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Results tagged with descriptive-set-theory
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user 8133
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.
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When can I "draw" a topology in Baire space?
The motivation for this question is a bit convoluted, so in the interests of conciseness I'm just asking it as a curiosity (and I do find it interesting on its own); if anyone is interested, feel free …
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Open games formed by pasting together infinitely many clopen games
Throughout, I think of games and their underlying trees as the same: so a "clopen game" and a "well-founded tree" mean the same thing.
Fix a sequence of clopen games $\lbrace T_i: i\in\omega\rbrace$. …
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Variously pointed closed sets
A tree $A\subseteq \omega^{<\omega}$ - possibly with dead ends - is pointed iff every path $p\in[A]$ has $p\ge_TA$. This lifts to two distinct notions of pointedness for closed sets in Baire space:
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Ensuring nonempty lightface Borel sets have elements via theories of second-order arithmetic
This question is an outgrowth of this MathSE question: https://math.stackexchange.com/questions/276068/members-of-lightface-borel-sets.
A Borel set $X\subseteq 2^\omega$ is a member of the smallest …
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Reducing largeness notions, uniformly
This question is a particular take on the following theme. Suppose $A$ and $B$ are two notions of "large subset of $\omega^\omega$;" when is there a uniform method for turning an element of $A$ into a …
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Does the boldface class $\Delta^1_2$ have the uniformization property? (assuming $V=L$)
DISCLAIMER: All pointclasses considered here are boldface.
Most of the time, when doing descriptive set theory, we want the projective sets to "behave well;" for example, maybe we don't want there to …
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Perfectly transversable theories
Fix some countable language $\Sigma$, and some reasonable way of interpreting reals as $\Sigma$-structures with domain $\omega$. Let $T$ be a complete $\Sigma$-theory with continuum-many isomorphism t …
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Is the Banach game quantifier "intractable"? (Becker's guess)
(This is a revised version of the original question. Below I work in $\mathsf{ZF+DC+AD}$, but I would be happy to add further axioms if appropriate: $\mathsf{ZF+DC+AD_\mathbb{R}}$, for example, seems …
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Is there a Hausdorff space whose "covering problem" has intermediate complexity?
For a "reasonable" pointclass ${\bf \Gamma}$, say that a second-countable space $(X,\tau)$ is ${\bf \Gamma}$-describable iff for some (equivalently, every) enumerated subbase $B=(B_i)_{i\in\omega}$ we …
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Complexity of winning strategies for open games (for open player)
If $G\subseteq\omega^{<\omega}$ is a computable clopen game, then $G$ has a winning strategy which is hyperarithmetic $(\Delta^1_1)$, by an inductive ranking process. The key observation here is that …
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Conflating reals and sets of countable ordinals "nicely"
It is consistent with ZFC that $2^{\aleph_1}=2^{\aleph_0}$. This can be gotten easily via forcing; more interestingly, it is a direct consequence of forcing axioms (which also set this value at $\alep …
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How much can complexities of bases of a "simple" space vary?
Given a countable subbase of a topology, we can consider its complexity in terms of the difficulty of determining whether one family of basic open sets covers another basic open set. My question is ab …
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2-colorings of the reals
It's easy to prove that, if $\mathbb{R}$ is well-orderable, then there is a 2-coloring of pairs of reals with no uncountable homogeneous set, i.e., there is an $m: [\mathbb{R}]^2\rightarrow 2$ such th …
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Can $\mathbb{C}$ have a "doppelganger" in $L(\mathbb{R})$ with countable automorphism group?
Working in $\mathsf{ZFC}$ + large cardinals (a proper class of Woodins, to be precise), is there a field $F\in L(\mathbb{R})$ such that $V\models F\cong\mathbb{C}$ and $L(\mathbb{R})\models\vert\mathi …
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"Clubiness" of projective sets of ordinals
I'm sure this is just my google-fu failing me, but: what are sufficient, non-overkill large cardinal axioms which guarantee "Every (boldface) $\Pi^1_n$ set of (real codes for) countable ordinals conta …
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