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2
votes
3answers
391 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
361 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
524 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
262 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
475 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
512 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 ...
10
votes
3answers
871 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
3answers
604 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} = ...
20
votes
0answers
879 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
423 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 ...
8
votes
2answers
908 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
456 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
182 views

The projection of a weakly homogeneous tree is determined

Where can I read a proof of this?
3
votes
1answer
321 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
339 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
233 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 ...
30
votes
4answers
7k 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
498 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
348 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 ...