1. What is the precise statement of SIMH#(omega_1)?
I am surprised that anybody cares about this! For what it’s worth, it’s just the SIMH# but restricted to the parameter omega_1 (i.e. #-generation plus if a sentence with parameter omega_1 holds in a cardinal-preserving, #-generated outer model then it holds in an inner model). I thought that I had an argument for its compatibility with arbitrary LCs, but I was wrong, and was tricked by the fact that the IMH# (without the “S”) is indeed so compatible.
Thank you for this clarification. Your exchange with Hugh went like this:
I think it is worth pointing out to everyone that , and even the weaker which we know to be consistent, implies that there is a real x such that does not exist.
No, that is not true. The is compatible with all large cardinals. So is the .
I wasn’t aware that you had changed your mind. My impression is that this exchange is what initially generated interest in the exact statement of .
2. Why should we think the study of countable models will shed light on V?
Pen, have you also missed the e-mail that I sent to Geoffrey on 24.September?
Thank you for reminding me of your letter to Geoffrey. I confess I have some trouble digesting this (which is probably why I forgot about it), but perhaps others would care to comment.
PS: I am excited to hear the answer to my AC question:
Is the prevalent view among philosophers of set theory that AC is not derivable from the MIC?
I can only speak for myself. My own view is that AC is at least compatible with the MIC, and that MIC took over from an ill-determined conglomeration of different thoughts about ‘collections’ partly because the extrinsic evidence for Choice (from set theory and from the rest of mathematics, as documented in Moore’s history) forced out any ‘determined by a rule’ or ‘extension of a predicate’-type notion.