Timeline for Universally Baire Tree Representation of Projective Sets
Current License: CC BY-SA 3.0
17 events
| when toggle format | what | by | license | comment | |
|---|---|---|---|---|---|
| May 1, 2015 at 4:47 | vote | accept | William | ||
| Apr 30, 2015 at 21:38 | answer | added | Trevor Wilson | timeline score: 5 | |
| Apr 30, 2015 at 20:18 | comment | added | Trevor Wilson | Okay, I will write an answer. | |
| Apr 30, 2015 at 20:17 | comment | added | William | @TrevorWilson Okay I think what you mentioned above may be what I am looking for. I wanted to know whether under any large cardinal assumption, can every $\Pi_3^1$ set have trees which continue to represent it in all generic extensions. In fact, for what I want to do, I do not really need trees that form the universally Baire pair, but just that there is a tree that continue to represent the formula in all generic extensions. Could you elaborate on the your comments and give some references, perhaps as an answer? | |
| Apr 30, 2015 at 19:51 | comment | added | Trevor Wilson | I think the editing makes the question less clear. Are you assuming large cardinals now? If there is a proper class of Woodin cardinals then the answer is yes, but if your assumption is just that $\Pi^1_3$ sets are universally Baire (as in the original question) then I think it may be open. Also, instead of "$1_\mathbb{Q} \Vdash_\mathbb{Q}$... the same two statements" did you mean "$1_\mathbb{Q} \Vdash_\mathbb{Q} p[\check{T}] = \mathbb{R} \setminus p[\check{U}]$"? | |
| Apr 30, 2015 at 19:40 | comment | added | Yizheng Zhu | @William Yes. See Steel's online notes "the derived model theorem". | |
| Apr 30, 2015 at 18:30 | history | edited | William | CC BY-SA 3.0 |
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| Apr 30, 2015 at 18:28 | comment | added | William | @YizhengZhu What do you mean by "It is true under large cardinals"? Do you mean that there exists some tree $T$ such that $V \models A = p[T]$ and in all generic extensions $A = p[T]$? | |
| Apr 30, 2015 at 18:26 | comment | added | William | @AsafKaragila I have elaborated further on what "continues to represent" should mean. | |
| Apr 30, 2015 at 18:23 | history | edited | William | CC BY-SA 3.0 |
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| Apr 30, 2015 at 17:02 | comment | added | Trevor Wilson | @Asaf I think that "$T$...continues to represent a given $\Pi^1_3$ set in all generic extensions" means (by analogy with the first paragraph) that one continues to have $A = p[T]$ in all generic extensions where $A$ is defined by the given $\Pi^1_3$ formula. The idea of restricting one's attention to a particular forcing notion $\mathbb{P}$ does not appear until the very last sentence (and in my opinion is peripheral to the main question.) | |
| Apr 30, 2015 at 16:51 | comment | added | Asaf Karagila♦ | @Trevor: It might be the underlying issue allowing us to prove this for the case of $\mathbf\Pi^1_2$; but this is really the question as far as I read it. | |
| Apr 30, 2015 at 16:36 | comment | added | Trevor Wilson | @Asaf I think that "for some forcing" versus "for every forcing" is not the main point, but rather whether the way of extending the set to a generic extension using trees agrees with the way of extending using the syntactical definition. | |
| Apr 30, 2015 at 10:27 | comment | added | Asaf Karagila♦ | If I understand correctly, you are asking the following: UB sets are such that for every forcing there are two trees whose branches in the generic extension form the UB set and its complement. For $\mathbf\Pi^1_2$ we can reverse the quantifiers and find trees fitting for any forcing. Can we do it for higher projective sets, assuming these sets are UB? | |
| Apr 30, 2015 at 9:51 | comment | added | Yizheng Zhu | It is true under large cardinals. So the question is asking if the assumption can be weakened to universal Baireness. Do I understand the question correctly? | |
| Apr 30, 2015 at 6:16 | comment | added | Trevor Wilson | As far as I know (for what it's worth) these questions are still open. If I'm wrong, I'd be very interested to hear about it. | |
| Apr 30, 2015 at 5:00 | history | asked | William | CC BY-SA 3.0 |