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9
votes
1answer
440 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 $\mathbb{R}$, ...
3
votes
1answer
212 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
2answers
307 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
0answers
323 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
1answer
208 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
1answer
177 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
0answers
138 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 ...
6
votes
3answers
651 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) ...
3
votes
1answer
220 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
2answers
262 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
1answer
430 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
2answers
655 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
2answers
832 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
0answers
133 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
0answers
562 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 ...
26
votes
0answers
901 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
1answer
500 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
0answers
196 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
2answers
277 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
1answer
272 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
1answer
231 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 ...
44
votes
2answers
3k 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
1answer
406 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
3answers
373 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
1answer
306 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
1answer
465 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
2answers
994 views

Why does inner model theory needs 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
2answers
287 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
1answer
313 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 ?
3
votes
2answers
498 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
2answers
623 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
1answer
322 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
1answer
307 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
2answers
441 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
4answers
569 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
2answers
231 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
3answers
961 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
2answers
520 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
1answer
651 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
1answer
231 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
2answers
809 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
3answers
389 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
1answer
356 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
1answer
518 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
0answers
258 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
1answer
472 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
1answer
284 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
0answers
508 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
1answer
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 ...