From: Mike Oliver Subject: Re: large cardinals Date: Wed, 19 Jan 2000 14:56:37 -0800 Newsgroups: sci.math Summary: [missing] Allan Adler wrote: > > Mike Oliver writes: > > > Woodin's P_max > > What is it? Wish I could tell you more about it. I attended one colloquium where he talked about it; in one hour he probably got through 40 slides, each of which might have taken a good half-hour to really understand. What I can tell you is that he comes up with a model in which true Pi^2_1 sentences are maximized (CH, being Sigma^2_1, is therefore false in the model). It has some sort of stability under forcing, reminiscent of the way in which, once L(R) |= AD, the theory of L(R) can no longer be changed by forcing. Now the latter strikes me as a good reason to believe that AD in fact holds in L(R) (there's a sense that you've reached the end of the line for L(R), and it makes sense that there should be such an end because of the restricted way in which sets may be formed in L(R) ). So I am disposed to believe that an approach *like* P_max could plausibly tell us whether we ought to believe CH. Now, that's different from saying "We ought to believe 2^Aleph_0=Aleph_2 because of P_max". For that, the details of the argument obviously matter, and they are far beyond me at the moment; if I wanted to understand them I would have to devote quite some time to it.