Timeline for Number of transitive models of set theory
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11 events
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| Jun 15, 2020 at 7:27 | history | edited | CommunityBot |
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| Nov 18, 2012 at 7:09 | vote | accept | Andrés E. Caicedo | ||
| Sep 21, 2010 at 23:27 | answer | added | Amit Kumar Gupta | timeline score: 0 | |
| Sep 17, 2010 at 20:53 | comment | added | Andrés E. Caicedo | Joel, about which construction methods I would expect to play a role, here is one possibility: Say from your assumptions you manage to define a set $A$ of codes for models, and that $A$ is of low complexity (descriptive set theoretically). Say also that `not many' codes correspond to the same model. If you manage to argue that $A$ is uncountable, then this should give you a "perfect set" of different models. (Roughly, I would expect a solution to question 3 would go along these lines. It is also why I think the key question is 1.) | |
| Sep 17, 2010 at 16:12 | answer | added | Andreas Blass | timeline score: 16 | |
| Sep 15, 2010 at 0:30 | comment | added | Andrés E. Caicedo | Hi Dave. Yes, you are correct. We could assume $T$, the theory ZFC+V=L+there is only one $\alpha$ with $L_\alpha\models$ZFC. In $T$, I do not see any way of proving there are two transitive models of set theory other than by appealing to a forcing argument. So unless we come up with something very very ingenious, we would like some additional assumption that ensures the universe is not tiny. However, perhaps there is already a clever argument in $T$+there is a countable transitive model of $V\ne L$ that gives us infinitely many such models. (We know via forcing that there are continuum many.) | |
| Sep 15, 2010 at 0:07 | comment | added | Joel David Hamkins | Dave, that model has continuum many transitive models, since it can see the minimal model, and therefore knows that the minimal model is countable, and so has a perfect set of forcing extensions of it. | |
| Sep 14, 2010 at 22:43 | comment | added | Dave Marker | Without some kind of large cardinal assumption this seems unlikely as you could be in a model of ZFC+V=L where there are few transitive models of ZFC. Like if $\beta$ is the second countable ordinal where $L_\beta\models$ ZFC. Does this make sense? | |
| Sep 14, 2010 at 21:06 | comment | added | Andrés E. Caicedo | Hi Joel. I think that is part of the question, I am not sure what methods we have for dealing with these problems, other than forcing. Of course, it is not that there are any issues with forcing, or any inherent difficulties to it (proof theoretic or otherwise). However, it is puzzling to me that I do not see how to show 1, for example, without some forcing construction used somewhere. I would like to tell you "descriptive set theoretic methods" or somesuch, but I don't want to overlook something valuable by adding this restriction. | |
| Sep 14, 2010 at 19:58 | comment | added | Joel David Hamkins | Andres, I'm not quite clear on which construction methods you are allowing here; after all, the forcing arguments are not so difficult... | |
| Sep 14, 2010 at 19:24 | history | asked | Andrés E. Caicedo | CC BY-SA 2.5 |