**5**

votes

**0**answers

349 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

235 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

128 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

214 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

504 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

449 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

208 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

233 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

317 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 ...

**5**

votes

**0**answers

333 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

218 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

144 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 ...

**5**

votes

**3**answers

727 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

267 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

284 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

462 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

774 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

850 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

144 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

585 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 ...

**27**

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

578 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

204 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 ...

**6**

votes

**2**answers

292 views

### Are Vitali-type nonmeasurable sets determinate?

Here, by a Vitali set, I mean the following. Call $f_1,f_2:\omega\rightarrow 2$ tail-equivalent if {$n| f_1(n)\not=f_2(n)$}$<\infty$. Vitali sets (existence via AC) contain one such $f$ from ...

**5**

votes

**1**answer

289 views

### Higher computability : Constructive ordinal and $\Delta^1_1$ predicates

Everything I know on this subject comes from Sacks book : "Higher recursion theory"
Let $\mathcal{O^Y}$ be the set of codes for ordinals constructive in $Y$.
We should have the result that $A ...

**6**

votes

**1**answer

233 views

### Kunen tree and Martin tree

Do we know under which conditions the Kunen tree (Recall the Kunen tree provides an analysis of the equivalence classes of functions $f: \omega_1 \to \omega_1$ with respect to the normal measure ...

**49**

votes

**2**answers

4k views

### Is every sigma-algebra the Borel algebra of a topology?

This question arises from the excellent question posed on math.SE
by Salvo Tringali, namely, Correspondence
between Borel algebras and topology.
Since the question was not answered there after some ...

**9**

votes

**1**answer

416 views

### A question on infinite dimensional Gaussian measure and affine tranformations.

Let $\gamma_\infty$ denote the product Gaussian measure on $\mathbb{R}^\mathbb{N}$. Which $a,b \geq 0$ satisfy that for every Borel set $K\subseteq \mathbb{R}^\mathbb{N}$ of positive measure, $a K + ...

**7**

votes

**3**answers

379 views

### Question of combinatorics in the lower part of the Borel hierarchy.

Let $S^\omega$ denote either $\omega^\omega$ or $2^\omega$.
Let's call a function $f: S^\omega \rightarrow$ {0,1} 'nice' if
there exists a function $g_f: S^{\lt \omega} \rightarrow 2$ such that for ...

**5**

votes

**1**answer

348 views

### Borel reduction/Wadge hierarchy

An equivalence relation $E$ is Borel-reducible to an equivalence relation $F$ if there is a $\Delta^1_1$ function $f$ such that $xEy$ holds iff $f(x)Ff(y)$ holds. A set $A\subset \omega^{\omega}$ is ...

**12**

votes

**1**answer

488 views

### 2-colorings of the reals

It's easy to prove that, if $\mathbb{R}$ is well-orderable, then there is a 2-coloring of pairs of reals with no uncountable homogeneous set, i.e., there is an $m: [\mathbb{R}]^2\rightarrow 2$ such ...

**14**

votes

**2**answers

1k views

### Why does inner model theory need so much descriptive set theory (and vice versa)?

I am curious about how much descriptive set theory is involved in inner model theory.
For instance Shoenfield's absoluteness result is based on the construction of the Shoenfield tree which ...

**1**

vote

**2**answers

293 views

### Models of $AD$ different from $L(\mathbb{R})$

Today it is known that $AD$ (the axiom of determinacy of games played with integers) is true in $L(\mathbb{R})$. Has it been proven that this is the only model in which $AD$ is true? Have other models ...

**3**

votes

**1**answer

320 views

### Action on a compact group

If $G$ is an infinite compact group, how many orbits can $G$ have under the group action of its continuous automorphisms ?

**4**

votes

**2**answers

519 views

### Question about 0-dimensional Polish spaces

Hello everybody,
I'm stuck with proving (or disproving) the following statement.
Statement:
For every $0$-dimensional Polish space $(X,\mathcal{T}\ )$, and a countable basis of clopen sets ...

**5**

votes

**2**answers

630 views

### Theorems proved with AD whose proof is also known in the ZF world

This question arises from discussions with my professor and from Todd Eisworth comments in this question Large cardinal axioms and the perfect set property
In $L(\mathbb{R})$ we have $AD$ and it is a ...

**2**

votes

**1**answer

334 views

### Large cardinal axioms and the perfect set property

It is known that if there is a measurable cardinal then every $\Pi_1^1$ set has the perfect set property (i.e it is either countable or contains a copy of $2^{\omega}$). Also if we have ...

**2**

votes

**1**answer

320 views

### Moschovakis Coding Lemma

I trying to study the Coding Lemma (in descriptive Set Theory) and there is a small point in the proof that I don't understand. Let me first recall the version I'm studying ( there are different ...

**8**

votes

**2**answers

454 views

### Exact consistency-strength of “all projective sets are Ramsey”

I wonder if the exact consistency strength of
"All projective sets have the Ramsey property"
is still open.
In Solovay's model, all sets have the Ramsey property, so the consistency strength of this ...

**15**

votes

**4**answers

586 views

### Continuously selecting elements from unordered pairs

The symmetric square of a topological space $X$ is obtained from the usual square $X^2$ by identifying pairs of symmetric points $(x_1,x_2)$ and $(x_2,x_1)$. Thus, elements of the symmetric square can ...

**0**

votes

**2**answers

235 views

### When can the one-one continuous image of a perfect set fail to be perfect?

Let $\mathfrak{M}$ and $\mathfrak{N}$ be perfect Polish spaces, $P$ a nonempty perfect subset of $\mathfrak{M}$, and $f: \mathfrak{M} \rightarrow \mathfrak{N}$ a continuous surjection that's injective ...

**12**

votes

**3**answers

1k views

### Partitioning $\mathbb{R}$ into $\aleph_1$ Borel sets

I just ran into this deceptively simple looking question.
Is it always possible to partition $\mathbb{R}$ (or any other standard Borel space) into precisely $\aleph_1$ Borel sets?
On the one ...

**7**

votes

**2**answers

553 views

### Perfect set property for projective hierarchy

Is there any paper discussing the consistency strength (or possible equivalents, maybe large cardinals) of just assuming the perfect set property for certain levels of the projective hierarchy?

**8**

votes

**1**answer

754 views

### Vitali Sets vs Bernstein Sets…

AC is enough to guarantee the existence of both Bernstein Sets and Vitali Sets...
However is the existence of Vitali Sets strictly weaker than that of Bernstein Sets?
What about the other way round?
...

**6**

votes

**1**answer

240 views

### A restatement, in terms of the semi-group product of the left-invariant completion of a Polish group, of http://mathoverflow.net/questions/71389

This is a re-statement, of sorts, of Is there a relational countable ultra-homogeneous structure whose countable substructures do not have the amalgamation property?, so far unanswered.
Let $G$ be a ...

**7**

votes

**2**answers

962 views

### Definable Wellordering of the Reals

Why are we interested in definable wellordering of the reals? For instance, we have
Con(ZFC) $\Rightarrow$ Con(ZFC + there is a $\Delta^1_2$-wellordering of $\mathbb{R}$),
Con(ZFC + there is a ...

**2**

votes

**3**answers

410 views

### Code universal arithmetical sets by a hyperarithmetical set?

For each n, there is a (lightface) Σ0n set Sn ⊆ ω2 that's universal for the Σ0n subsets of ω. Since {n} × Sn is Σ0n, there is a union R of arithmetical ...

**10**

votes

**1**answer

387 views

### Consistency strengths related to the perfect set property

I want a model of $\mathrm{MA}_{\sigma\mathrm{-centered}}+\neg\mathrm{CH}$ in which every set of reals in $L(\mathbb{R})$ has the perfect set property. In terms of consistency strength, it is known ...

**8**

votes

**1**answer

552 views

### $\Delta^0_{\alpha}$ universal sets does not exist

I am taking a course in descriptive set theory, and the exam is approaching on Sunday. In the framework of proving that for an uncountable Polish space $X$ the following holds: ...