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9
votes
1answer
287 views

Concerning Silver's result

Jack Silver proved that if $x$ is a real so that every $x$-admissible ordinal is a cardinal in $L$, then $0^{\sharp}$ exists. I wonder whether various weaker or stronger versions of Silver's result ...
5
votes
2answers
230 views

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 ...
7
votes
1answer
223 views

Analytic uniformization

Suppose I am given a subset of $2^\omega\times\omega^\omega$ of some bounded Borel rank. Can I get an analytic uniformization of this set?
5
votes
0answers
203 views

Existence of an universally measurable pullback

Let $X,Y$ and $Z$ be standard Borel spaces: topological spaces homeomorphic to Borel subsets of complete separable metric spaces. Let $K\subseteq X\times Y$ be analytic. Assume that $K_x$ is not ...
9
votes
1answer
647 views

Generalizations of the Tietze extension theorem (and Lusin's theorem)

I am reasking a year-old math.stackexchange.com question asked by someone else. (For my needs every space $X$ and $Y$ will be Polish---that is a completely separably metrizable space.) The Tietze ...
6
votes
2answers
241 views

Borel kernel over an analytic set implies existence of a Borel map

Let $X$ and $Y$ be standard Borel spaces, and let $A\subseteq X\times Y$ be an analytic set with a full projection on $X$: that is $\pi_X(A) = X$. Suppose that there exists a Borel-measurable kernel ...
3
votes
1answer
124 views

Maps that are a.e. equal have almost the same graphs

Let $X$ and $Y$ be two measurable spaces, and let $p$ be a probability measure on $X\times Y$. Denote by $p_X$ the marginal of $p$ on $X$, that is an image of $p$ under projection on $X$. Consider two ...
2
votes
1answer
145 views

Universally measurable map coincides a.e. with a Borel map

Let $X$ be a standard Borel space: that is, a topological space equivalent to a Borel subset of $\Bbb R$. It is known that for any probability measure $p$ on $X$ and any universally measurable set ...
16
votes
1answer
461 views

Question about product topology

Suppose $S\subset\mathbb{R}$ is dense without interior point, and for every open interval $I,J\subset\mathbb{R}$, $I\cap S$ is homeomorphic to $J\cap S$. Is $S\times S$ homeomorphic to $S$? By Luzin ...
10
votes
1answer
244 views

Consistency strength of projective determinacy (PD)

Let PD stand for projective determinacy, and consider the two claims: (1) For each n=1,2,..., Con(ZFC+PD) implies Con(ZFC + there are n Woodin cardinals) (2) Con(ZFC+PD) implies Con(ZFC + there are ...
3
votes
1answer
151 views

Is the ideal of compact operators strongly Borel?

Let $H$ be a separable infinite dimensional Hilbert space. Denote by $\mathcal{B}(H)$ the space of bounded operators on $H$, and $\mathcal{K}(H)$ the ideal of compact operators. When endowed with the ...
5
votes
1answer
242 views

Is the equivalence between a $\Sigma^0_1$ and a $\Pi^0_1$ formula defining the same recursive set provable in a sufficiently strong arithmetic ?

Let $A$ be a recursive set. $A$ is recursively enumerable, so $A$ may be defined by a $\Sigma^0_1$ formula, i.e. by $\exists \overrightarrow{a} \phi (\overrightarrow{a}, n)$, where $\phi$ contains no ...
4
votes
1answer
212 views

From universal measurability to measurability

Let $(\Omega,\Sigma)$ be a measurable space and $K$ be a compact metrizable space endowed with its Borel $\sigma$-algebra $\mathcal{B}(K)$. Let $A\subseteq\Omega\times K$ be universally ...
10
votes
1answer
399 views

Restrictions of null/meager ideal

Let I denote the null (resp. meager) ideal on reals. Is it consistent that for any pair of non null (resp. meager) sets A and B, there is a null (resp. meager) preserving bijection between A and B? In ...
7
votes
1answer
442 views

How long can it take to generate a $\sigma$-algebra?

I want to know if there is a $\sigma$-algebra such that for every countable ordinal $\alpha$ the $\sigma$-algebra can be generated in more than $\alpha$ steps but less than $\omega_{1}$ steps. Given ...
3
votes
1answer
120 views

Borel ideals on $\omega$ are meager?

Let $\mathcal{I}$ be a proper ideal on $\omega$. If $\mathcal{I}$ is Borel as a subset of $2^\omega$, does it follow that $\mathcal{I}$ is meager? Edit: What if $\mathcal{I}$ contains all finite ...
5
votes
1answer
125 views

Basis theorem (due to Solovay?)

I'm finishingg up my bibliography and I'm looking for a reference for the statement that, working in $L(\R)$, the $\Delta^2_1$ sets form a basis for the $\Sigma^2_1$ predicates. I believe that it is ...
3
votes
2answers
317 views

Cohen algebra (generalization)

Let Bor($X$) = class of all borel subsets of $X$. Cohen algebra is defined as Bor(X) modulo the ideal of meager sets. The Cohen algebra has a combinatorial : it is the unique atomless complete ...
4
votes
2answers
195 views

Obtaining conditional probabilities as pushforwards of [0,1]

It is standard that every Borel probability measure on a polish space $X$ can be obtained as pushforward of the uniform measure $\lambda$ on $[0,1]$ along an almost-everywhere-defined Borel-measurable ...
17
votes
2answers
596 views

Image of L^1 under the Fourier Transform

The Fourier Transform $\mathcal{F}:L^1(\mathbb{R})\to C_0(\mathbb{R})$ is an injective, bounded linear map that isn't onto. It is known (if I remember correctly) that the range isn't closed, but is ...
20
votes
0answers
798 views

Godel on recursion-theoretic hierarchies

At the end of his excellent article, "The Emergence of Descriptive Set Theory" (http://math.bu.edu/people/aki/2.pdf), Kanamori writes: "Another mathematical eternal return: Toward the end of his ...
0
votes
1answer
137 views

Product of Baire sigma-algebras

Suppose that $X$ is a Polish space and $\mathcal{E}$ is the $\sigma $-algebra of subsets of $X$ with the property of Baire. Consider the product $\sigma $-algebra $\mathcal{E}\otimes \mathcal{E}$ on ...
4
votes
1answer
294 views

Models of Determinacy

Today we have that $L(\mathbb{R}) \models AD$ (assuming there are $\omega$ many Woodin cardinals and a measurable above them all). I was wondering what other models of $AD$ might look like and if it ...
3
votes
1answer
152 views

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 ...
2
votes
0answers
106 views

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 $\lbrace T_i: i\in\omega\rbrace$. ...
4
votes
1answer
163 views

When do substructures have computable copies?

Say that a class $\mathcal{C}$ of countable first-order structures in some finite signature has the effective substructure property if $\mathcal{C}$ is closed under isomorphism and whenever $A\in ...
9
votes
1answer
183 views

Ensuring nonempty lightface Borel sets have elements via theories of second-order arithmetic

This question is an outgrowth of this MathSE question: http://math.stackexchange.com/questions/276068/members-of-lightface-borel-sets. A Borel set $X\subseteq 2^\omega$ is a member of the smallest ...
4
votes
1answer
375 views

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 $L_{\omega_1^{CK}}[a]$ ...
6
votes
0answers
167 views

$\omega$-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 ...
4
votes
2answers
303 views

Connectedness of the complement of small subsets (extended question)

The following questions occurred to me while browsing this site and looking at Exercise 20 here. Question 1. Let $n>1$. Does there exist a countable dense subset $A\subset\mathbb{R}^n$ for which ...
7
votes
0answers
251 views

Uniform closure of subspaces of Baire class 1

Describe a uniformly closed linear subspace $A \subset C([0,1])$ such that the space $B_1(A)$ is not uniformly complete. Here $B_1(A)$ is the set of all bounded functions $f$ which are pointwise ...
12
votes
1answer
467 views

The Practical Impact of Set-Theoretic Axioms on Measure Theory

The set-theoretic evidence is that we could probably safely add axioms to make many more sets measurable. For example, we could add axioms that would make projective sets measurable. I'm curious ...
6
votes
1answer
156 views

$\Delta^1_2$-well ordering vs $\Delta^1_3$

It is a classical result that if $0^{\sharp}$ exists, then there is a model of $ZFC$ in which there is a $\Delta^1_3$ well ordering of reals but no $\Delta^1_2$-well ordering. My question is: Is ...
2
votes
0answers
247 views

Descriptive set theory on $\mathbb{R}^\mathbb{N}$

The short version of my question is, What is a good source for learning about descriptive set theory on the space $\mathbb{R}^\mathbb{N}$, under the product topology coming from the discrete topology ...
4
votes
2answers
273 views

Is the generalized Baire space complete?

I want to see whether the fact that the Baire space $\omega^\omega$ is a complete (metrizable) space generalizes to $\kappa^\kappa$ being a complete (topological) space. I think this is an easy ...
10
votes
3answers
879 views

Games that never begin

Games that never end play a major role in descriptive set theory. See for example Kechris' GTM. Question: Does there exist a literature concerning games that never begin? I have in mind two ...
7
votes
6answers
759 views

The reals as continuous image of the irrationals

In the wikipedia article about descriptive set theory I read that $\mathbb{R}$ (with its usual topology) is a Polish space, and that every Polish space 1) can be obtained as a continuous image of ...
4
votes
0answers
135 views

Almost universal properties

Suppose that a small category $C$ has sets of objects and arrows that carry the structure of, say, Polish spaces, in some appropriately compatible way. For example $C$ might be an internal category in ...
5
votes
0answers
332 views

Proof of “AD + every set of reals is Suslin” implies AD$_\mathbb{R}$

Could someone point me toward a proof that "ZF + AD + every set of reals is Suslin" (+ $\mathsf{DC}\_\mathbb{R}$?) implies $\mathsf{AD}\_\mathbb{R}$, either with a reference or a hint? I am ...
7
votes
1answer
274 views

Determinacy and definable ultrafilters

It is a simple consequence of AD that there are no non-principal ultrafilters on $\omega$: for $U$ an ultrafilter on $\omega$, consider the game $G_U$ where players I and II play natural numbers $x_0$ ...
10
votes
1answer
228 views

Which forcings preserve (some) determinacy?

The question is exactly as in the title. I'm interested in general in all questions of the form "which forcings preserve property P?" for any P, but determinacy assumptions occupy a special place in ...
8
votes
0answers
467 views

Is every $\sigma$-algebra of sets *abstractly* the Borel algebra of a topology on perhps some other set?

Is every sigma-algebra the Borel algebra of a topology? inspires the present question which asks for less. Question: Given a $\sigma$-algebra ${\cal A}$ on a set $X$, does there exist a topology ...
5
votes
1answer
426 views

A question about Q?

Let A=$\{a_n : n\in \omega \}\subset 2^{\omega\times\omega}$ be nonempty countable without isolated points (i.e. homeomorphic to $\mathbb{Q}$), and satisfy $ \forall n\in \omega \exists^\infty ...
5
votes
0answers
304 views

Closure properties of familes of $G_\delta$ sets.

Given a family of sets $G\subset P(X)$, can one characterize by "closure properties" alone whether or not $G$ arises as the family of all $G_\delta$ for some topology on $X$? some Polish space ...
3
votes
1answer
214 views

Sets of reals and absoluteness

Schoenfield's absoluteness states that if $\phi$ is $\Sigma^1_2$ then $V\models \phi$ iff $L\models \phi$. The set of reals in $L$ is $\Sigma^1_2$ and it is the largest countable $\Sigma^1_2$ set of ...
3
votes
0answers
123 views

Theory of (definable) ideals on a multi-dimensional countable set

I'm wondering if there are any results about definable ideals/filters on $\omega\times\omega$, $\omega\times\omega\times\omega$, etc. To give a sense of the kind of results I might be looking for: ...
5
votes
0answers
197 views

Infinity-Borel sets in ZFC

The notion of an $\infty$-Borel set of reals is useful in the study of AD. Under ZFC it becomes trivial: every set of reals is $\infty$-Borel. However, the notion of an $\infty$-Borel code is still ...
9
votes
0answers
464 views

Existence (or non) of “definable” ultrafilters

This is a question which I suspect has an absurdly easy answer, but I'm not seeing it. Let $\langle\cdot,\cdot\rangle:\omega^2\rightarrow\omega$ be your favorite pairing map (for me, this is the ...
9
votes
1answer
439 views

Constructing an injective reduction of equivalence relations

[Metastuff: I asked this question in a slightly different way on mathSE last week, and it didn't go anywhere, which is why I am asking here. I added the DST tag because it's basically a problem about ...
5
votes
1answer
202 views

Companion of the pointclass of inductive sets

This question is about the notion of a companion for a Spector class, as defined in Moschovakis's book Elementary Induction on Abstract Structures. I am interested in Spector classes on $\mathbb{R}$, ...