Computability and Logic

18
The Unprovability of Consistency
According to Gödel’s second incompleteness theorem, the sentence expressing that a
theory like P is consistent is undecidable by P, supposing P is consistent. The full proof
of this result is beyond the scope of a book on the level of the present one, but the overall
structure of the proof and main ingredients that go into the proof will be indicated in
this short chapter. In place of problems there are some historical notes at the end.
Officially we defined T to be inconsistent if every sentence is provable from T ,
though we know this is equivalent to various other conditions, notably that for some
sentence S, both S and ∼S are provable from T .IfT is an extension of Q, then since
0 ≠ 1 is the simplest instance of the first axiom of Q, 0 ≠ 1 is provable from T , and
if 0 = 1 is also provable from T , then T is inconsistent; while if T is inconsistent,
then 0 = 1 is provable from T , since every sentence is. Thus T is consistent if
and only if 0 = 1 is not provable from T . We call ∼Prv T ( 0 = 1 ), which is to
say ∼∃y Prf T ( 0 = 1 , y), the consistency sentence for T . Historically, the original
paper of Gödel containing his original version of the first incompleteness theorem
(corresponding to our Theorem 17.9) included towards the end a statement of a
version of the following theorem.
18.1 Theorem* (Gödel’s second incompleteness theorem, concrete form). Let T be
a consistent, axiomatizable extension of P. Then the consistency sentence for T is not
provable in T .
We have starred this theorem because we are not going to give a full proof of it. In
gross outline, Gödel’s idea for the proof of this theorem was as follows. The proof of
Theorem 17.9 shows that if the absurdity 0 = 1 is not provable in T then the Gödel
sentence G T is not provable in T either, so the following is true: ∼Prv T ( 0 = 1 ) →
∼Prv T ( G T ). Now it turns out that the theory P of inductive arithmetic, and hence
any extension T thereof, is strong enough to ‘formalize’ the proof of Theorem 17.9,
so we have
⊢ T ∼Prv T ( 0 = 1 ) →∼Prv T ( G T ).
But G T wasaGödel sentence, so we have also
⊢ T G T ↔∼Prv T ( G T ).
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