Timeline for What ccc forcings add a Suslin tree?
Current License: CC BY-SA 3.0
13 events
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| Apr 13, 2017 at 12:58 | history | edited | CommunityBot |
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| Oct 13, 2016 at 10:33 | answer | added | Stefan Hoffelner | timeline score: 8 | |
| Jul 15, 2016 at 1:15 | comment | added | Ashutosh | @MihaHabič Tennenbaum's poset for adding a Suslin tree is same as adding $\aleph_1$ Cohen reals. | |
| Jul 14, 2016 at 12:38 | comment | added | Asaf Karagila♦ | @Stefan: That is a plausible idea. Assaf Rinot mentioned this might be a consistent negative answer. This is also why there's that last comment about a consistency result when assuming no Suslin trees exist. | |
| Jul 14, 2016 at 12:32 | comment | added | Stefan Hoffelner | How about the following idea: Let $T$ be a Suslin tree such that forcing with its nodes as conditions will not only kill the Susliness of $T$, but will introduce a fresh Suslin tree $S$ to the universe. (I think such $T$ should exist if we twist the usual '$\diamondsuit$ yields Suslin tree' construction). This would give us a negative solution to the question as the forcing does not add any reals at all. | |
| Jul 13, 2016 at 15:13 | comment | added | Miha Habič | @MohammadGolshani I don't know whether Tennenbaum's poset for a Suslin tree always adds a Cohen real, but it can consistently do so. Specifically, by a theorem of Pawlikowski, any poset of size $<\mathbf{add}(\mathcal{M})$ that adds a real adds a Cohen real. So over a model of MA where CH fails, for example, Tennenbaum's poset will add a Cohen real. | |
| Jul 13, 2016 at 7:37 | comment | added | Mohammad Golshani | Another possibility is to make a look at Tennenbaum's forcing for adding a Souslin tree. It is $c.c.c,$ but it may add a Cohen real (I did not check it). | |
| Jul 13, 2016 at 7:35 | comment | added | Mohammad Golshani | By a result of Laver, if MA holds, then adding a Random real does not add a Souslin tree. But maybe one can show the consistency of "there are no Souslin trees and adding a random real adds a Souslin tree"? If so, we can get a consistent negative answer to your question! | |
| Jul 13, 2016 at 3:38 | comment | added | Asaf Karagila♦ | Yeah, that would be sort of a cop out, I guess. Unless of course the random real defines a truly fresh Suslin tree. | |
| Jul 13, 2016 at 0:56 | comment | added | user3462 | About your choice of the word 'add', it seems conceivable that if we are in a model with a Suslin tree, then adding a random real (say) gives you a Suslin tree which is not in the ground model. But I guess this does not give a satisfactory answer to your questions? | |
| Jul 11, 2016 at 16:58 | history | edited | Asaf Karagila♦ | CC BY-SA 3.0 |
added 171 characters in body
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| Jul 11, 2016 at 16:49 | comment | added | Asaf Karagila♦ | Along with Miha's question, it seems that today is "ccc day"! :-P | |
| Jul 11, 2016 at 16:48 | history | asked | Asaf Karagila♦ | CC BY-SA 3.0 |