Timeline for Forcing PFA with ccc forcing
Current License: CC BY-SA 3.0
17 events
| when toggle format | what | by | license | comment | |
|---|---|---|---|---|---|
| Aug 30, 2016 at 6:22 | vote | accept | Miha Habič | ||
| Jul 14, 2016 at 16:07 | answer | added | Paul Larson | timeline score: 13 | |
| Jul 13, 2016 at 19:03 | comment | added | Asaf Karagila♦ | @Paul: That was an off hand suggestion of a way to kill PFA in a "relatively" tame way. Whether or not you can restore PFA after adding a Cohen real without collapsing cardinals, I don't know. I wouldn't be surprised in most "naive" approaches would not work as they might add some robust enough statements. But what happens if you add some real, change some cardinal characteristics, and then "fix it by forcing MA", as Miha suggested? Again, I don't know. I'd expect the answer to still be negative; and if that doesn't work, I wouldn't expect anything to work (so the answer would be negative). | |
| Jul 13, 2016 at 17:17 | comment | added | Paul McKenney | Ah, I misunderstood; I thought you were suggesting adding a Suslin tree by Cohen forcing or something like that. | |
| Jul 13, 2016 at 16:52 | comment | added | Asaf Karagila♦ | @Miha: If anything works, my guess would be that that approach would work. Still seems a bit unlikely. I talked with Assaf Rinot and he pointed that for him, and I guess this should be a key point for the rest of the discussion, the difference between MA and PFA is that PFA kills club guessing. And ccc forcings cannot kill club guessing. So when you break PFA you need to ensure that you haven't added any club guessing principles. What exactly that means for the approach we bring up here? I'm not sure, I'm not particularly knowledgeable about club guessing. | |
| Jul 13, 2016 at 16:46 | comment | added | Miha Habič | @AsafKaragila So you would start with a model of PFA, add a Suslin tree (not via ccc forcing, so maybe using Jech's poset) to kill PFA and then try to resurrect it by ccc forcing? Could it be that if you just force MA and $\mathfrak{c}=\omega_2$ over that you recover PFA? | |
| Jul 12, 2016 at 19:58 | comment | added | Asaf Karagila♦ | @Paul: How? Miha pointed out that you must add some subsets of $\omega_1$ (either reals or Suslin-branches); tci pointed out that forcing with just one Suslin tree cannot force PFA back (and the same can be said about a lot of ccc forcings); I suggested that perhaps doing something like that, but then finding some sufficiently long (read: $\omega_2$-length) iteration of ccc forcings that fix all the problems we caused might work, but that sounds implausible indeed. So I think that Miha's question and my suggested strategy are quite distinct. No? | |
| Jul 12, 2016 at 19:47 | comment | added | Paul McKenney | @Asaf, didn't Miha already address this strategy in his question? | |
| Jul 12, 2016 at 17:44 | comment | added | Asaf Karagila♦ | @tci: But what happens when you force with a long iteration of Suslin tree and other ccc forcings? | |
| Jul 12, 2016 at 17:41 | comment | added | user3462 | @AsafKaragila Forcing with a Suslin tree forces $\mathfrak{t}= \omega_1$. This is Lemma 2 in Farah, 'OCA and towers in P(N)/fin', but there it is mentioned that Booth would have known of this result. | |
| Jul 11, 2016 at 16:43 | comment | added | Asaf Karagila♦ | (And yes, that does seem a bit implausible!) | |
| Jul 11, 2016 at 16:43 | comment | added | Asaf Karagila♦ | The only "nontrivial" that a ccc forcing can really do and not destroy PFA is to shoot branches through Suslin trees. (Or so I think, anyway.) Which is of course consistent with your observation from the paper of Andres and Boban. But it seems really weird if that might happen. E.g. if you add a Cohen real to a model of PFA and somehow revive PFA by shooting branches through the Suslin trees you've added and restoring all sort of cardinal invariants to their rightful size. | |
| Jul 11, 2016 at 14:52 | history | edited | Miha Habič | CC BY-SA 3.0 |
added 126 characters in body
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| Jul 11, 2016 at 14:36 | comment | added | Miha Habič | @PaulMcKenney I've added a reference. | |
| Jul 11, 2016 at 14:36 | history | edited | Miha Habič | CC BY-SA 3.0 |
added 358 characters in body
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| Jul 11, 2016 at 14:22 | comment | added | Paul McKenney | Do you have a reference for the Caicedo-Velickovic result? | |
| Jul 11, 2016 at 13:50 | history | asked | Miha Habič | CC BY-SA 3.0 |