Timeline for Is every class that does not add sets necessarily added by forcing?
Current License: CC BY-SA 3.0
8 events
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| Jan 10, 2013 at 18:22 | comment | added | François G. Dorais | I expected that this would be possible but Ali pointed out that it might not be. Satisfaction classes are examples of $\Pi^1_1$ singletons but I wanted to avoid these since they can add new sets. It would be interesting to know which models have $\Pi^1_1$ singletons. | |
| Jan 10, 2013 at 16:39 | comment | added | Joel David Hamkins | François, do you expect that we should be able to add a $\Pi^1_1$ singleton class over any model of ZFC? My answer seems to illustrate your plan, for models that admit a satisfaction class, since "being a satisfaction class" is even first-order expressible in the language with a predicate for the class. | |
| Jan 10, 2013 at 16:23 | vote | accept | jonasreitz | ||
| Jan 10, 2013 at 12:04 | answer | added | Joel David Hamkins | timeline score: 15 | |
| Jan 10, 2013 at 5:29 | comment | added | Ali Enayat | @Jonas: I deleted my answer since it did not take your requirement that $G$ not be a member of the GB-model. On the other hand, I have an argument that shows your hunch is right if we also insist that the notion of forcing $P$ be Ord-closed in the sense of $M$; and I will try to see if this condition can be removed. | |
| Jan 9, 2013 at 22:58 | comment | added | François G. Dorais | A $\Pi^1_1$-singleton class could not be added by forcing. I haven't worked out how to make this work, but here is the plan... Find a formula $\forall Z \phi(X,Z)$ where $\phi$ only has set quantifiers but $Z$ ranges over classes which provably has at most one solution $X$ but no solution definable by comprehension using only set quantifiers. The unique solution cannot be added by forcing over the model of NBG where all classes are defined by comprehension using only quantification over sets. | |
| Jan 9, 2013 at 21:14 | history | edited | Asaf Karagila♦ | CC BY-SA 3.0 |
added 1 characters in body
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| Jan 9, 2013 at 20:38 | history | asked | jonasreitz | CC BY-SA 3.0 |