Computability and Logic

236 THE UNPROVABILITY OF CONSISTENCY
From (5) we obtain
(6)
‘If S is true, then Santa Claus exists’ is true.
By the logic of identity again it follows that
(7)
S is true.
And from (5) and (7) we infer, without any special assumptions, the conclusion that
(8)
Santa Claus exists.
18.4 Theorem (Löb’s theorem). If B(x) is a provability predicate for T , then for any
sentence A, if⊢ T B( A ) → A, then ⊢ T A.
Proof: Suppose that B is a provability predicate for T and that
(1)
⊢ T B( A ) → A.
Let D(y) be the formula (B(y) → A), and apply the diagonal lemma to obtain a
sentence C such that
(2)
⊢ T C ↔ (B( C ) → A).
So
(3)
⊢ T C → (B( C ) → A).
By virtue of property (P1) of a provability predicate,
(4)
⊢ T B( C → (B( C ) → A)).
By virture of (P2),
(5)
⊢ T B( C → (B( C ) → A) ) → (B( C ) → B( B( C ) → A )).
From (4) and (5) it follows that
(6)
⊢ T B( C ) → B( B( C ) → A ).
By virtue of (P2) again,
(7)
⊢ T B( B( C ) → A ) → (B( B( C ) ) → B( A )).
From (6) and (7) it follows that
(8)
⊢ T B( C ) → (B( B( C ) ) → B( A )).
By virtue of (P3),
(9)
⊢ T B( C ) → B( B( C ) ).
From (8) and (9) it follows that
(10)
⊢ T B( C ) → B( A ).
From (1) and (10) it follows that
(11)
⊢ T B( C ) → A.