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