Timeline for Obtaining a lightface pointclass from a boldface one
Current License: CC BY-SA 3.0
10 events
| when toggle format | what | by | license | comment | |
|---|---|---|---|---|---|
| Mar 28, 2014 at 21:13 | vote | accept | Trevor Wilson | ||
| Mar 28, 2014 at 19:21 | answer | added | user48871 | timeline score: 4 | |
| Feb 27, 2014 at 3:53 | comment | added | Trevor Wilson | @CarloVonSchnitzel Yes, it does sound like a reasonable idea although it seems like one might need an effective way of finding an index of a $\bf \Gamma$-scale on a given $\bf \Gamma$ set. | |
| Feb 27, 2014 at 3:33 | comment | added | Rachid Atmai | Assuming some amount of determinacy probably, if $\Sigma_1^1(U)$ has the scale property then $\forall^{\mathbb{R}} \Sigma_1^1(U)$ will have the scale property since $\forall^{\mathbb{R}} \Sigma_1^1(U) = \Game \Sigma_1^1(U)$ and the game quantifier propagates scales (using some local determinacy, if I recall correctly this is in chapter 6 of Moschovakis). So maybe using the scale on the universal set $U$ (as $\bf \Gamma$ is inductive-like) one can build a scale for any $\Sigma_1^1(U)$ set. I trying to write this down rigorously. (Also maybe $\Sigma_1^1(U)$ is $\omega$-parametrized). | |
| Feb 27, 2014 at 2:33 | comment | added | Trevor Wilson | I should have said $\Gamma = \forall^\mathbb{R} \Sigma^1_1(U)$ in my last comment, I think, in order to ensure that $\Gamma$ is closed under $\forall^\mathbb{R}$. However the scale property still seems trickier to obtain. | |
| Feb 27, 2014 at 2:24 | comment | added | Trevor Wilson | @CarloVonSchnitzel I think in the definition of $\Sigma^1_1(U)$ one should also require closure under preimages by recursive functions. If so, then it looks to me like ${\bf \Gamma} = \bigcup_{x\in \mathbb{R}} \Gamma(x)$ where $\Gamma = \Sigma^1_1(U)$. Probably $\Sigma^1_1(U)$ is $\omega$-parameterized (although at the moment I have only tracked down the corresponding result that ${\bf \Sigma}^1_1(U)$ is $\mathbb{R}$-parameterized.) However, it is not clear to me whether $\Sigma^1_1(U)$ will have the scale property. | |
| Feb 27, 2014 at 1:47 | comment | added | Rachid Atmai | Let $\bf \Gamma$ be any inductive-like pointclass (say this is on some product space $X$). Let $U \subset \mathbb{R} \times X$ be a universal set for $\bf \Gamma$ subsets of $X$. Define a lightface pointclass: $\Sigma_1^1 (U)$, the least lightface pointclass containing $U$ and closed under $\cup$, $\cap$, integer quantification and $\exists^{\mathbb{R}}$. Can it be that $\bf \Gamma$ = $\cup_{x \in \mathbb{R}} \forall^{\mathbb{R}} \Sigma_1^1 (U)(x)$? Or maybe $\bf \Gamma$ = $\cup_{x \in \mathbb{R}} \exists^{\mathbb{R}} (\Sigma_1^1 (U) \wedge \Pi_1^1 (U))(x)$. These are random guesses. | |
| Feb 26, 2014 at 22:15 | history | edited | Trevor Wilson | CC BY-SA 3.0 |
added 120 characters in body
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| Feb 26, 2014 at 22:09 | history | edited | Trevor Wilson | CC BY-SA 3.0 |
added 120 characters in body
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| Feb 26, 2014 at 21:54 | history | asked | Trevor Wilson | CC BY-SA 3.0 |