On Saturday, I unfortunately missed Ulrich Kohlenbach’s talk, since the banquet the night before went a little long and I overslept. That was a pity, since I really like and admire his work. The second talk was by Harvey Friedman, “My 40 Years on His Shoulders”. He as the title suggests, he gave a survey of his work on finding mathematically “natural” or “interesting” statements which are independent of strong theories. The three areas he focussed on was the theory of well quasi-orders (Kruskal’s theorem, the graph minor theorem), Borel selection theory, and Boolean relation theory. If you’ve followed Harvey’s posts on FOM, this was familiar territory. It was too bad that there wasn’t more of a Friedman-MacIntyre discussion at the end. Then came the session on set theory: first, Paul Cohen reminisced about how he proved the independence of CH and his relationship to Gödel. He related how his interest in number theory as a graduate student at Chicago led him to think about developing a decision procedure for Diophantine equations. His colleagues there mentioned that what he tried to do onflicted with results of a certain Gödel–and this was confirmed by Kleene, when he gave a talk at Chicago during that time. This led Cohen to read Gödel’s paper. He mentioned the story that circulates about how he [Cohen] allegedly asked Hartley Rogers in Princeton around 1958 what the hardest open problem in logic was, and that Rogers said it was the independence of CH–which Cohen then set to prove. Cohen said it rings a dim bell, but even if it hadn’t happend the way it’s been told, it would still continue to be told that way, because it’s a good story. In any case, he attempted to read the Princeton monograph on the consistency of CH, but said he didn’t understand it. He also said that he once asked Gödel if he had proved the independence of AC, and that Gödel had said, yes. Asked by what method, Gödel replied, “Well, it was somewhat similar to yours.” At that time, Cohen reported, Gödel already seemed quite exhausted, and never wanted to talk about details of proofs. Cohen did want to debunk one claim made about him: that he at one point attempted to show that arithmetic was inconsistent (claimed, e.g., by Anil Nerode here)–he tried to find a decision procedure for Diophantine equations, which there isn’t (see above). Dana Scott was up next. He gave an explanation for why Gödel’s Princeton monograph is so hard to read: According to Kreisel, Gödel began the lectures upon which the monograph was based with a longish introduction where he explained the idea of the proof–but the notetaker though it was just chit chat and didn’t write it down! Scott’s talk was technical, on parametric sets and geometry, and I didn’t quite follow it, I must admit. The third talk in the set theory session was Hugh Woodin on the large cardinal program. That was exciting and interesting, but I don’t trust my notes enough to confidently report what he said. I think I’ve heard him give something like it at Berkeley already, so maybe there’s a written version of it somewhere? In any case, here’s a way to get a job in the Berkeley math department: Hugh bet that in the next 10,000 years no-one will prove ZFC + “there are infinitely many Woodin cardinals” inconsistent. If he loses, he will resign his job and insist that whever finds the contradiction be appointed in his place. In the afternoon, Avi Wigderson gave a talk on complexity, randomness, and game theory. It’s up at his website. The last event of the meeting was Roger Penrose’s public lecture at City Hall. The place was packed–probably 600 people or more. Since it was aimed at a general audience, there wasn’t much detail. As far as I could tell, there wasn’t that much new, either. Funnily enough, he seemed not to remember his own argument against strong AI using Gödel’s theorems. Penrose’s talk was followed by a screening of Jimmy Schimanovich and Peter Weibel’s movie about Gödel’s life and work, “Kurt Gödel–Ein mathematischer Mythos.” The End.