that x is the Gödel
number of a sentence that is provable in T . This is conspicuously not the case for
Prv* T (x), which means or says that x the Gödel number of a sentence that is provable
in T and T is consistent.
The thought that whatever is provable had better be true might make it surprising
that a further condition was not included in the definition of provability predicate,