Computability and Logic

PROBLEMS 185
unformalized mathematics of a proof of a theorem from a set of axioms. For in later
chapters we are going to be establishing results about the scope and limits of formal
deducibility whose interest largely depends on their having something to do with proof
in a more ordinary sense (just as results about the scope and limits of computability
in one or another formal sense discussed in other chapters depend for their interest
on their having something to do with computation in a more ordinary sense).
We have already mentioned towards the end of Chapter 10 that theorems and
axioms in ordinary mathematics can virtually always be expressed as sentences of
a formal first-order language. Suppose they are so expressed. Then if there is a
deduction in the logician’s formal sense of the theorem from the axioms, there will
be a proof in the mathematician’s ordinary sense, because, as indicated earlier, each
formal rule of inference in the definition of deduction corresponds to some ordinary
mode of argument as used in mathematics and elsewhere. It is the converse assertion,
that if there is a proof in the ordinary sense, then there will be a deduction in our very
restrictive format, that may well seem more problematic. This converse assertion is
sometimes called Hilbert’s thesis.
As the notion of ‘proof in the ordinary sense’ is an intuitive, not a rigorously defined
one, there cannot be a rigorous proof of Hilbert’s thesis. Before the completeness
theorem was discovered, a good deal of evidence of two kinds had already been
obtained for the thesis. On the one hand, logicians produced vast compendia of
formalizations of ordinary proofs. On the other hand, various independently proposed
systems of formal deducibility, each intended to capture formally the ordinary notion
of provability, had been proved equivalent to each other by directly showing how to
convert formal deductions in one format into formal deductions in another format;
and such equivalence of proposals originally advanced independently of each other,
while it does not amount to a rigorous proof that either has succeeded in capturing
the ordinary notion of provability, is surely important evidence in favor of both.
The completeness theorem, however, makes possible a much more decisive argument
in favor of Hilbert’s thesis. The argument runs as follows. Suppose there is a
proof in the ordinary mathematical sense of some theorem from some axioms. As
part-time orthodox mathematicians ourselves, we presume ordinary mathematical
methods of proof are sound, and if so, then the existence of an ordinary mathematical
proof means that the theorem really is a consequence of the axioms. But if the theorem
is a consequence of the axioms, then the completeness theorem tells us that, in
agreement with Hilbert’s thesis, there will be a formal deduction of the theorem from
the axioms. And when in later chapters we show that there can be no formal deduction
in certain circumstances, it will follow that there can be no ordinary proof, either.
Problems
14.1 Show that:
(a) Ɣ secures if and only if Ɣ ∪∼ is unsatisfiable.
(b) Ɣ secures if and only if some finite subset of Ɣ secures some finite
subset of .
14.2 Explain why the problems following this one become more or less trivial if
one is allowed to appeal to the soundness and completeness theorems.