Timeline for Can we always permute Cohen reals?
Current License: CC BY-SA 3.0
12 events
| when toggle format | what | by | license | comment | |
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| Nov 12, 2013 at 18:20 | comment | added | Asaf Karagila♦ | Andreas thanks for the answer, although earlier today I realized that my situation is in fact much simpler than this case and solved it. Thanks for the answer, though! | |
| Nov 12, 2013 at 18:15 | comment | added | Andreas Blass | @AsafKaragila So your $\dot c$ is not just some real but the generic real directly added by the forcing. You also seem to be taking the forcing to be just the set of finite partial functions $\omega\to2$, not the completion mentioned in the last sentence of the question. Now if the automorphism $\pi$ is in $L$, then $L[c]=L[\pi c]$, but if not 9so $\pi$ is in the ground model but not in $L$) then I would think one could code non-constructible information into $\pi$ and thus make the models different. | |
| Nov 11, 2013 at 21:02 | comment | added | Asaf Karagila♦ | By canonical name I meant $\{(p,\check n)\mid p(n)=1\}$, of course. | |
| Nov 11, 2013 at 20:51 | comment | added | Andreas Blass | @Asaf What exactly do you mean by "canonical"? I know of canonical names for ground model reals, and for the generic real directly added by the forcing, and for some other specific reals (or, rather, descriptions of reals), but I don't see a clear notion of canonical names for just any real in the extension. | |
| Nov 11, 2013 at 18:53 | comment | added | Asaf Karagila♦ | I have a related question which I don't think merits a whole new thread. Let $\dot c$ be the canonical name for a real. Can we find a permutation $\pi$ of the forcing such that $\Bbb R^{L[c]}$ and $\Bbb R^{L[\pi c]}$ are different? | |
| Sep 28, 2013 at 12:55 | vote | accept | Asaf Karagila♦ | ||
| Sep 28, 2013 at 12:53 | comment | added | Andreas Blass | @AsafKaragila I've just edited my answer to address the edited version of the question. The answer is still negative, for essentially the same reason. | |
| Sep 28, 2013 at 12:51 | history | edited | Andreas Blass | CC BY-SA 3.0 |
added answer to new version of the question
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| Sep 28, 2013 at 7:56 | comment | added | Asaf Karagila♦ | Sorry for the very late edit, but it arrived. | |
| Sep 27, 2013 at 21:51 | comment | added | Asaf Karagila♦ | Right, I was just thinking about adding a clause regarding this. But posting and editing from your phone is really annoying. I apologize for the trouble. although it seems like the idea can be generalized further to the interesting case. | |
| Sep 27, 2013 at 21:47 | comment | added | Andreas Blass | Addendum: Not even if both $\dot x$ and $\dot y$ are Cohen-generic over the ground model will they necessarily be connected by an automorphism of the forcing. For example, $\dot x$ might generate your whole Cohen extension, while $\dot y$ generates some intermediate submodel. | |
| Sep 27, 2013 at 21:45 | history | answered | Andreas Blass | CC BY-SA 3.0 |