**5**

votes

**3**answers

468 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

240 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

822 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 ...

**28**

votes

**0**answers

942 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

178 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

379 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

164 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

172 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

204 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

409 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

183 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

346 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

277 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

507 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

188 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

286 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

341 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

967 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

**6**answers

918 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

**0**answers

140 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

**0**answers

353 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

**1**answer

309 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 $x_0$ ...

**11**

votes

**1**answer

297 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 ...

**11**

votes

**0**answers

515 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

**1**answer

437 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

**0**answers

366 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

**1**answer

243 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

**0**answers

129 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

**0**answers

223 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 ...

**10**

votes

**0**answers

516 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

**1**answer

457 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

**1**answer

211 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 $\mathbb{R}$, ...

**4**

votes

**1**answer

236 views

### sigma-algebra generated by OD sets

Assume $V=L(\mathbb{R})$ and the Axiom of Determinacy. Is every set of reals generated by ordinal-definable sets of reals under the operations of countable union and intersection?
The class of sets ...

**6**

votes

**2**answers

325 views

### Weakly homogeneous trees under AD

If AD$_\mathbb{R}$ holds and $\kappa < \Theta$ then every tree $T$ on $\kappa$ is weakly homogeneous (Martin–Woodin, "Weakly homogeneous trees.") I recall hearing that the hypothesis can be ...

**6**

votes

**0**answers

344 views

### Sets of reals amenable to each L[x]

If $A$ is a set of reals such that $A \cap L[x] \in L[x]$ for each real $x$, is there a real $z$ such that $A \cap L[x] \in OD_z^{L[x]}$ for a cone of $x$?
This can be proved under the Axiom of ...

**3**

votes

**1**answer

225 views

### Wadge Determinacy of some related classes.

Let $A, B \subset \omega^\omega$. The Wadge Game $(A, B)$ is played as followed: Player I plays $a_0 \in \omega$. Then Player II plays $b_0 \in \omega$. Then Player I choose $a_1 \in \omega$; ...

**0**

votes

**1**answer

184 views

### Trees on $\omega$

I am going to give a construction of a tree on $\omega$ that at first appears as though it is well founded. However, this tree cannot be well-founded because, using the rank function on finite ...

**7**

votes

**0**answers

148 views

### How about the Wadge rank of A union B when A and B are Wadge equivalent?

Hello,
We know that A is Wadge reducible to B if there is a continuous map $f$ such that A is the preimage of B via $f$, and the Wadge order is defined by $A\leq_{w}B$ if $A$ is Wagde reducible to ...

**7**

votes

**3**answers

880 views

### About the axiom of choice, the fundamental theorem of algebra, and real numbers

About fundamental theorem of algebra, there is a large collection different demonstrations.
I ask: is there some proof that avoids AC (choice axiom)?
In a general topos (with natural number object) ...

**4**

votes

**1**answer

277 views

### Indeterminacy of long games

Hello, all,
Several months ago I sat in on a seminar on AD+, which was incredibly wonderful even though I could barely follow it at all. AD+ is a technical variant of AD, the axiom of determinacy, ...

**4**

votes

**2**answers

298 views

### Is the Turing equivalence relation the orbit equiv. relation of the action of a countable group?

The Turing equivalence relation on $\cal P(\mathbb{N})$ is defined by $A\equiv_T B$ iff $A\leq_T B$ and $B\leq_T A$. This is a countable Borel equivalence relation on the polish space $\cal ...

**11**

votes

**1**answer

479 views

### How badly does compactness fail in $\mathcal{L}_{\omega_1\omega}$?

I would like to get a better idea of how badly compactness fails in $\mathcal{L}_{\omega_1\omega}$.
Let $\Gamma$ be an arbitrary set of sentences from $\mathcal{L}_{\omega_1\omega}$. Let the ...

**8**

votes

**2**answers

813 views

### Good source for Effective Descriptive Set Theory

I just finished a first course in Descriptive Set Theory using Kechris' "Classical Descriptive Set Theory" and was hoping to find a good source for learning some of the Effective DST. Kechris doesn't ...

**14**

votes

**2**answers

863 views

### Decomposing $\mathbf{\Pi}^1_1$ sets into closed sets

It is well known that every $\mathbf{\Pi}^1_1$-set is a union of $\aleph_1$-many Borel sets. I wonder whether it can be improved under certain reasonable set theory axioms assumption.
For example, ...

**2**

votes

**0**answers

148 views

### Unbounded Class of Orbit Equivalence Relations

In their paper titled "The Classification of Hypersmooth Borel Equivalence Relations" Alexander Kechris and Alain Louveau quote the following (Theorem 5.2 in the article) as "Harrington, unpublished": ...

**5**

votes

**0**answers

629 views

### ZFC+``every analytical set is measurable"

I know that "ZFC+the existence of an inaccessible cardinal" is equconsistent to
"ZFC + every $\mathbf{\Sigma}^1_3$ set is measurable".
Then how about the light face case?
Without large cardinal ...

**29**

votes

**0**answers

1k views

### Concerning the various proofs from the axiom of choice that R^3 admits of surprising geometrical decompositions into circles, skew lines and so on: can we prove in any instance that there are no Borel such decompositions? Or that AC is required?

This question follows up on a comment I made on Joseph O'Rourke's
recent question, one of several questions here on mathoverflow
concerning surprising geometric partitions of space using the axiom
of ...

**5**

votes

**1**answer

612 views

### Countable admissible ordinals

Jensen claimed that for any finite increasing sequence countable admissible ordinals $\omega= \alpha_0<\alpha_1\cdots <\alpha_n$, there is a real $x$ so that, for each $m\leq n$, $\alpha_m$ is ...

**0**

votes

**0**answers

212 views

### Universally measurable sets and the perfect set property

Is it true that all universally measurable sets (say on $[0,1]$ ) have the perfect set property?
I am not an expert in this at all and the answer may be known, but I was not able to find it.
I ...