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Results tagged with descriptive-set-theory
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user 23648
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.
10
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1
answer
459
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What is the least $\alpha$ such that $L_\alpha$ contains a non-measurable set
What is the least level of the constructable hierarchy that contains a non-measurable (Lebesgue) subset of $2^\omega$. If it makes a difference assume we are working inside L (V=L).
I'm pretty sure it …
- 3,029
6
votes
0
answers
248
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$\omega$-models of $\mathbf{\Sigma^1_1}-DC$ and $\mathbf{\Delta^1_1}-CA$
So what is needed to demonstrate something (say like $L_{\omega_1^{CK}}[a]$ is a $\omega$-models of $\mathbf{\Sigma^1_1}$-$DC$ or $\mathbf{\Delta^1_1}$-$CA$? It's not like I don't understand what the …
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4
votes
1
answer
568
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Definition of HYP in $L_{\omega_1^{CK}}[a]$?
The structure $L_{\omega_1^{CK}}$ consists of only HYP sets (I believe) and HYP in this structure is the same as the actual hyperaritmetic sets. Now if I move to the structure $L_{\omega_1^{CK}}[a]$ …
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4
votes
1
answer
530
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Complexity of |a| < |b| for ordinal notations?
What is the complexity (e.g. is it $\Sigma^0_1$, arithmetic, fully $\Pi^1_1$) of the relation $|a| < |b|$ given two notations $a, b \in \mathscr{O}$ (Kleene's O)?
What about the case where only one of …
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3
votes
1
answer
130
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A $\Pi^0_1$ class of $\alpha$-subgenerics for all $\alpha < w_1^{CK}$
In Harrington's mimeographed notes (see here) solving McLaughlin's conjecture he builds reals $f \in \omega^\omega$ which have the property of being $\alpha$-subgeneric defined as follows. He does th …
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2
votes
1
answer
116
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$\Pi^0_2$ singleton forming minimal pair with $0''$
Is there a $\Pi^0_2$ singleton that forms a minimal pair with $0''$? That is, is there a set $X$ such that $X$ is the unique solution to $\forall x \exists y \phi(X|_y, x)$, $X$ and $0''$ are incompa …
- 3,029
2
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Hyperarithmetically least elements in $\Pi^1_1$ sets
I'm pretty sure the claim isn't even true for every $\Pi^0_1$ class (working in $\omega^\omega$ or $\Pi^0_2$ if working in $2^\omega$). It's well known that one can produce a recursive tree in $\omeg …
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2
votes
1
answer
139
views
Are Cohen Generics Minimal Covers?
Are Cohen generics (in $2^\omega$) minimal covers?
I'm ultimately interested in this question for some more effective notion of forcing but I realized I wasn't sure how to show this even assuming full …
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1
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Finding 1-generic paths through a tree $T \subseteq 2^{<\omega}$
I don't think you were asking for 1-genericity relative to $T$ but just plain old normal 1-genericity. I’m going to assume $T$ has no terminal nodes since if it doesn't things get more messy (though …
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1
vote
1
answer
94
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Intersection of (relativized/preimage) measure 0 with every hyperarithmetic perfect set
Given a perfect tree $T$ on $2^{<\omega}$ viewed as a function from $2^{<\omega}$ to $2^{<\omega}$ define the measure of a subset of $[T]$ to be the measure of it's preimage under the usual measure on …
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1
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A $\Pi^0_1$ class of $\alpha$-subgenerics for all $\alpha < w_1^{CK}$
So I emailed back and forth with Leo about this and, after the usual part where I drag the conversation into confusion by getting overly specific, I believe I understand how this is supposed to work.
…
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1
vote
0
answers
97
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Name For Effective Cantor-Bendixsonish Derivitive
When dealing with a tree (substring closed subset of $\omega^{< \omega})$ a useful operation will frequently be to remove any nodes with finite ordinal rank (i.e., all nodes whose extensions on the tr …
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0
votes
1
answer
85
views
Kurtz randomness and supermartingales with infinite *limit*
Suppose you replace the usual success conditions for a supermartingale (lim sup is infinite) with the requirement that the actual limit is infinite, e.g. a supermartingale $B$ succeeds on $X \in 2^\om …
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