5
votes
0answers
36 views

Sets that are not $\infty$-Borel

I have seen a few techinques for proving that certain sets of real numbers are $\infty$-Borel (definition) but it just occurred to me that I don't know of any way to prove that a set of real numbers ...
5
votes
3answers
110 views

Bijection between closed uncountable sets and R?

This is probably a really stupid question but I'm hoping it's true: can we find a bijection between every closed uncountable set and R (real number line)?
3
votes
1answer
60 views

Maximal pruned subtree - an absolute notion?

Fix a tree $ p $ over $ \omega $. Let $ [p] $ denote the set of all branches of $ p $. Given a set of reals $ F \subseteq \omega^\omega $, let $ T(F) := \{ x \mathord{\upharpoonright} n : x \in F ...
8
votes
1answer
89 views

Consistency strength of Turing measurability

This is probably well-known to recursion theorists, but as google didn't help me, I'll ask it here. Convention: All sets of reals in the following discussion are assumed to be closed under Turing ...
1
vote
1answer
71 views

Is any subset of the open ordinal space $[0,\Omega)$ $G_\delta$?

Consider the open ordinal space $[0,\Omega)$, where $\Omega$ is the first uncountable ordinal. Can I say that every subset of $[0,\Omega)$ is $G_\delta$? If yes, does this imply that $[0,\Omega)$ is ...
1
vote
1answer
69 views

Banach Mazur game - Oxtoby - Thm 6.1

I have asked about this theorem before but found lately that I still don't fully understand its proof. Here are the rules of the game described. A closed interval in $\mathbb{R}$ denoted $I_0$ is ...
0
votes
1answer
51 views

Oxtoby - Thm 6.2 - There exists a strategy in which (A) wins iff $I_1 \cap B$ of first category

I am reading the book Measure and Category of Oxtoby. In Theorem 6.2 it is stated that There exists a strategy in which (A) wins iff $I_1 \cap B$ of first category As in the picture below. My ...
10
votes
3answers
293 views

Can sets of cardinality $\aleph_1$ have nonzero measure?

$\aleph_1$ is the cardinality of the countable ordinals. It is the least cardinal number greater than $\aleph_0$, and assuming the continuum hypothesis it's equal to $\mathfrak{c}$, the cardinality of ...
8
votes
3answers
287 views

Finitely Additive not Countably Additive on $\Bbb N$

Does there exist a function defined on the power set of the natural numbers to the interval from $0$ to $1$, $p:2^{\Bbb N}\rightarrow [0,1]$, such that $p$ is finitely additive, i.e. ...
0
votes
0answers
64 views

Borel set without Borel uniformization

What is an example of a Borel subset $X$ of $\omega^\omega\times\omega^\omega$, such that $$ \forall f\in\omega^\omega\exists g\in\omega^\omega((f, g)\in X)$$ which has no Borel uniformization? That ...
4
votes
1answer
179 views

Question on constructing an auxiliary game of a closed game

I'm trying to understand Lemma 20.7, page 142, Classical Descriptive Set Theory(Kechris) by working on a concrete example. Lemma 20.7:Let $T$ be a non-empty pruned tree, and let $X \subseteq [T]$ ...
4
votes
1answer
175 views

Prerequisites for understanding Borel determinacy

I have just learned that Gale-Stewart game is determined for open and closed sets, so naturally I'm interested to understand Borel determinacy which seems to be on a totally different level. What's ...
11
votes
1answer
172 views

Well-orderings and the perfect set property

From a wellordering of an uncountable set of reals, Bernstein constructed a set of reals without the perfect set property. My question is, does an uncountable well-ordering itself violate the perfect ...
4
votes
1answer
177 views

Set of real whose image under any continuous function is a perfectly meager nullset

Sierpinski showed in Hypothèse du Continu that there exist a set $A \subseteq \mathbb{R}$ with the cardinality of the continuum whose image under any continuous function is a nullset. (Proposition ...
9
votes
1answer
239 views

From universal measurability to measurability

Let $(\Omega,\Sigma)$ be a measurable space and $K$ be a compact metrizable space endowed with its Borel $\sigma$-algebra $\mathcal{B}(K)$. Let $A\subseteq\Omega\times K$ be universally ...
9
votes
0answers
157 views

Restrictions of null/meager ideal

Let I denote the null/meager ideal on reals. Is it consistent that for any pair of non null/meager sets A and B, there is a null/meager preserving bijection between A and B? In particular, is this ...
12
votes
3answers
201 views

Axiom of Choice and Determinacy

In my set theory course we have been talking about the axiom of determinacy. One of the first things we showed was that $AD$ and $AC$ are incompatible. We later showed that $ZF+AD$ implies the ...
2
votes
1answer
89 views

What would be arithmetic hierarchy of $\Sigma_1^0 \wedge \Pi_1^0$?

What would be arithmetic hierarchy of the form of formula like $\phi \wedge \psi$ where $\phi$ is $\Sigma_1^0$ and $\psi$ is $\Pi_1^0$? Prenex normal form seems to give me no answer for this.
5
votes
1answer
113 views

Show that $\bf AD_2 \Leftrightarrow \bf AD_{\omega} \nRightarrow\bf AD_{\mathbb{R}}$

How to show that $\bf AD_2 $ is equivalent to $\bf AD_{\omega}$, but not equivalent to $\bf AD_{\mathbb{R}}$? $\bf AD$ is abbreviated for axiom of determinacy.$\bf AD_2 $, $\bf AD_{\omega}$, and $\bf ...
4
votes
1answer
132 views

$\bf AD$ implies $\bf AC_{\omega}(\mathbb{R})$?

How to show that $\bf AD$ implies $\bf AC_{\omega}(\mathbb{R})$? $\bf AD$ is abbreviated for axiom of determinacy. $\bf AC_{\omega}(\mathbb{R})$ states that for each family $(X_i)_{i∈\omega}$, in ...
2
votes
0answers
105 views

Classification of $\omega$-games of different fixed action spaces

Suppose, a two-player game $$G(\omega,(X_n)_{n \in\omega},(Y_n)_{n \in\omega},A)$$ where $\omega$ is the number of the moves of each of the two players I and II. $(X_n)_{n \in \omega}$ (or $(Y_n)_{n ...
1
vote
1answer
74 views

Why is the ground model set of reals nonmeager in the Cohen extension?

To be more precise, I want to add a Cohen real to a ground model $V$ of ZFC and then show that for each open interval $(a,b)$, the set $V \cap (a,b)$ is nonmeager in the extension. Thanks in advance. ...
1
vote
2answers
323 views

Cantor-Bendixson theorem proof

I am looking for a proof of Cantor-Bendixson theorem involving transfinite numbers (I am interested only in the case of real line). I fact, I have already seen one but I have a trouble in ...
5
votes
2answers
253 views

Is there any generalization of the hyperarithmetical hierarchy using the analytical hierarchy to formulas belonging to third-order logic and above?

As I understand, hyperarithmetical sets are defined according to the analytical hierarchy, that is, second-order-logic formulas. There is a generalization of hyperarithmetic theory named α-recursion ...
1
vote
0answers
64 views

Borel sets used in Model Theory

In Lemma 4.4.13 (page 161) of D.Marker- Model Theory book it is proved that $D(F,T)$ (the set of all possible F-diagrams of models of T) is a Borel set. Can someone explain to me the proof given a ...
18
votes
1answer
545 views

What are some good intuitions for understanding Souslin's operation $\mathcal{A}$?

What are some good intuitions for understanding Souslin's operation $\mathcal{A}$? Recall the definition: Let $S = \mathbb{N^{<N}} = \bigcup_{n = 1}^\infty \mathbb{N}^n$ be the set of ...
4
votes
2answers
250 views

Which are “big theorems” of descriptive set theory?

Question: If one were to fully understand 10 theorems in DST, or 15,20,25,30 theorems, which ones would be the most important to understand in order to work towards an understanding of descriptive set ...
4
votes
1answer
350 views

What is the generating set of the Vietoris topology?

Here is what Kechris says about the Vietoris topology, Let X be a top. space. We denote by K(X) the space of all compact subsets of X equipped with the Vietoris topology, i.e., the one generated by ...
20
votes
1answer
602 views

Universally measurable sets of $\mathbb{R}^2$

$$\text{Is }{{\cal B}(\mathbb{R}^2})^u={{\cal B}(\mathbb{R}})^u\times {{\cal B}(\mathbb{R}})^u\,?\tag1$$ Is the $\sigma$-algebra of universally measurable sets on $\mathbb{R}^2$ equal to the product ...
3
votes
2answers
205 views

Proof of Incompatibility of Axioms of Determinacy and Choice

I'm working through some lecture notes on the axiom of determinacy, and have run into some trouble with the proof of the incompatibility of the axiom of determinacy with the axiom of choice. ...
0
votes
1answer
122 views

Simple Question about Borel Hierarchy

I'm a bit confused about the basics of the borel hierarchy. My question is this: if i have a closed set P and I make the set $\forall^\omega P$, is that $\Pi^0_3$? Similarly, if I have an open set P ...
1
vote
2answers
141 views

Measure on Baire space

Inspired by the first parenthetical sentence of Joel's answer to this question, I have the following question: is there any useful notion of measurability in the Baire space $\omega^\omega$? Some ...
2
votes
3answers
319 views

Importance of free ultrafilter

What would be the usage of free ultrafilter? And how is it important? BTW, I know the concept of free ultrafilter, so I only need explanation on usage and importance.
8
votes
1answer
248 views

Definable order types without infinity axiom.

Denote by $ZF^\times$ the theory of $ZF$ without the axiom of infinity. We know that $V_\omega$, the set of all hereditarily finite sets in a model of $ZF$, is a model of $ZF^\times$. We further know ...
4
votes
2answers
199 views

Analytic versus Analytical Sets

Browsing MathOverflow I came across a question about analytical sets. Through the discussion following a comment made by our very own Asaf, I learned that bold face $\mathbf{\Sigma^1_1}$ and light ...
2
votes
0answers
106 views

constructive ordinal and $\Delta^1_1$ predicate

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 ...
10
votes
1answer
423 views

“Nice” well-orderings of the reals

I have a question which I believe could be easily resolved if I happened to look at the right source - hence my asking it here as opposed to at MathOverflow. I've tried googling it, but I haven't been ...
4
votes
1answer
149 views

Does there exist a subset of $\mathbb{R}$ of cardinality $2^{\aleph_0}$ that has no perfect subset?

Assuming the axiom of choice, is there a way to construct a subset of the reals of cardinality $2^{\aleph_0}$ that has no perfect subset?
18
votes
1answer
3k views

Cardinality of Borel sigma algebra

It seems it's well known that if a sigma algebra is generated by countably many sets, then the cardinality of it is either finite or $c$ (the cardinality of continuum). But it seems hard to prove it, ...
4
votes
1answer
247 views

Separable metric space with 0-dimensional kernel

Suppose $X$ is a separable metric space, let $D(X)$ denote the Cantor-Bendixson derivative of $X$, and $D_\alpha(X)$ the $\alpha$-th derivative of $X$. We denote $\operatorname{Ker}(X)$ the kernel of ...
4
votes
1answer
192 views

$\Sigma_{\alpha+1}^0(X)$ universal set based on the Baire space

In my attempts to write a cleaner proof (compared to what I was taught) to the fact that for uncountable Polish spaces the Borel hierarchy does not stop until $\omega_1$ I am trying to prove the ...
0
votes
1answer
185 views

Sets invariant under sections

Let $X$ be a compact in the Polish space (metric, complete, separable) and $G\subseteq X\times X$ is open. For $x\in X$ we define the section of $G$: $$ s(x) = \{y\in X|\langle x,y \rangle \in ...
8
votes
1answer
263 views

Universal sets in metric spaces

Today I saw in the class the theorem: Suppose $X$ is a separable metric space, and $Y$ is a polish space (metric, separable and complete) then there exists a $G\subseteq X\times Y$ which is open and ...
10
votes
1answer
339 views

Why is the real line not used in Descriptive Set Theory?

In most Descriptive Set Theory books, the rationale for working with the Baire space ($\mathbb{N}^{\mathbb{N}}$) as opposed to the real line ($\mathbb{R}$) is that the connectedness of the latter ...
8
votes
1answer
710 views

Polish Spaces and the Hilbert Cube

I've been trying to prove that every Polish Space is homeomorphic to a $G_\delta$ subspace of the Hilbert Cube. There is a hint saying that given a countable dense subset of the Polish space $\{x_n : ...
4
votes
1answer
173 views

Where in the analytic hierarchy does V=L start having consequences?

I note that the ordinals of L are the same as V, so I guess that it has no $\Pi_1^1$ consequences. On the other hand Wikipedia tells me that it asserts the existance of a $\Delta_2^1$ non-measurable ...