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Results tagged with descriptive-set-theory
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user 3859
Descriptive Set Theory is the study of definable subsets of Polish spaces, where definable is taken to mean from the Borel or projective hierarchies. Other topics include infinite games and determinacy, definable equivalence relations and Borel reductions between them, Polish groups, and effective descriptive set theory.
3
votes
Accepted
Martin-Solovay Tree of Weakly Homogeneous Tree under $\mathsf{AD}_\mathbb{R}$
It is a theorem of $\text{ZF+DC}$ that if $T$ is a weakly homogeneous tree on $\omega\times \kappa$ some $\kappa$ with homogeneity system of measures $\vec{\mu}$ and $ms(T,\vec{\mu})$ is the Martin-So …
- 3,804
2
votes
Sets that are not $\infty$-Borel
This is a nice question Trevor. This is not an answer but it is a bit too long for a comment. If we assume that every set is $\infty$-Borel (say we're just assuming $AD^+$), then there is no proper ($ …
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1
vote
$\Delta^0_{\alpha}$ universal sets does not exist
Hi. Here's the proof I've learned. It does not separate the cases $\alpha=2$ and $\alpha>2$. I hope I am not misunderstanding your question.
Theorem: Let $X$ be an uncountable Polish space. Then for …
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3
votes
Kunen tree and Martin tree
Let me answer my question which I stumbled on while looking at my profile. The answer is yes. Actually assuming $AD$, if $\kappa$ is less than the supremum of the Suslin cardinals then every tree $T$ …
- 3,804
1
vote
Accepted
Absoluteness and Tree Representations
Assuming $\lambda$ is a limit of Woodin cardinals, $\delta_0,...,\delta_n,...$, then the pointclass $Hom_{<\lambda}$ is closed under $\forall^{\mathbb{R}}$. This indeed follows from Martin-Steel.
I …
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3
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Does the boldface class $\Delta^1_2$ have the uniformization property? (assuming $V=L$)
Assuming $V=L$ then we have $AC$ and $CH$, so every set of reals is at most $\aleph_1$ Suslin. So we can find scales for them and uniformize them.In particular every $\Delta^1_2$ set of reals can be u …
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1
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Decomposing $\mathbf{\Pi}^1_1$ sets into closed sets
There is a theorem of my teacher Steve Jackson which says that assuming $ZFC + AD^{L(\mathbb{R})}$ every projective set is $\aleph_{\omega}$-Borel. So in particular this holds for $\Pi^1_1$ sets. The …
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7
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Limits of determinacy on reals
$\text{AD}_{\mathbb{R}}$ is equiconsistent with the existence of a $\lambda$ which is a limit of Woodin cardinals and cardinals which are $< \lambda$-strong. This large cardinal hypothesis is also kno …
- 3,804
1
vote
n odd: $\bf\Delta^1_n$ wadge degrees are $< \bf\delta^1_{n+1}$
Cody, in the 4th paragraph, when you consider a universal $\bf \Pi^1_n$ universal set $U$ (or $\bf \Pi^1_n$-complete like you wrote), for $n$ odd, I think you meant to say that $\phi$ is a norm corres …
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5
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Weakly homogeneous trees under AD
You indeed don't need $AD_{\mathbb{R}}$ and $\kappa < \Theta$ and it is correct this can be weakened. The argument was improved by Woodin : Assume $AD$ and assume that $\kappa$ is less than the suprem …
- 3,804
2
votes
Accepted
Countable $\mathbf\Sigma^1_2$ equivalence relations
We will show the generalization of the Feldman-Moore theorem on countable equivalence relations to the context of thin $\kappa$-Suslin equivalence relations where $\kappa$ is any infinite cardinal. Un …
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4
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Universal $(\Sigma^2_1)^{\text{Hom}_{\mathord{<}\lambda}}$ set
I don't know if this answer will address your specific problem directly, but under AD, Wadge's Lemma implies that every non-selfdual pointclass has a universal set. There is an argument in Jackson's a …
- 3,804
4
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$\omega$ universally Baire sets, tree representations
Concerning your first question, under $AD$, not every pointclass has the scale property. Recall that having a tree representation or as it is called in descriptive set theory, being $\kappa$-Suslin, f …
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1
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A version of the Martin–Solovay tree for $H_\kappa$
Assume $AD+V=L(\mathbb{R})$. Let $\kappa\leq\delta^2_1$ be a regular reliable cardinal (for example a projective ordinal of the form $\delta^1_{2n+1}$) and let $u_{\alpha}^{(\kappa)}$ be the $\kappa^{ …
- 3,804
13
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Accepted
The Axiom of Determinacy and the Banach-Mazur game
The claim is false. The Banach-Mazur game, also known as the $**$-game shows (and is equivalent to) that every set of reals has the Baire property. What is true, as you've noted, is that if one has a …
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