I can half-jokingly refer to Gregory Chaitin as my colleague. Actually, he *is* my colleague - we both work for IBM. We're even in the same division of IBM - research. But here the similarity ends. I'm one of those thousands of people in research who just do their work to bring profits to big blue - develop and maintain software that is used throughout the company, and is sold to external customers. Chaitin is one of those whom IBM keeps just to have his name in its ranks !

And for a reason ! Chaitin is one of the pioneers of modern mathematics. His results may be as famous in half a century as Godel's and Turing's results are famous now, and IBM will always gladly brag that he did his research in its Watson center in New York.

I mentioned Godel and Turing for a purpose. Chaitin's research involves some of both, using Turing's ideas (as long as others') to prove results that have a very high correlation with what Godel is most known for - incompletness of formal axiomatic systems.

In the book, Chaitin strolls the readers through a lot of interesting stuff - beginning with prime numbers and various proofs of their infinity, through countable vs. uncountable sets, through doubts of the "reality" of real numbers, proofs that most real numbers are transcendental, Turing's halting problem, algorithmic complexity, the concepts of "elegant" and self-delimiting programs and up to a philosophical discussion of real and random numbers, touching the various problems researchers face with the slippery concept of infinity.

The philosophical discussions are, in fact very interesting. I've never faced the topic of "realness" of real number. The question is - if we can never calculate them fully, are they really real ? Chaitin presents the difference between meaningful reals and random reals. Most reals are meaningless, but some reals are full of meaning, though they're completely random and un-calculable - like his Omega number (the probability that a random program halts).

There are a lot of beautiful (and I'd even admit "exciting") mathematical proofs in the book. The author told about his discontent with Godel's proof of the incompleteness theorem - it's too long and complicated. The proofs presented in the book are mostly short - easy to understand and feel "natural".

The only bad side is the writing style. I'm not closed in on my feelings about it, but Chaitin, it seems to me, is not a very good author. He's a brilliant researcher and writes about exciting things, but his writing style is lacking. Maybe it's just because I recently finished GEB, and Hofstadter's writing is a true piece of art, so Chaitin loses in comparison.

By the way, this is a free ebook you can download from Chaitin's website !