Graduate Seminar Topic - Gödel's Theorems

I took this course because it was required and because I didn't like any of the other courses offered. The easiest way to describe the course is that it's an independent study but with a group of students. Everyone picks their own topic and is required to give four half-hour presentations and write four 10-page papers. The topic I picked was Gödel's incompleteness theorems. A topic from what some people call logic and others call metamathematics. It's a great subject that few math majors even know about. I knew about it from the classic book "Gödel, Escher, Bach : an Eternal Golden Braid" by Douglas Hofstadter which was given to me as a Christmas present when it was first published in 1979. Of course I didn't read it then. Heck I was single and who in their right mind reads a 700+ page technical book?

Now I must pick the main thrust of the rest of this posting. I could: 1) describe my presentations and try to teach you Gödel's results 2) give short reviews of the books that I read for the class 3) describe my experience of the class. The last choice might be amusing since the other students' topics ranged far and wide. Some topics were poorly chosen and equally poorly presented. A couple of the topics were very amusing like the construction of magic squares or the history of computational devices including finger counting to 100,000. I don't think that I can really explain the theorems before you get bored so I'll review my sources for the presentations and the papers.

"Gödel, Escher, Bach" or GEB is certainly a classic. Only a portion is about Gödel's theorems, of course. In general it's hard to say what this book is about because it's about so many different things. Many Escher drawings are reproduced and the text explains their significance in context of recursion, self-reference, and self-replication. Bach's fugues are analyzed in a similar way. And the book amusingly explains basic logic and Gödel's theorems. The author develops some fantastic analogies to give you different ways to understand what's going on. He compares the Gödel sentence to a vinyl record that's specifically designed to break the record player that will play it (nowadays, think of an mp3 file that's designed to break the player that will play it). And he invents amusing, yet deeply philosophical, dialogs between Achilles, the Tortoise, and the Crab. And let's not overlook the theme of Zen Buddhism that runs through the book.

My favorite takeaway from this book is a diagram that Hofstadter devised that explains Gödel's theorems visually. Here it is:

Gödel basically proved that a formal system designed to encompass arithmetic cannot prove all arithmetic truths. There are unreachable truths that are beyond the formal system. That's the left side of the picture. Since the book covers Eastern thought, or at least Zen thinking, he included the opposite of formal arithmetic on the right. In that system you start with negative axioms and prove the negation of theorems. But that side too is incomplete since it has unreachable falsehoods.

So now you do know Gödel's results. I bit the bullet and read the entire book (742 pages) and recommend it to anyone (who really, really likes math, logic, programming, or artificial intelligence).

My primary source was "A Profile of Mathematical Logic" by Harold DeLong. This book is a readable history of logic from the ancient Greeks up through the middle of the 20th century. DeLong gives a good treatment of Gödel's theorems adhering very closely to Gödel's research paper that was published in 1931. I "borrowed" the whole progression for my presentations/papers from this book. The scholarship behind this book is superb as the bibliography is 24 pages long. Highly recommended for the philosophy or math lover.

In 1958 a small paperback entitled "Gödel's Proof" was written by Nagel and Newman. This book is highly readable. If you had to read a book on Gödel this should be the one.

Next is a collection of scholarly papers on metamathematics titled "From Frege to Gödel: A Source Book of Mathematical Logic, 1879 – 1931." This collection includes two of Gödel's most famous papers translated into English. One proves that first order predicate calculus is complete. The other that formal arithmetic is incomplete.

My last source was a brand new book "An Introduction to Gödel's Theorems." This is a serious textbook designed for a one or two semester course leading up to the theorems. It is as much fun to read as a standard logic textbook. In fact I found a number of other books like this that were written for a course taught by the book's author.

About Kurt Gödel

To say that Gödel was a genius is an understatement. The originality of his first incompleteness proof is breathtaking. He has been called the greatest logician since Aristotle. Unfortunately he suffered from paranoia and hypochondria most of his adult life. In his 72nd year he literally starved himself to death fearing that his food was poisoned.