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Results tagged with descriptive-set-theory
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user 160347
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.
2
votes
Proof of the uniform boundedness theorem for analytic subsets of WO
Presuming that $\Sigma^1_1(Z)$ denotes the collection of all boldface-$\Sigma^1_1$ subsets of $Z$, it's basically like the Kunen-Martin theorem. I'm going to use $Y=\omega^\omega$ is Baire space inste …
- 10k
7
votes
Accepted
Uniformization under AD
Here's an argument under the further assumption that $V=L(\mathbb{R})$. Something like this is presumably recorded somewhere. The main point comes from Steel's paper "Scales in $L(\mathbb{R})$", but I …
- 10k
8
votes
Accepted
Existence of Borel uniformization for coanalytic set with non-$K_\sigma$ sections
No.
For let $A\subseteq{^\omega}\omega\times{^\omega}\omega$ be the set of pairs $(x,c)$
such that $c=(f,t)$ where $f\in{^\omega}\omega$ and
$t\in{^\omega}2$ codes the theory of $L_\alpha[x]$ (in the …
- 10k
9
votes
Accepted
Existence of a winning strategy in the $\boldsymbol{\Delta}_2^0$ game
Player I has a winning strategy: First play a singleton $A_0=A_1=\ldots=\{z_0\}$, for some real $z_0$, and the $x_n$'s consistent with $z_0$, until player II plays their first 1, if they ever do. Afte …
- 10k
8
votes
Accepted
Can you fit a $G_\delta$ set between these two sets?
No, not for $\alpha\geq\omega$.
For let $A$ be $G_\delta$ and suppose that WO$_\alpha\subseteq G_\delta$. Let's show that $A\not\subseteq$ WO. Fix a sequence $\left<A_n\right>_{n<\omega}$ of open sets …
- 10k
4
votes
Accepted
Hyperarithmetically least elements in $\Pi^1_1$ sets
I suppose by $a\leq_Hb$, you mean that $a$ is $\Delta^1_1(\{b\})$. And I suppose in the question, $A=X$. Under this interpretation, the answer is no; in fact, there is a $\Delta^1_1$ set $X$ for which …
- 10k
14
votes
Accepted
Detecting uncountable cardinals in $(\mathbb{R};+,\times,\mathbb{N})$
For the first question (distinct regular cardinals $>\aleph_1$): Force ZFC + MA + $2^{\aleph_0}=\aleph_3$ over $L$ in the usual way (see Jech, Theorem 16.13; note the forcing is ccc and it forces MA + …
- 10k
3
votes
Accepted
When is $A$ "$L$-ish" whenever $B$ is "$L$-ish"?
Remarks:
(i) I'm interpreting the definition of $\leq_{L,\mathrm{end}}$ as quantifying over set models $W'$, not proper classes.
(ii) I'm considering the main question (comparing the two orders), par …
- 10k
9
votes
Accepted
How big is the least non-$\Sigma^1_1$-pointwise-definable ordinal?
Claim: Let $\kappa$ be least such that $L_\kappa$ is admissible and
$L_\kappa\models$``$\omega_1$ exists''
and let $\alpha=\omega_1^{L_\kappa}$. Then $\alpha$ is the least
non-$\Sigma^1_1$-pd ordinal. …
- 10k
4
votes
Accepted
Does $\mathit{Aut}(\mathbb{R};+)$ have a copy in $L(\mathbb{R})$ granting large cardinals?
Assuming also CH, the answer to the more general question is yes,
there is a structure in $L(\mathbb{R})$ (in fact, just the set $\mathbb{R}$, with no additional structure), whose automorphism group i …
- 10k
13
votes
How much Dependent Choice is provable in $Z_2$? And what about Projective Determinacy?
Here is a natural model $M$ of $Z_2$ where projective DC fails. Starting with a model of ZFC + $V=L$, force over $L$ with the Levy collapse $\mathrm{Coll}(\omega,{<\aleph_\omega})$ collapsing all ordi …
- 10k
8
votes
Accepted
Inner model for KP and a Well-Ordering of the Reals
Re the first question with KP, yes, $L_{\omega_1^{\mathrm{ck}}}$.
(Use the order of constructibility and Ville's Lemma.)
The second question is a bit vague. But for each ordinal $\alpha<\omega_1^{\ma …
- 10k