Friday, March 10, 2006

To begin ... a Gödel talk at CUSPOMMS

To blog or not to blog? I'm in two minds. But why not just dive in and see how it goes?

Today was my second outing this academic year to talk to non-philosophers in Cambridge about Gödel, incompleteness and the like. The first time was at a meeting of the Trinity Math. Society. Rather staggeringly, there were more than eighty people there. Perhaps not a brilliantly judged talk, but I did have good fun e.g. telling them about Goodstein's Theorem. (Having a lot of bright mathmos getting the point and smiling at the cheek of the Goodstein proof made a nice change.)

Today's outing was to give a talk at CMS to the slightly unfortunately named CUSPOMMS. A very mixed audience, we meet there approximately fortnightly for talks on the philosophy of mathematics, broadly construed. Rather perversely, I suppose, there was less philosophy than in the Trinity talk. In the event, I was explaining one pretty way of proving (a version of) Gödel's First Theorem without explicitly constructing a Gödel sentence that codes up 'I am unprovable'. The point of doing this is to counteract that familiar tendency to think that the Gödelian result must be fishy because it depends on something too close to the Liar paradox for comfort.

Paul Erdös had the fantasy of a Book in which God records the smartest and most elegant proofs of mathematical results (have a look at the terrific Proofs from the Book by Aigner, Hofmann and Ziegler). So I was aiming to outline one Book proof: a version of the talk can be downloaded here


jafisherton said...

Would you mind briefly describing how you were able to prove "Gödel's First Theorem without explicitly constructing a Gödel sentence that codes up 'I am unprovable'". I am very interested to see how you do this. Thanks!

Peter Smith said...

Essentially the argument of Ch.6 of my Gödel book! (The book also contains a number of other proofs that don't directly depend on the construction of Gödel sentences.)