Gödel's proof

I first heard of Gödel's proof while reading GEB (I've not finished it yet, to my great chagrin). GEB as you might know is a book that defies genre boundaries. It's popular science, music, art and just beautiful passages and dialogs rolled into one, and then something more. Words (my words, by any count) will fail to catch its essence. But more to the point, I first came to know about Gödel's proof in its introduction. This little book was cited as the main source of inspiration for the author of GEB, Douglas Hofstadter. At that time, I did not take much notice of this seemingly trivial tidbit. But a few days later, I stumbled upon the book and decided to give it a try. The effort was well worth the time. It'a a pinnacle of clear and consise exposition. The main focus is explaining Gödel's incompleteness theorem to the non specialist. In simple words (my own), it reads

For any axiomatic system rich enough to express the arithmetic of whole numbers, there are truths which are not provable/disprovable in the system, which means the system is incomplete. What's more, this incompleteness is inherent to the axiomatic method. Also the proof of consistency of the system cannot be given inside the system.

Gödel proved this by discovering a way to mirror statements about arithmetical truths (so called meta-mathematical statements) into arithmetic (Gödel numbering) and then proceeded to obtain the aforementioned result, all at the young age of 25, in 1931. Considered to be one of the most important advances in Logic since Aristotle, it means that mathematics cannot be axiomatized (and hence meachanized, not at least with current models of computation). Some took this as a sign to despair and some as a way to promote mystical mumbo jumbo. IIRC, Freeman Dyson was one of those taking an optimistic view, and seeing Gödel's theorem as a sign that there'll always be a place for human ingenuity.

In later life, Gödel joined the Institute, where he was a close friend of Einstein.


But the end was not so good. From Wikipedia,

In later life, Gödel suffered periods of mental instability and illness. He had an obsessive fear of being poisoned; he wouldn't eat unless his wife, Adele, tasted his food for him. Late in 1977, Adele was hospitalized for six months and could not taste Gödel's food anymore. In her absence, he refused to eat, eventually starving himself to death.

That aside, it was a privilege to be able to glimpse at one of the most important intellectual advances of the twenteith century. I think this book is a must read.

[Note for AT guys: It's there in the office library].

Enjoy!!