Computability and Logic

THE UNPROVABILITY OF CONSISTENCY 235
namely, that for every sentence A we have
(P5)
⊢ T B( A ) → A.
But in fact, as we also show below, no provability predicate fulfills condition (P5)
unless T is inconsistent.
Our next theorem will provide answers to three questions. First, just as the diagonal
lemma provides a sentence, the Gödel sentence, that ‘says of itself’ that it is unprovable,
so also it provides a sentence, the Henkin sentence, that ‘says of itself’ that it
is provable. In other words, given a provability predicate B(x), there is a sentence
H T such that ⊢ T H T ↔ B( H T ). Gödel’s theorem was that, if T is consistent, then
the Gödel sentence is indeed unprovable. Henkin’s question was whether the Henkin
sentence is indeed provable. This is the first question our next theorem will answer.
Second, call a formula Tr(x)atruth predicate for T if and only if for every sentence
A of the language of T we have ⊢ T A ↔ Tr ( A ). Another question is whether, if T
is consistent, there can exist a truth predicate for T . (The answer to this question is
going to be negative. Indeed, the negative answer can actually be obtained directly
from the diagonal lemma of the preceding chapter.) Third, if B(x) is a provability
predicate, call ∼B( 0 = 1 ) the consistency sentence for T [relative to B(x)]. Yet
another question is whether, if T is consistent, the consistency sentence for T can be
provable in T . (We have already indicated in Theorem 18.3 that the answer to this
last question is going to be negative.)
The proof of the next theorem, though elementary, is somewhat convoluted, and
as warm-up we invite the reader to ponder the following paradoxical argument, by
which we seem to be able to prove from pure logic, with no special assumptions, the
existence of Santa Claus. (The argument would work equally well for Zeus.) Consider
the sentence ‘if this sentence is true, then Santa Claus exists’; or to put the matter
another way, let S be the sentence ‘if S is true, then Santa Claus exists’.
Assuming
(1)
S is true
by the logic of identity it follows that
(2)
‘If S is true, then Santa Claus exists’ is true.
From (2) we obtain
(3)
If S is true, then Santa Claus exists.
From (1) and (3) we obtain
(4)
Santa Claus exists.
Having derived (4) from the assumption (1) we infer that without the assumption (1),
indeed without any special assumption, that we at least have the conditional conclusion
that if (1), then (4), or in other words
(5)
If S is true, then Santa Claus exists.