By G J Chaitin
The aim of this book is to present the strongest possible version of G¨odel's incompleteness theorem, using an information-theoretic approach based on the size of computer programs.
One half of the book is concerned with studying , the halting probability of a universal computer if its program is chosen by tossing a coin. The other half of the book is concerned with encoding as an algebraic equation in integers, a so-called exponential diophantine equation.
G¨odel's original proof of his incompleteness theorem is essentially the assertion that one cannot always prove that a program will fail to halt. This is equivalent to asking whether it ever produces any output. He then converts this into an arithmetical assertion. Over the years this has been improved; it follows from the work on Hilbert's 10th problem that G¨odel's theorem is equivalent to the assertion that one cannot always prove that a diophantine equation has no solutions if this is the case.
In our approach to incompleteness, we shall ask whether or not a program produces an in nite amount of output rather than asking whether it produces any; this is equivalent to asking whether or not a diophantine equation has in nitely many solutions instead of asking whether or not it is solvable.
If one asks whether or not a diophantine equation has a solution for N dierent values of a parameter, the N dierent answers to this question are not independent; in fact, they are only log2 N bits of information. But if one asks whether or not there are in nitely many solutions for N dierent values of a parameter, then there are indeed cases in which the N dierent answers to these questions are independent mathematical facts, so that knowing one answer is no help in knowing any of the others. The equation encoding has this property.
When mathematicians can't understand something they usually assume that it is their fault, but it may just be that there is no pattern or law to be discovered!
How to read this book: This entire monograph is essentially a proof of one theorem, Theorem D in Chapter 8. The exposition is completely self-contained, but the collection Chaitin (1987c) is a useful source of background material. While the reader is assumed to be familiar with the basic concepts of recursive function or computability theory and probability theory, at a level easily acquired from Davis (1965) and Feller (1970), we make no use of individual results from these elds that we do not reformulate and prove here. Familiarity with LISP programming is helpful but not necessary, because we give a selfcontained exposition of the unusual version of pure LISP that we use, including a listing of an interpreter. For discussions of the history and signi cance of metamathematics, see Davis (1978), Webb (1980), Tymoczko (1986), and Rucker (1987).
Although the ideas in this book are not easy, we have tried to present the material in the most concrete and direct fashion possible. We give many examples, and computer programs for key algorithms. In particular, the theory of program-size in LISP presented in Chapter 5 and Appendix B, which has not appeared elsewhere, is intended as an illustration of the more abstract ideas in the following chapters.