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5
votes
1answer
263 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
227 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 ...
41
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
400 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
371 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
293 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
450 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
953 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
277 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
311 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
484 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
620 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
310 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
297 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
434 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
565 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
230 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
931 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
486 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
625 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? ...
5
votes
1answer
223 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
757 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
386 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
345 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
510 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
254 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
465 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
279 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
495 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
980 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
3answers
832 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 ...
7
votes
3answers
585 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} = ...
19
votes
0answers
847 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
1answer
406 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
3answers
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 ...
4
votes
0answers
168 views

Borel selections of partitions of the space of probability measures

The environment: Suppose that $X$ is a Polish space and $\Delta(X)$ the set of Borel probability measures over $X$ (given the topology of weak convergence, with the Prohorov metric). Let ...
8
votes
2answers
836 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
1answer
431 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
1answer
179 views

The projection of a weakly homogeneous tree is determined

Where can I read a proof of this?
3
votes
1answer
308 views

Question about Woodin's paper “On the consistency strength of projective uniformization”

In the paper "On the consistency strength of projective uniformization" Woodin proves a lemma "Assume $M$ is a model of ZFC that is $\Sigma^{1}_{3}$-absolute. Then $M\vDash\forall ...
2
votes
1answer
330 views

Descriptive complexity of Hamel bases of R^ω

(base theory = ZFC) Are any Hamel bases for the vector space $\mathbb{R}^{\omega}$ in the 1. analytical hierarchy?2. projective hierarchy? In any of the above cases where the answer is not simply ...
3
votes
1answer
231 views

Related Open Game in Analytic Determinacy

For this question, please refer to Chapter 33 page 638, Set Theory Millennium Edition, by Thomas Jech. The proof of analytic games $G_A$ is converted into an open game $G^\ast$ on some suitable ...
23
votes
4answers
6k views

Non Borel sets without axiom of choice

This is a simple doubt of mine about the basics of measure theory, which should be easy for the logicians to answer. The example I know of non Borel sets would be a Hamel basis, which needs axiom of ...
4
votes
3answers
487 views

disjoint translates of a dense uncountable set

If {c(n)} is an arbitrary sequence of irrational numbers converging to 0 then Q + c(n), the set obtained by adding c(n) to the set of rational numbers Q, is clearly disjoint from Q for each n. Is ...
6
votes
1answer
340 views

A subset of Baire space Wadge incomparable to a Borel set?

Let $\omega^\omega$ be Baire space. If $A,B\subseteq\omega^\omega$ we say that $A$ is Wadge reducible to $B$ (written $A\leq_w B$) if there is a continuous function ...