Timeline for Universal $(\Sigma^2_1)^{\text{Hom}_{\mathord{<}\lambda}}$ set
Current License: CC BY-SA 3.0
10 events
| when toggle format | what | by | license | comment | |
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| Aug 20, 2013 at 2:29 | comment | added | Trevor Wilson | Sorry, it's my fault that the original question wasn't clearer. I should probably add some motivation also. | |
| Aug 20, 2013 at 2:28 | comment | added | Rachid Atmai | Ah yes indeed, so I am misunderstanding the whole issue... | |
| Aug 20, 2013 at 2:27 | comment | added | Trevor Wilson | $\omega$-parameterization is what I am trying to show. (I had forgotten the terminology.) | |
| Aug 20, 2013 at 2:21 | comment | added | Rachid Atmai | I was looking at Moschovakis' Book right now and I think the issue about having a non-recursive real to reduce a given set to the universal set seems to be addressed in 3H.1 (The good parametrization theorem), but under the hypothesis that the $\Gamma$ is $\omega$-parametrized. Basically all sets in $\Gamma$ are coded by recursive reals. Is $\Sigma^2_1$ $\omega$-parametrized? (I guess it is because it has to be a Spector pointclass). | |
| Aug 20, 2013 at 1:58 | comment | added | Trevor Wilson | Ok, thanks for your help. I should edit the question to add clarification. | |
| Aug 20, 2013 at 1:49 | comment | added | Rachid Atmai | I don't know if we would or wouldn't need a non-recursive real to reduce a given set to the universal one. | |
| Aug 20, 2013 at 1:38 | comment | added | Trevor Wilson | I wasn't concerned about the difference between $X$ and $\mathbb{R}$ (by which I mean $\omega^\omega$) so much as I was about the difference between $\mathbb{R}$ and $\omega$. That is to say, to reduce another lightface set to this universal (or just complete) lightface set, how do we know we don't need a non-recursive real as a parameter? | |
| Aug 20, 2013 at 1:34 | comment | added | Rachid Atmai | Actually that's a good question. I think in this case instead of a general space $X$ like in the above argument, we need to only look at type 1 spaces, because Wadge's Lemma is formulated for type 1 space (see Moschovakis section 7). So I guess we need to restrict to sets of reals here instead of subsets of a general space $X$ and then the argument would be OK. | |
| Aug 20, 2013 at 1:15 | comment | added | Trevor Wilson | Ok, this might be the better route, to forget about the syntactic notion and to use general pointclass arguments in the derived model. However, I probably should have clarified that I want a lightface $\Sigma^2_1$ subset of $\omega \times \mathbb{R}$ that is universal for subsets of $\mathbb{R}$. Does this argument give such a thing? It would also be enough for my purposes to have a lightface $\Sigma^2_1$ subset of $\mathbb{R}$ from which every other lightface $\Sigma^2_1$ subset of $\mathbb{R}$ can be obtained via recursive substitution. | |
| Aug 20, 2013 at 0:47 | history | answered | Rachid Atmai | CC BY-SA 3.0 |