**5**

votes

**1**answer

304 views

### Failure of Shoenfield's Absoluteness

Shoenfield's absoluteness states that if $M \subseteq N$ are models of $ZF$ and $M \supseteq \omega_1^N$, then every $\Sigma^1_2$ formula with parameters in $M$ is absolute between $M$ and $N$. In ...

**1**

vote

**0**answers

205 views

### Comparing two metrics on the space of infinite sequences and relating open and closed sets

Let $X = \{ 0, 1 \}$ and $X^{\mathbb N_0} = \{ x_0 x_1 x_2 \ldots : x_i \in X \}$ be the space of all infinite sequences, then a metric could be defined on it
$$
d(u,v) := \frac{1}{2^r} \mbox{ with } ...

**2**

votes

**1**answer

175 views

### Particular neighborhoods of analytical sets

Let $X$ be a standard Borel space: a topological space isomorphic to a Borel subset of a complete separable metric space. Denote by $\mathcal P(X)$ the set of all Borel probability measures over $X$ ...

**9**

votes

**1**answer

296 views

### Are there trees for $(\Sigma^2_1)^{\text{uB}}$?

If there is a proper class of Woodin cardinals, then Woodin showed (using stationary towers) that $(\Sigma^2_1)^{\text{uB}}$ statements are generically absolute, where $\text{uB}$ denotes the ...

**16**

votes

**1**answer

616 views

### Meager subspaces of a Banach space and weak-* convergence

I previously asked a version of this question on Math.SE, but didn't receive an answer. (But there is a bounty there if you want to claim it!)
Let $X$ be a Banach space. (If it helps, feel free to ...

**8**

votes

**2**answers

215 views

### Natural $\Pi^1_2$ (or worse) classes of structures?

(To clarify, my interest is mainly lightface, that is, $\Pi^1_2$ instead of $\bf \Pi^1_2$, although it doesn't particularly matter.)
This is just an idle curiosity. In logic, I find myself frequently ...

**13**

votes

**4**answers

546 views

### Continuity on a measure one set versus measure one set of points of continuity

In short: If $f$ is continuous on a measure one set, is there a function $g=f$ a.e. such that a.e. point is a point of continuity of $g$?
Now more carefully, with some notation: Suppose $(X, d_X)$ ...

**7**

votes

**1**answer

227 views

### Is the consistency of $\mathcal{L}_{\infty\omega}$-sentences absolute?

The question is exactly that of the title. Suppose $\varphi\in V$ is an $\mathcal{L}_{\infty\omega}$-sentence, and $W$ is an inner model of $V$ such that $\varphi\in W$. Is the statement
...

**3**

votes

**1**answer

195 views

### Product of Topological Measure Spaces

Def. A Radon measure $\mu$ on a compact Hausdorff space $X$ is uniformly regular if there is a countable family $\mathcal{A}$ of compact $G_\delta$-subsets of $X$ such that for every open set ...

**9**

votes

**1**answer

173 views

### n odd: $\bf\Delta^1_n$ wadge degrees are $< \bf\delta^1_{n+1}$

My adviser is out of town and there is a comment in the Van Wesep paper "wadge degrees and descriptive set theory" that I can't figure out.
Work in ZF+AD throughout.
As stated in the title, the ...

**8**

votes

**1**answer

332 views

### Partition $\Bbb{R}$ into a family of sets each one homeomorphic to the Cantor set

It is known that there is no (nontrivial) partition of $\Bbb{R}$ into a countable number of closed set. But is there a partition of $\Bbb{R}$ into sets, each one homeomorphic to the cantor ternary ...

**3**

votes

**1**answer

497 views

### Different Metrics for Baire Space and their induced Topologies

The Baire-Space is the set of all infinite sequences of integers, i.e.
$$
\mathcal N = \omega^{\omega}.
$$
On this space usually the following metric is given
$$
d(\alpha, \beta) = \left\{ ...

**11**

votes

**0**answers

253 views

### Absoluteness of “$\kappa$-homogeneously Suslin” for sets of reals

What is known about the absoluteness, or lack thereof, of the notion of "$\kappa$-homogeneously Suslin" for sets of reals?
For example, if $A$ is $\kappa$-homogeneously Suslin and $\lambda > ...

**9**

votes

**1**answer

287 views

### Universal $(\Sigma^2_1)^{\text{Hom}_{\mathord{<}\lambda}}$ set

Does anyone know of a reference for the fact that if $\lambda$ is a limit of Woodin cardinals, then the pointclass $(\Sigma^2_1)^{\text{Hom}_{\mathord{<}\lambda}}$ is $\omega$-parameterized? By ...

**9**

votes

**1**answer

307 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

**2**answers

249 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

**1**answer

228 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

**0**answers

222 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 ...

**11**

votes

**2**answers

1k 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

**2**answers

254 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

**1**answer

130 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

**1**answer

193 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

**1**answer

487 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

**1**answer

325 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 ...

**4**

votes

**1**answer

160 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

**1**answer

329 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

**1**answer

237 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

**1**answer

470 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

**1**answer

471 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

**1**answer

144 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

**1**answer

131 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 ...

**5**

votes

**3**answers

445 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

**2**answers

232 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

**2**answers

772 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 ...

**24**

votes

**0**answers

896 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

**1**answer

164 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

**1**answer

360 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

**1**answer

162 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

**0**answers

115 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

**1**answer

171 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

**1**answer

201 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

**1**answer

404 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

**0**answers

180 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

**2**answers

333 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

**0**answers

273 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

**1**answer

497 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

**1**answer

182 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

**0**answers

279 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

**2**answers

327 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 ...

**13**

votes

**3**answers

961 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 ...