Timeline for Are there trees for $(\Sigma^2_1)^{\text{uB}}$?
Current License: CC BY-SA 3.0
13 events
| when toggle format | what | by | license | comment | |
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| May 5, 2015 at 18:14 | vote | accept | Trevor Wilson | ||
| Jun 4, 2014 at 14:46 | answer | added | Trevor Wilson | timeline score: 4 | |
| S Dec 2, 2013 at 23:57 | history | bounty ended | CommunityBot | ||
| S Dec 2, 2013 at 23:57 | history | notice removed | CommunityBot | ||
| S Nov 24, 2013 at 21:58 | history | bounty started | Trevor Wilson | ||
| S Nov 24, 2013 at 21:58 | history | notice added | Trevor Wilson | Draw attention | |
| Nov 20, 2013 at 0:57 | comment | added | Trevor Wilson | Also, note that for any given $\lambda$ the tree $T_{\varphi,\lambda}$ is a set, so in sufficiently large generic extensions $V[g]$ it is countable and its projection is analytic and therefore too simple to be the desired $(\Sigma^2_1)^{\text{uB}_\lambda}$ set. But one approach would be trying to show that this analytic set (the projection) is always contained in the desired $(\Sigma^2_1)^{\text{uB}_\lambda}$ set. I have no idea whether this is true. (The analogous containment is true if you consider various sizes of Shoenfield tree for $\Sigma^1_2$, so maybe there is hope.) | |
| Nov 20, 2013 at 0:51 | comment | added | Trevor Wilson | ...but I don't know of any general construction of a tree whose projection is the limit (or lim sup or lim inf, if we don't want to assume that the limit exists) of the projections of a given uncountable sequence of trees, however. | |
| Nov 20, 2013 at 0:49 | comment | added | Trevor Wilson | @CarloVonSchnitzel Thanks. Fixing any particular generic extension $V[g]$, the trees $T_{\varphi,\lambda}$ for sufficiently large $\lambda$ all have the correct projection. (This is because in $V[g]$ we have $\text{uB}_\lambda = \text{uB}$ for all sufficiently large $\lambda$. Unfortunately the meaning of "sufficiently large" depends on $g$.) So if there were a tree whose projection in any generic extension was the limit of the projections of the trees $T_{\varphi,\lambda}$ as $\lambda \to \text{Ord}$ then the answer to my question would be "yes". | |
| Nov 20, 2013 at 0:39 | history | edited | Trevor Wilson | CC BY-SA 3.0 |
typo
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| Nov 19, 2013 at 23:51 | comment | added | Rachid Atmai | Is there a way to define "bigger and bigger" derived models for larger and larger $\lambda$ limits of Woodin cardinals and look at the corresponding trees? Along the way the derived models would have to cohere in some specific way as to ensure that the successive projections of the trees agree. Basically more and more statements would have to be verified. The final tree could be a lim inf of the construction. This just a quick guess which might turned out to be naive. In the 5th line there is a small typo: you meant to write "where $UB_{\lambda}$ denotes". Nice question by the way. | |
| Nov 19, 2013 at 19:15 | history | edited | Trevor Wilson | CC BY-SA 3.0 |
added 247 characters in body
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| Nov 19, 2013 at 19:05 | history | asked | Trevor Wilson | CC BY-SA 3.0 |