Re: Paper and slides on indefiniteness of CH

Dear Sy,

There is a great deal of disconnect between your reaction to Hugh’s letter and the impression I got from his letter, so much so that it feels like we read different letters.

Right at the start, Hugh’s letter has the line “I want to emphasize that what I describe below is just my admittedly very optimistic view” and later it has the line “Now comes (really extreme) sheer speculation.” It is thus clear that the letter is presenting an optimistic view.

Now, on the one hand, you seem to realize it is the presentation of an optimistic scenario — e.g. when you speak of “fantasy” and the difficulty of solving some of the underlying conjectures on which it rests, like, the iterability problem — but then later you switch and write:

You give the feeling that you are appearing at the finish line without running the race. … It gives the false impression that you have figured everything out, while in fact there is a lot not yet understood even near the beginning of your story.

I didn’t get that impression at all and I don’t know how you got it. I got the impression of someone presenting an “very optimistic view”, one that is mathematically precise and has the virtue of being sensitive to mathematical conjectures. ["There are rather specific conjectures which if proved would, I think argue strongly for this view. And if these conjectures are false then I would have to alter my view".] Far from getting the impression of someone who made it look like he was “at the finish line without running the race” I got the impression of someone who had a clear account of a finish line, was working hard to get there, realized there was a lot to do, and even thought that the finish line could disappear if certain conjectures turned out to be false.

(The mathematics behind this is considerable. In addition to the massive amount of work in inner model theory over the last forty years the new work is quite involved. E.g. even the monographs “Suitable Extender Models” 1 and 2 and the monograph on fine structure, alone amount to more than 1000 pages of straight mathematics, which, given my experience with the “expansion factor” in this work, is a misleadingly small number.)

It is remarkable to me that we now have such a scenario in inner model theory, one that is mathematically precise and has mathematical traction in that if certain conjectures turn out to be true one would have a strong case for it. The point is that given the incremental nature of inner model theory, a decade ago no one would have advocated such a view since, e.g., once one reached one supercompact the task of reaching a huge cardinal would not thereby be solved (any more than solving the inner model problem for strong cardinals also solved the inner model problem for Woodin cardinals). But now there has been a shift in landscape — a shift due to mathematical discoveries, showing that in a precise sense one just has to reach one supercompact and that at that point there is `overflow’ — and one can articulate such a scenario in a mathematically precise manner.

It is a virtue of a foundational program if it can articulate such a scenario. A foundational program should be able to list a sequence of conjectures which if true would make a case for the program and which if false would be a mark against the program, and even, in an extreme case set one back to square one. To do this is not to indulge in sheer fantasy. It is to give a program mathematical traction.

I would like to see you do the same for your program. You really should, at some stage, be able to do this. There must be a line of conjectures that you can point to which if true would make a strong case for the program and if false would be a setback; otherwise, it is not open to certification or refutation and one starts to wonder whether it is infinitely revisable and so “not even wrong”. I’m sure you agree. So please tell us whether you are at the stage where you can do that and if you are then I for one would like to hear some of the details (or be pointed to a place where I can find them).

Best,
Peter

P.S. I owe you a response to your request for feedback on one of the points at issue between you and Pen. I’m sorry for not doing that yet. The semester started. I’ll send something soon.

Also, a high bar must be met to send an email to so many people and I doubt I will meet that high bar. I’ll send it here since it was requested here. But eventually I think this should all be moved to a blog or FOM, something where people can subscribe.

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