Computability and Logic

THE UNPROVABILITY OF CONSISTENCY 233
And so we have
⊢ T ∼Prv T ( 0 = 1 ) → G T .
So if we had ⊢ T ∼Prv T ( 0 = 1 ), then we would have ⊢ T G T , which by Proposition
17.9 we do not.
Of course, the key step here, of which we have not given and are not going to
be giving the proof, is the claim that a theory like P is strong enough to ‘formalize’
the proof of a result like Theorem 17.9. Gödel’s successors, beginning with Paul
Bernays, have analyzed just what properties of Prv T are actually essential to get the
second incompleteness theorem, finding that one does not really have to ‘formalize’
the whole proof of Theorem 17.9, but only certain key facts that serve as lemmas in
that proof. We summarize the results of the analysis in the next two propositions.
18.2 Lemma*. Let T be a consistent, axiomatizable extension of P, and let B(x) be
the formula Prv T (x). Then the following hold for all sentences:
(P1)
(P2)
(P3)
If ⊢ T A then ⊢ T B( A )
⊢ T B( A 1 → A 2 ) → (B( A 1 ) → B( A 2 ))
⊢ T B( A ) → B( B( A ) ).
Again we have starred the lemma because we are not going to give a full proof.
First we note a property not on the above list:
(P0)
If ⊢ T A 1 → A 2 and ⊢ T A 1 , then ⊢ T A 2 .
This is a consequence of the Gödel completeness theorem, according to which the
theorems of T are just the sentences implied by T , since if a conditional A 1 → A 2
and its antecedent A 1 are both implied by a set of sentences, then so is its consequent
A 2 . Whatever notion of proof one starts with, so long as it is sound and complete, (P0)
will hold. One might therefore just as well build it into one’s notion of proof, adding
some appropriate version of it to the rules of one’s proof procedure. Of course, once it
is thus built in, the proof of (P0) no longer requires the completeness theorem, but becomes
comparatively easy. [For the particular proof procedure we used in Chapter 14,
we discussed the possibility of doing this in section 14.3, where the version of (P0)
appropriate to our particular proof procedure was called rule (R10).]
(P1) holds for any extension of Q, since if ⊢ T A, then Prv T ( A ) is correct, and
being an ∃-rudimentary sentence, it is therefore provable in Q. (P2) is essentially the
assertion that the proof of (P0) (which we have just said can be made comparatively
easy) can be ‘formalized’ in P. (P3) is essentially the assertion that the (by no means
so easy) proof of (P1) can also be ‘formalized’ in P. The proofs of the assertions
(P2) and (P3) of ‘formalizability’ are omitted from virtually all books on the level
of this one, not because they involve any terribly difficult new ideas, but because the
innumerable routine verifications they—and especially the latter of them—require
would take up too much time and patience. What we can and do include is the proof
that the starred lemma implies the starred theorem. More generally, we have the
following: