sentence F), a necessary and sufficient condition for A to be a theorem of T is that B( A ) → A is a theorem of T . Now for the promised derivation of the three results mentioned earlier.

18.5 Corollary. Suppose that B(x) is a provability predicate for T . Then if T H ↔

B( H ), then T H .

Proof: Immediate from Lob’s¨ theorem.

18.6 Corollary. If T is consistent, then T has no truth predicate.

Proof: Suppose that Tr(x) is a truth predicate for T . Then a moment’s thought shows that Tr(x) is also a provability predicate for T . Moreover, since Tr(x) is a truth

predicate, for every A we have T Tr( A ) → A. But then by Lob’s¨ theorem, for every A we have T A, and T is inconsistent.

And finally, here is the proof of Theorem 18.3.

Proof: Suppose T B( 0 = 1 ). Then T B( 0 = 1 ) → F for any sentence F, and in particular T B( 0 = 1 ) → 0 = 1, and hence T 0 = 1, and since T is an extension of Q, T is inconsistent.

It is characteristic of important theorems to raise new questions even as they answer old ones. Godel’s¨ theorems (as well as some of the major recursion-theoretic and model-theoretic results we have passed on our way to Godel’s¨ theorems) are a case in point. Several of the new directions of research they opened up will be explored in the remaining chapters of this book. One such question is that of how far one can go working just with the abstract properties (P1)–(P3), without getting involved in the messy details about a particular predicate PrvT (x). That question will be explored in the last chapter of this book.

Historical Remarks

We alluded in passing in an earlier chapter to the existence of heterodox mathematicians who reject certain principles of logic. More specifically, in the late nineteenth and early twentieth centuries there were a number of mathematicians who rejected ‘nonconstructive’ as opposed to ‘constructive’ existence proofs and were led by this

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 Godel’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 Godel¨ proved was that, even using the full strength of P, one cannot prove the consistency of P itself, let alone anything stronger.

In the course of this essentially philosophically motivated work, Godel¨ introduced the notion of primitive recursive function, and established the arithmetization of syntax by primitive recursive functions and the representability in formal arithmetic of primitive recursive functions. But though primitive recursive functions were thus originally introduced merely as a tool for the proof of the incompleteness theorems, it was not long before logicians, Godel¨ himself included, began to wonder how far beyond the class of primitive recursive functions one had to go before one arrived at a class of functions that could plausibly be supposed to include all effectively computable functions. Alonzo Church was the first to publish a definite proposal. A. M. Turing’s proposal, involving his idealized machines, followed shortly thereafter, and with it the proof of the existence of a universal machine, another intellectual landmark of the last century almost on the level of the incompleteness theorems themselves.

Godel¨ and others went on to show that various other mathematically interesting statements, besides the consistency statement, are undecidable by P, assuming it to be consistent, and even by stronger theories, such as are introduced in works on set theory. In particular, Godel¨ and Paul Cohen showed that the accepted formal set theory of their day and ours could not decide an old conjecture of Georg Cantor, the creator of the theory of enumerable and nonenumberable sets, which Hilbert in 1900 had placed first on a list of problems for the coming century. The conjecture, called the continuum hypothesis, was that any nonenumerable set of real numbers is equinumerous with the whole set of real numbers. Mathematicians would be, according to the results of Godel¨ and Cohen, wasting their time attempting to settle this conjecture on the basis of currently accepted set-theoretic axioms, in the same way people who try to trisect the angle or square the circle are wasting their time. They must either find some way to justify adopting new set-theoretic axioms, or else give up on the problem. (Which they should do is a philosophical question, and like other philosophical questions, it has been very differently answered by different thinkers. Godel¨ and Cohen, in particular, arrayed themselves on opposite sides of the question: Godel¨ favored the search for new axioms, while Cohen was for giving up.)

Further Topics