Computability and Logic

HISTORICAL REMARKS 237
From (2) and (11) it follows that
(12)
⊢ T C.
By virtue of (P1) again,
(13)
⊢ T B( C ).
And so finally, from (11) and (13), we have
(14)
⊢ T A.
Since the converse of Löb’s theorem is trivial (if ⊢ T A, then ⊢ T F → A for any
sentence F), a necessary and sufficient condition for A to be a theorem of T is that
B( A ) → A is a theorem of T . Now for the promised derivation of the three results
mentioned earlier.
18.5 Corollary. Suppose that B(x) is a provability predicate for T . Then if ⊢ T H ↔
B( H ), then ⊢ T H.
Proof: Immediate from Löb’s theorem.
18.6 Corollary. If T is consistent, then T has no truth predicate.
Proof: Suppose that Tr(x) is a truth predicate for T . Then a moment’s thought
shows that Tr(x) is also a provability predicate for T . Moreover, since Tr(x) is a truth
predicate, for every A we have ⊢ T Tr( A ) → A. But then by Löb’s theorem, for every
A we have ⊢ T A, and T is inconsistent.
And finally, here is the proof of Theorem 18.3.
Proof: Suppose ⊢ T ∼B( 0 = 1 ). Then ⊢ T B( 0 = 1 ) → F for any sentence F,
and in particular ⊢ T B( 0 = 1 ) → 0 = 1, and hence ⊢ T 0 = 1, and since T is an
extension of Q, T is inconsistent.
It is characteristic of important theorems to raise new questions even as they
answer old ones. Gödel’s theorems (as well as some of the major recursion-theoretic
and model-theoretic results we have passed on our way to Gödel’s theorems) are
a case in point. Several of the new directions of research they opened up will be
explored in the remaining chapters of this book. One such question is that of how
far one can go working just with the abstract properties (P1)–(P3), without getting
involved in the messy details about a particular predicate Prv T (x). That question will
be explored in the last chapter of this book.
Historical Remarks
We alluded in passing in an earlier chapter to the existence of heterodox mathematicians
who reject certain principles of logic. More specifically, in the late nineteenth
and early twentieth centuries there were a number of mathematicians who rejected
‘nonconstructive’ as opposed to ‘constructive’ existence proofs and were led by this