On Incompleteness
Gregory Chaitin has put up on the web his wonderful new book “Meta Math! – The Quest for Omega”:http://www.cs.auckland.ac.nz/CDMTCS/chaitin/omega.html (via “Tim Bray”:http://www.tbray.org/ongoing/When/200x/2004/06/11/Chaitin ). Some random thoughts … apart from the fact that the book itself is brilliant.
1. I’ve always wondered why Gˆdel’s work on Incompleteness and Turing’s work on Uncomputability isn’t as well known (from my purely layman’s perspective of course) as I would imagine it should be. This lack of popularity is strange – in its time, Russel’s paradox and Hilberts challenges were very big deals, and therefore, many thought that Gˆdel’s proof destroyed the raison d’Ítre for pure mathamatics. Chaitin however provides a plausible reason… by the time these theories were catching on, in the late thirties, the world that more serious things to be worried about than the incompleteness of formal systems.
2. Chaitin’s book is a very rare example of “popular science from the horse’s mouth!” .. of an extremely accessible account from the very originator of the theory. (As an aside, wonder why Russians are so prolific in writing amazing popular science books … eg. “Perelman”:http://www.cut-the-knot.org/books/perelman/index.shtml and “Gamow”:http://medlem.spray.se/gamow/georgegamow.html were my favourites when I was a kid, among many other memorable books and authors.)
3. And finally, right at the “beginning”:http://www.cs.auckland.ac.nz/CDMTCS/chaitin/omega.html#ch1 of Chaitin’s book, there are three proofs for infinitely many primes. Read them, chew them, roll them in your head … and if you want to know if there is beauty in mathematics, look no further!