**7**

votes

**3**answers

894 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) ...

**5**

votes

**1**answer

286 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

308 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

483 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

852 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

868 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

152 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Relations" Alexander Kechris and Alain Louveau quote the following (Theorem 5.2 in the article) as "Harrington, unpublished": ...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

635 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

619 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

213 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

302 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

316 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

237 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 ...

**56**

votes

**2**answers

5k 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

421 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

380 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

393 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

507 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 ...

**16**

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

299 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

326 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

536 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

641 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

335 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

341 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

465 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

594 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

240 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 ...

**14**

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

586 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

864 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

252 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

1k 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

419 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

410 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

569 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: ...

**3**

votes

**0**answers

266 views

### For METRIZABLE spaces, do the Banach classes and Baire classes coincide?

In this paper: 'Borel structures for Function spaces' by Robert Aumann,
http://projecteuclid.org/euclid.ijm/1255631584
Aumann claims that when X and Y are metric spaces (among other things), the ...

**9**

votes

**1**answer

491 views

### Images of Borel subsets of non-metric compact spaces

The following question was noted by Jan Pachl in connection with the
study of Arens products and he has not received a satisfactory
answer from the various experts he has asked. Let $X$ and $Y$ be
...

**3**

votes

**1**answer

293 views

### Any subset of Baire space is a union of a boldface $\Delta_2^0$ set and a set with no isolated points. Anybody know how to prove this?

I'm trying to do due diligence and determine whether this is known, trivial, original, etc. I have a proof of:
Theorem: If $S\subseteq \mathbb{N}^{\mathbb{N}}$ then $S=X\cup Y$ for some $X$ which is ...

**4**

votes

**0**answers

530 views

### Is this observation about the Borel Hierarchy trivial?

Hello, consider the following theorem. Is it trivial? Is it interesting? Is it worth including in a paper if I can prove it in 1 line as a corollary?
Theorem: Suppose $n>0$ is a natural. ...

**5**

votes

**1**answer

1k views

### Does recent work of Woodin clash with an older result in Descriptive Set Theory?

Background/Motivation
First time posting here, so I give the motivation for the question.
Early on in Descriptive Set Theory Sierpinski proved every
${\Sigma}^1_2$ set (PCA set in the older ...

**10**

votes

**3**answers

977 views

### Universal sets in metric spaces

(I am cross-posting this from math.SE as it seems to be slightly over the top for that site.)
I saw in the class the theorem:
Suppose $X$ is a separable metric space, and $Y$ is a polish space ...

**8**

votes

**3**answers

645 views

### A compactness property for Borel sets

Is the following generalised compactness property of Borel sets in a Polish space consistent with ZFC?
(*) Let $\mathcal{B}$ be a family of $\aleph_1$-many Borel sets. If $\bigcap \mathcal{B} = ...

**24**

votes

**0**answers

975 views

### Do all possible trees arise as orbit trees of some permutation groups?

I.Motivation from descriptive set theory
(Contains some quotes from Maciej Malicki's paper.)
The classical theorem of Birkhoff-Kakutani implies that every metrizable topological group G admits a ...

**1**

vote

**1**answer

461 views

### Choice function on the countable subsets of the reals

It is easy to find a choice function on all finite subsets of $\mathbb R$, but without using the axiom of choice, not on all subsets. Is there an "explicit" choice function on the countable subsets of ...

**14**

votes

**3**answers

1k views

### Parts of Set Theory immune to independence

The motivation for asking this question is a passage (3.2) in an article by Greg Hjorth where he said that "...it is also an attractive feature of the theory of Borel cardinalities and of the theory ...

**8**

votes

**2**answers

1k views

### Wanted: chain of nowhere dense subsets of the real line whose union is nonmeagre, or even contains intervals

Let $X$ be a topological space. When I call a set nowhere dense, meagre or similar without qualification, I mean that it has this property as a subset of $X$. Call a subset of $X$ weager (for weakly ...

**3**

votes

**1**answer

482 views

### Question about John Steel's “The derived model theorem”

In John Steel's paper "The derived model theorem",
http://math.berkeley.edu/~steel/papers/dm.ps
John Steel asserts that it is clear that $\mathrm{Hom}^{Y}_{\kappa}$ is closed downward under ...

**0**

votes

**1**answer

194 views