Computability and Logic

134 THE UNDECIDABILITY OF FIRST-ORDER LOGIC
11.3 Lemma
(a) If f i is obtained by composition from functions f k and f j1 , ... f jp for which Ɣ is
adequate, then Ɣ is adequate also for f i .
(b) If f i is obtained by primitive recursion from functions f j and f k for which Ɣ is
adequate, then Ɣ is adequate also for f i .
Proof: We leave (a) to the reader and do (b). Given a, b, and c with f i (a, b) = c, for
each p ≤ b let c p = f i (a, p), so that c b = c. Note that since f i is obtained by primitive
recursion from f j and f k ,wehave
and for all p < b we have
c 0 = f i (a, 0) = f j (a)
c p ′ = f i (a, p ′ ) = f k (a, p, f i (a, p)) = f k (a, p, c p ).
Since Ɣ is adequate for f j and f k ,
(6a)
(6b)
f j (a, 0) = c 0
f k (a, p, c p ) = c p ′
are consequences of Ɣ. But (6a) and (5a) imply
(7a)
f i (a, 0) = c 0
while (6b) and (5b) imply
(7b)
fi(a, p) = c p → f i (a, p ′ ) = c p ′.
But (7a) and (7b) for p = 0 imply f i (a, 1) = c 1 , which with (7b) for p = 1 implies
f i (a, 2) = c 2 , which with (7b) for p = 2 implies f i (a, 3) = c 3 , and so on up
to f i (a, b) = c b = c, which is what needed to be proved to show Ɣ adequate for f i.
Since every f i is either a basic function or obtained from earlier functions on our
list by the processes covered by Lemma 11.3, the lemma implies that Ɣ is adequate
for all the functions on our list, including f r = f . In particular, if f (m, n) = 0, then
f r (m, n) = 0 is implied by Ɣ, and hence so is ∀y f r (m, y) = 0, which is D(m).
Thus we have reduced the problem of determining whether for some n we have
f (m, n) = 0 to the problem of determining whether Ɣ implies D(m). That is, we
have established that if the decision problem for logical implication were solvable,
the nullity problem for f would be solvable, which it is known, as we have said, that
it is not, assuming Church’s thesis. Hence we have established the following result,
assuming Church’s thesis.
11.4 Theorem (Church’s theorem). The decision problem for logical implication is
unsolvable.
Problems
11.1 The decision problem for validity is the problem of devising an effective procedure
that, applied to any sentence, would in a finite amount of time enable
one to determine whether or not it is valid. Show that the unsolvability of