Timeline for A “paradox” about the inner model problem
Current License: CC BY-SA 4.0
11 events
| when toggle format | what | by | license | comment | |
|---|---|---|---|---|---|
| Nov 9, 2020 at 18:24 | comment | added | Monroe Eskew | But would such generic ultrapowers really be thought of as canonical models? | |
| Nov 9, 2020 at 13:52 | comment | added | Gabe Goldberg | Sorry for the delay. If you take a stationary tower ultrapower of a countable mouse, adding a real that is not OD, then it contains a mouse as a countable elementary substructure (the original mouse), but it is not a mouse. Something similar can be done with proper class inner models if you start with a proper class mouse $M$ with a Woodin cardinal $\delta$ such that $\delta^{+M}$ is countable. | |
| Nov 3, 2020 at 18:04 | comment | added | Monroe Eskew | To clarify for (4), I was wondering if you could point me to an example where someone discusses constructing a premouse like you describe, which is not an iterate of a mouse but has a mouse as a countable elementary substructure. | |
| Nov 3, 2020 at 15:51 | comment | added | Gabe Goldberg | Sorry, what you mean by “such usage?” I’m not offended, but you’d have to ask an inner model theorist. The Inner Model Problem is definitely not a precise problem (not sure whether that means it’s not mathematical), but the test questions are precise, so there are clearly defined goals. Even the IMP is not as vague as your characterization suggests: there are lots of precise criteria people hope the models will satisfy, and if a model were proposed, I bet there would be general agreement on whether it solves the problem. At least this has been true in the past. | |
| Nov 3, 2020 at 14:46 | comment | added | Monroe Eskew | Thanks for your responses. On (4), can you give an example of such usage? Regarding (3), my concern is that it becomes not really a mathematical problem per se, but just a kind of label for an area of research, whose goals are basically to keep doing the kind of thing they’re doing, and whose particular concrete objectives shift as needed. (Is that offensive? I think it characterizes most math research actually.) | |
| Nov 3, 2020 at 13:53 | comment | added | Gabe Goldberg | (1) No, I meant just that when people say canonical inner model, they include all Mitchell-Steel models. (2) The WEM result does say something new about inner models at the level of supercompact cardinals, but first of all, that was about 15 years ago, and second it did not rule out using the same basic methodology (comparison of backgrounded extender models via some kind of iteration trees) in the construction. (3) Yes, although then you’d have to say what has already been done. (4) No it is not internally expressible, although I still don’t think it will meet all your canonicity criteria. | |
| Nov 3, 2020 at 9:58 | comment | added | Monroe Eskew | Is being a Mitchell-Steel proper class premouse with all countable $\Sigma_1$-elementary substructures iterable internally expressible? (Or do we want to consider countable subsets and iteration strategies that are not members?) If so, this would seem to fall prey to the anti-canonicity observations linked above. | |
| Nov 3, 2020 at 7:51 | comment | added | Monroe Eskew | Three questions. (1) Are you saying that it is plausible that all large cardinals can exist in Mitchell-Steel premice whose $\Sigma_1$ hull is iterable? (2) Do you not think that the rigidity result about weak extender models already says something in the direction of “our current conception must be drastically altered”? (3) Is it fair to say reword the Inner Model Problem as, “Develop analogues to what we’ve already done for larger cardinals”? | |
| Nov 3, 2020 at 0:15 | history | edited | Gabe Goldberg | CC BY-SA 4.0 |
added 96 characters in body
|
| Nov 2, 2020 at 22:14 | comment | added | Andrés E. Caicedo | I think one could add that our understanding of canonicity gets revised as we climb up. This is perhaps tautological but deserves discussing precisely because our understanding can radically change our goals. For instance, before reaching Woodin cardinals, common wisdom was that canonical inner models had a $\Sigma^1_3$ well-ordering of their reals. The issue was that we really did not understand iterability well enough at that point, and the missing ingredient (iteration trees) had not manifested itself yet. | |
| Nov 2, 2020 at 20:28 | history | answered | Gabe Goldberg | CC BY-SA 4.0 |