**7**

votes

**1**answer

171 views

### On thin $\Sigma^1_2$ equivalence relations

This question is regarding Hjorth's paper "Some applications of coarse inner model theory", J. Symbolic Logic 62 (1997), no. 2, 337–365.
Hjorth claims that if $E$ is a thin $\Sigma^1_2$ equivalence ...

**4**

votes

**1**answer

89 views

### Measurable $\epsilon$-optimal selection with an analytically measurable stochastic kernel

Let $(X, \mathcal{X})$ and $(A, \mathcal{A})$ be standard Borel spaces, $D \subseteq X \times A$ be an analytic set, and $D_x := \{a \in A : (x, a) \in D\}$ denote the $x$-section of $D$ at $x \in X$.
...

**7**

votes

**0**answers

102 views

### Choice and the Baire property in non-separable complete metric spaces

It's known to be consistent with ZF+DC that every subset of $\mathbb{R}$ has the Baire property (BP). (E.g. Shelah's model). If so, then every subset of every complete separable metric space has ...

**5**

votes

**1**answer

277 views

### Comparing the sizes of uncountable sets of reals under AD

Working in ZF+AD, let $$\theta_0(X)=\min\{\alpha\in ON: \not\exists f: X\rightarrow \alpha\mbox{ surjective and OD}\}$$ be the least ordinal onto which $X$ does not surject in an OD way, for $X\...

**5**

votes

**1**answer

150 views

### Spreading sets - especially without choice

For what follows, I work in ZF+AD+DC. However, the questions below are not obviously trivial in ZFC, so I'm also interested in results in that system.
Suppose I have a set $X\subseteq \mathbb{R}$. ...

**9**

votes

**1**answer

954 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

**1**answer

186 views

### Ordinal-definable witnesses to the perfect set property?

This possibly a very basic descriptive set-theory question; if it is too basic for MO, feel free to migrate.
Throughout we work in ZF+AD. My question is:
If $A$ is an uncountable OD set of reals,...

**9**

votes

**4**answers

766 views

### Is it inconsistent for a model of set theory to contain its own first order theory?

I am wondering if it is inconsistent to have a model of set theory V such that V contains an $A\subset \omega$ that codes its first order theory.I.e. for all $\{\underline\epsilon\}$-sentences $\phi$, ...

**6**

votes

**0**answers

111 views

### Has a continuous map from $\kappa^\omega$ to $[0,1]^\omega$ a non-scattered fiber?

Question. Let $\kappa>\mathfrak c$ be a cardinal endowed with the discrete topology and $f:\kappa^\omega\to[0,1]^\omega$ be a continuous map. Is there a point $y\in[0,1]^\omega$ whose preimage $f^{-...

**6**

votes

**1**answer

161 views

### Bernstein sets of large cardinality

A metrizable space $X$ will be called a generalized Bernstein set if every closed completely metrizable subspace $C$ of $X$ has cardinality $|C|<|X|$.
It is well-known that the real line contains ...

**18**

votes

**2**answers

735 views

### Meager subspaces of a Banach space and weak-* convergence

I previously asked a version of this question on Math.SE, but didn't receive an answer. (But there is a bounty there if you want to claim it!)
Let $X$ be a Banach space. (If it helps, feel free to ...

**7**

votes

**1**answer

147 views

### Is there a Fraisse limit whose automorphism group contains dense but not generic automorphisms?

It is well known that $\mathsf{Aut}(\mathbb{Q},<)$ has generic automorphisms (i.e., a comeagre conjugacy class under the diagonal action) but does not admit ample generics. The automorphism group $\...

**7**

votes

**1**answer

188 views

### Product of limit $\sigma$-algebras

Let $X$ and $Y$ be Polish (i.e. Borel subsets of separable completely metrizable) spaces. For a Polish space $Z$, let $\mathscr{S}(Z)$ denote the limit $\sigma$-algebra on $Z$, i.e. the smallest $\...

**2**

votes

**0**answers

72 views

### Do $G_\delta$-measurable maps preserve dimension?

This question (in a bit different form) I leaned from Olena Karlova.
Question. Let $f:X\to Y$ be a bijective continuous map between metrizable separable spaces such that for every open set $U\subset ...

**5**

votes

**1**answer

145 views

### Martin-Solovay Tree of Weakly Homogeneous Tree under $\mathsf{AD}_\mathbb{R}$

A tree $T$ on $\omega \times \lambda$ is weakly homogeneous if there is a countable set $\sigma$ of countably complete measures on ${}^{<\omega}\lambda$ so that $x \in p[T]$ if and only if there is ...

**3**

votes

**1**answer

247 views

### Representation of meager sets in Cohen extensions

Let $M$ be a transitive model of ZFC and $c\in {}^\omega2$ a Cohen real over $M$. Let $A$ be a meager Borel subset of $^\omega2$ in $M[c]$. I would like to prove that there exists a meager Borel set $...

**8**

votes

**2**answers

306 views

### “Clubiness” of projective sets of ordinals

I'm sure this is just my google-fu failing me, but: what are sufficient, non-overkill large cardinal axioms which guarantee "Every (boldface) $\Pi^1_n$ set of (real codes for) countable ordinals ...

**1**

vote

**0**answers

122 views

### Is every path connected space continuously path connected

Recall a topological space $X$ is path connected if for all $x,y \in X$ there is a continuous function $f\colon [0,1] \to X$ such that $f(0)=x$ and $f(1)=y$.
Say that $X$ is continuously path ...

**5**

votes

**1**answer

239 views

### Iteration of random reals

Consider two random reals $x, y$ over a transitive model $V$ of ZFC. More specifically, if $\mathcal C^V={}^\omega2$ is the Cantor space, composing the canonical homeomorphism with the projections $\...

**14**

votes

**1**answer

492 views

### Is there a class of mathematical structures with non-isomorphic natural representations as a standard Borel space?

Background. The field of Borel equivalence relation theory
provides a robust, unifying theory that organizes most of the
classification problems of classical mathematics into a hierarchy,
allowing us ...

**3**

votes

**1**answer

92 views

### Generic sections of non-null sets are non-null

Consider the Cantor space $\mathcal C={}^\omega2$ with the usual product measure, and let $r$ be a random real (over a transitive model $V$ of ZFC). Let $B\subset \mathcal C^V\times\mathcal C^V$ a ...

**3**

votes

**0**answers

174 views

### Borel equivalence relations in models of determinacy

The following appears as fact 3.1 in the slides from Hjorth's 2010 Tarski Lectures. Assume ${L(\mathbb R)} \models \mathrm{AD}$.
Fact 3.1 For E and F Borel equivalence relations one has
$$E \...

**9**

votes

**0**answers

321 views

### Uniform closure of subspaces of Baire class 1

Describe a uniformly closed linear subspace $A \subset C([0,1])$ such that the space $B_1(A)$ is not uniformly complete. Here $B_1(A)$ is the set of all bounded functions $f$ which are pointwise ...

**12**

votes

**2**answers

1k views

### Generalizations of the Tietze extension theorem (and Lusin's theorem)

I am reasking a year-old math.stackexchange.com question asked by someone else.
(For my needs every space $X$ and $Y$ will be Polish---that is a completely separably metrizable space.)
The Tietze ...

**5**

votes

**1**answer

236 views

### Is the set of subsequences of branches through a tree Borel?

Let $T$ be pruned subtree of $\omega^{<\omega}$. For my cases of interest, we may assume that $T$ is infinitely branching at every node, and consists of increasing sequences.
Let $A=\{x\in\omega^{\...

**0**

votes

**1**answer

147 views

### Countably generated $\sigma$-algebra

Let $(\Omega,\Sigma,\mu)$ be a countably generated probability space. Must $(\Omega,\Sigma,\mu)$ be isomorphic modulo null sets to a standard probability space?
I assume not, so here is a more ...

**4**

votes

**1**answer

120 views

### Is the following product-like space a Polish space?

Let $\mathcal{M}_1(\mathbb R)$ denote the space of Borel probability measures on $\mathbb R$. The space is a Polish space (a space which admits a complete, separable, metric) using, say the Levy-...

**6**

votes

**1**answer

191 views

### Basis theorem (due to Solovay?)

I'm finishing up my bibliography and I'm looking for a reference for the statement that, working in $L(\mathbb{R})$, the $\Delta^2_1$ sets form a basis for the $\Sigma^2_1$ predicates. I believe that ...

**1**

vote

**1**answer

156 views

### Absoluteness and Tree Representations

Suppose $T$ is a tree on $\omega \times \omega \times \delta$ for some ordinal $\delta$ is a homogeneous tree (with some coherent set of measures witnessing homoegeneity). ($T$ can have additional ...

**1**

vote

**1**answer

115 views

### A question about Borel sets on the unit interval

It is known that each non-decreasing continuous function $\phi$ induces a $\sigma$-additive measure $d\phi$ such that $\int_0^1 f(x) d\phi(x)$ exists for every bounded real-valued Baire function $f$. ...

**13**

votes

**0**answers

384 views

### Is $\mathbb{Z}^{\omega}$ ever the union of a chain of proper subgroups each isomorphic to $\mathbb{Z}^{\omega}$?

Recall that the covering number $cov(B)$ is the least cardinal $\kappa$ such that $\kappa$ meagre sets cover the real line. Andreas Blass and John Irwin http://www.math.lsa.umich.edu/~ablass/bb.pdf ...

**3**

votes

**2**answers

297 views

### Is there an uncountable Borel almost disjoint family?

Here we are considering subsets $\mathcal{F}$ of $2^\omega$, which are in correspondence with families of subsets of $\omega$ (sets of "reals"). Such a family is Borel if it is a Borel subset of $2^\...

**7**

votes

**3**answers

901 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) ...

**12**

votes

**1**answer

837 views

### Reverse-engineer forcing: am I reinventing the wheel?

In the course of a project I’m working on, I’ve started playing around with a sort of “reverse-engineering” forcing. It seems interesting, but
I have a sinking feeling I’m reinventing the wheel; does ...

**3**

votes

**0**answers

396 views

### “Nicely” strong measure zero sets

This question is essentially an expanded version of the unanswered half of Two strengthenings of "strong measure zero".
A set $X$ of reals is strong measure zero if, for any $f: \omega\...

**16**

votes

**1**answer

492 views

### Two strengthenings of “strong measure zero”

A set $X\subseteq\mathbb{R}$ is strong measure zero if, for every sequence $(\epsilon_i)_{i\in\mathbb{N}}$ of positive reals, there is a sequence $(I_i)_{i\in\mathbb{N}}$ of open intervals covering $X$...

**7**

votes

**0**answers

174 views

### Can you define a probability measure on the set of countable transitive models of ZFC?

It is well known that the set of hereditarily countable sets $H(\omega_1)$ —or, if you prefer, $H_{\omega_1}$— has cardinality $2^{\aleph_0}$, and I understand that every countable ...

**7**

votes

**1**answer

194 views

### Can Sacks forcing add a Cohen generic real over $L$?

Motivated by this question Forcing the negation of CH without adding Cohen reals over L and Todd Eisworth's comment, the question is the following:
1) Suppose $V$ has no Cohen generic reals over $L$. ...

**3**

votes

**1**answer

113 views

### Scott Rank of Models of Infinitary Sentences

Let $\mathscr{L}$ be a recursive language. Let $\varphi$ be a $\mathscr{L}_{\omega_1 \omega}$-sentence and $\varphi \in L_{\omega_1^\emptyset}$. (Let $\varphi$ be a computably infinitary formula.) Let ...

**7**

votes

**1**answer

374 views

### Can an ultrapower be undone by forcing?

I am not 100% certain this question is appropriate for MO; I may just be missing something obvious. Also, I vaguely recall a similar question being asked here a while ago, but I can't find it; if it ...

**2**

votes

**1**answer

141 views

### Classification of Lebesgue-Rokhlin spaces

I am currently trying to grasp some ideas on Lebesgue-Rokhlin spaces from Bogachev, "Measure Theory", vol. 2.
Such spaces are also known as standard probability spaces but the definitions are not ...

**5**

votes

**2**answers

111 views

### Relation between projective hierarchy and universally measurable sets

Let $X$ be Polish. It is known that every analytic and coanalytic subset of $X$ is universally measurable. The Wikipedia article about universally measurable sets notes that (assuming projective ...

**1**

vote

**1**answer

154 views

### Under what conditions can we put a complete norm on a linear subspace of a separable Banach space?

Question 1 Let $X$ a separable Banach Space and $Y\subset X$ linear subspace. When can we put a norm on $Y$ in such a way so that $Y$ is a Banach space?
Clearly if $Y$ is closed in the norm topology ...

**4**

votes

**1**answer

69 views

### Hurewicz versus Wadge hierarchy of zero-dimensional Borel sets?

Given two subsets $A,B$ of the Cantor cube $2^\omega$ we write $A\le_W B$ (resp. $A\le_H B$) if there is a continuous (and injective) function $f:2^\omega\to 2^\omega$ such that $f^{-1}(B)=A$. The ...

**9**

votes

**0**answers

229 views

### (A little bit) Beyond the E-recursive

The E-recursive functions are a particular generalization of classical recursion theory to the entire set-theoretic universe, $V$. They are defined via a schemes: see http://www.math.harvard.edu/~...

**7**

votes

**1**answer

345 views

### Is every ordinal potentially definable?

It is easy to see that, if $V\models\alpha>\omega_1^{CK}$, then $\alpha$ is not recursive in any forcing extension of $V$. The argument goes as follows:
The relation "$\Phi_e=r$" is $\Pi^0_2$.
...

**2**

votes

**0**answers

108 views

### A question regarding an analogue of the Kleene $T$-predicate for Koepke's ordinal computability

Does Koepke's notion of ordinal computability admit an analogue of the Kleene $T$-predicate? If so, is the existence of such a $T$-predicate independent of $ZFC$? Also, if one assumes the existence ...

**42**

votes

**4**answers

10k 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 ...

**8**

votes

**0**answers

254 views

### Outline of Generic Separable Banach Spaces don't have a Schauder Basis

So, I know P. Enflo showed that there is a separable Banach Space that doesn't satisfy the approximation property. My professor mentioned during class that in fact generic separable Banach Spaces don'...

**4**

votes

**0**answers

121 views

### Characterization of $L[T_{2n+1}]$ as a direct limit of mice

I am asking for a reference request/proof sketch for the result of Steel
that characterizes $L[T_{2n+1}]$ as a direct limit of mice.
Given that both $L[T_{2n+1}]$ and $M_{2n}$ have a $\Sigma_{2n+2}$ ...