This post is intended to accomplish several things at once. First and foremost, I want to explain (not just in the post) why I have been interested in Borel determinacy and in the natural proofs barrier. Roughly speaking (or should I say tl;dr?) I think that Martin’s proof of Borel determinacy has features that might just conceivably offer a way past that barrier.
As long-term readers of this blog will be aware, the P versus NP problem is one of my personal mathematical diseases (in Richard Lipton’s sense). I had been in remission for a few years, but last academic year I set a Cambridge Part III essay on barriers in complexity theory, and after marking the essays in June I thought I would just spend an hour or two thinking about the problem again, and that hour or two accidentally turned into about three months (and counting).
The trouble was that I had an idea that has refused to die, despite my best efforts to kill it. Like a particularly awkward virus, it has accomplished this by mutating rapidly, so that what it looks like now is very different from what it looked like at the beginning of the summer. (For example, at that stage I hadn’t thought of trying to model a proof on the proof of Borel determinacy.) So what am I to do?