Computability and Logic

238 THE UNPROVABILITY OF CONSISTENCY
rejection to reject the method of proof by contradiction, which has been ubiquitously
used in orthodox mathematics since Euclid (and has been repeatedly used
in this book). The most extreme critics, the ‘finitists’, rejected the whole of established
‘infinitistic’ mathematics, declaring not only that the proofs of its theorems
were fallacious, but that the very statements of those theorems were meaningless.
Any mathematical assertion going beyond generalizations whose every instance can
be checked by direct computation (essentially, anything beyond ∀-rudimentary sentences)
was rejected.
In the 1920s, David Hilbert, the leading mathematician of the period, devised a
program he hoped would provide a decisive answer to these critics. On the plane of
philosophical principle, he in effect conceded that sentences going beyond
∀-rudimentary sentences are ‘ideal’ additions to ‘contentful’ mathematics. He compared
this addition to the addition of ‘imaginary’ numbers to the system of real
numbers, which had also raised doubts and objections when it was first introduced.
On the plane of mathematical practice, Hilbert insisted, a detour through the ‘ideal’ is
often the shortest route to a ‘contentful’ result. (For example, Chebyshev’s theorem
that there is a prime between any number and its double was proved not in some
‘finitistic’, ‘constructive’, directly computational way, but by an argument involving
applying calculus to functions whose arguments and values are imaginary numbers.)
Needless to say, this reply wouldn’t satisfy a critic who doubted the correctness of
‘contentful’ results arrived at by such a detour. But Hilbert’s program was precisely
to prove that any ‘contentful’ result provable by orthodox, infinitistic mathematics is
indeed correct. Needless to say, such a proof wouldn’t satisfy a critic if the proof itself
used the methods whose legitimacy was under debate. But more precisely Hilbert’s
program was to prove by ‘finitistic’ means that every ∀-rudimentary sentence proved
by ‘infinitistic’ means is correct.
An important reduction of the problem was achieved. Suppose a mathematical
theory T proves some incorrect ∀-rudimentary sentence ∀xF(x). If this sentence is
incorrect, then some specific numerical instance F(n) for some specific number
n must be incorrect. Of course, if the theory proves ∀xF(x) it also proves each
instance F(n), since the instances follow from the generalization by pure logic.
But if F(n) is incorrect, then ∼F(n) is a correct rudimentary sentence, and as such
will be provable in T , for any ‘sufficiently strong’ T . Hence if such a T proves an
∀-rudimentary sentence ∀xF(x), it will prove an outright contradiction, proving both
F(n) and ∼F(n). So the problem of proving T yields only correct ∀-rudimentary
theorems reduces to the problem of showing T is consistent. Hilbert’s program was,
then, to prove finitistically the consistency of infinitistic mathematics.
It can now be appreciated how Gödel’s theorems derailed this program in its original
form just described. While it was never made completely explicit what ‘finitistic’
mathematics does and does not allow, its assumptions amounted to less than the assumptions
of inductive or Peano arithmetic P. On the other hand, the assumptions of
‘infinitistic’ mathematics amount to more than the assumptions of P. So what Hilbert
was trying to do was prove, using a theory weaker than P, the consistency of a theory
stronger than P, whereas what Gödel proved was that, even using the full strength of
P, one cannot prove the consistency of P itself, let alone anything stronger.