suggest that if one is ever going to study many important phenomena that occur in nature one will also inevitably run into them. But to traditional mathematics they seem uninteresting and quite alien.

As I said above, it is at some level not surprising that questions will be considered interesting in a particular field only if the methods of that field can say something useful about them. But this I believe is ultimately why there have historically been so few signs of undecidability or unprovability in mathematics. For any kinds of questions in which such phenomena appear are usually not amenable to standard methods of mathematics based on proof, and as a result such questions have inevitably been viewed as being outside what should be considered interesting for mathematics.

So how then can one set up a reasonable idealization for mathematics as it is actually practiced? The first step—much as I discussed earlier in this section—is to think not so much about systems that might be described by mathematics as about the internal processes associated with proof that go on inside mathematics.

A proof must ultimately be based on an axiom system, and one might have imagined that over the course of time mathematics would have sampled a wide range of possible axiom systems. But in fact in its historical development mathematics has normally stuck to only rather few such systems—each one corresponding essentially to some identifiable field of mathematics, and most given on pages 773 and 774.

So what then happens if one looks at all possible simple axiom systems—much as we looked, say, at all possible simple cellular automata earlier in this book? To what extent does what one sees capture the features of mathematics? With axiom systems idealized as multiway systems the pictures on the next page show some results.

In some cases the total number of theorems that can ever be proved is limited. But often the number of theorems increases rapidly with the length of proof—and in most cases an infinite number of theorems can eventually be proved. And given experience with mathematics an obvious question to ask in such cases is to what extent the system is consistent, or complete, or both.