**29**

votes

**0**answers

1k 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 ...

**28**

votes

**0**answers

963 views

### Godel on recursion-theoretic hierarchies

At the end of his excellent article, "The Emergence of Descriptive Set Theory" (http://math.bu.edu/people/aki/2.pdf), Kanamori writes:
"Another mathematical eternal return: Toward the end of his ...

**24**

votes

**0**answers

975 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 ...

**13**

votes

**0**answers

380 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 ...

**11**

votes

**0**answers

270 views

### Absoluteness of “$\kappa$-homogeneously Suslin” for sets of reals

What is known about the absoluteness, or lack thereof, of the notion of "$\kappa$-homogeneously Suslin" for sets of reals?
For example, if $A$ is $\kappa$-homogeneously Suslin and $\lambda > ...

**11**

votes

**0**answers

519 views

### Is every $\sigma$-algebra of sets *abstractly* the Borel algebra of a topology on perhps some other set?

Is every sigma-algebra the Borel algebra of a topology?
inspires the present question which asks for less.
Question: Given a $\sigma$-algebra ${\cal A}$ on a set $X$, does there exist a topology ...

**10**

votes

**0**answers

529 views

### Existence (or non) of “definable” ultrafilters

This is a question which I suspect has an absurdly easy answer, but I'm not seeing it.
Let $\langle\cdot,\cdot\rangle:\omega^2\rightarrow\omega$ be your favorite pairing map (for me, this is the ...

**9**

votes

**0**answers

226 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 ...

**9**

votes

**0**answers

157 views

### From interpolation to separation

Lusin's separation theorem states that, if $A$ and $B$ are disjoint analytic subsets of a Polish space, then there is a Borel set $X$ separating them ($A\subseteq X$, $B\cap X=\emptyset$). Craig's ...

**9**

votes

**0**answers

202 views

### The Chang model after collapsing an inaccessible limit of Woodins

If $\kappa$ is an inaccessible cardinal and $G \subset \operatorname{Col}(\omega,\mathord{<}\kappa)$ is a $V$-generic filter, then in $V[G]$ the Chang model $L(\text{Ord}^\omega)$ satisfies "every ...

**9**

votes

**0**answers

233 views

### Proving regularity properties from forcing axioms

It's well known that PFA implies projective determinacy. It's also well known that PD implies that all projective sets are Lebesgue measurable, have the Baire property, etc.
Is there a direct proof ...

**9**

votes

**0**answers

320 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 ...

**8**

votes

**0**answers

245 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 ...

**8**

votes

**0**answers

275 views

### A Banach-Tarski game

This is partially inspired by the question http://math.stackexchange.com/questions/1383397/cutting-a-banach-tarski-cake, which I find intriguing if unclearly written.
A paradoxical family of subsets ...

**7**

votes

**0**answers

170 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

**0**answers

163 views

### A jump operator for Borel equivalence relations

It is well-known that with respect to Borel reducibility the class of Borel equivalence relations on a standard Borel space does not admit a maximal element. We can use the well-known Friedman-Staley ...

**7**

votes

**0**answers

349 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 ...

**7**

votes

**0**answers

149 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

**0**answers

241 views

### The least admissible above a dominating real

Let $\mathbb{P}$ be the usual forcing which adds a dominating real: conditions in $\mathbb{P}$ are pairs $(p, f)$ with $p:\omega\rightarrow\omega$ finite partial and $f:\omega\rightarrow\omega$ total, ...

**6**

votes

**0**answers

270 views

### Reference for “if the set $A$ is Suslin, then every $\Sigma^1_1(A)$ set is Suslin”

Does anyone know of a reference for one or both of the following facts (in $\mathsf{ZF}$)?
If the set of reals $A$ is Suslin, then every $\Sigma^1_1(A)$ set of reals is Suslin.
If $T$ is a tree on ...

**6**

votes

**0**answers

187 views

### $\omega$-models of $\mathbf{\Sigma^1_1}-DC$ and $\mathbf{\Delta^1_1}-CA$

So what is needed to demonstrate something (say like $L_{\omega_1^{CK}}[a]$ is a $\omega$-models of $\mathbf{\Sigma^1_1}$-$DC$ or $\mathbf{\Delta^1_1}$-$CA$? It's not like I don't understand what the ...

**5**

votes

**0**answers

66 views

### Is there an analytic $\mathrm{P}$- ideal on $\omega$ which is not $\Sigma^0_2$ and not $\Pi^0_3$-complete?

Soleski proved that for any analytic $\mathrm{P}$-ideal on $\omega$ is $\Pi^0_3$.
The usually example , such as the density zero ideal $Z_0$ is $\Pi^0_3$-complete, $I_{\frac{1}{n}}$ is ...

**5**

votes

**0**answers

157 views

### Analytic equivalence relations whose classes are sometimes Borel

There are analytic equivalence relations for which the statement "All classes are Borel" is independent of $ZFC$. In all the examples I know about, the classes are non Borel in $L$ or $L[z]$ for some ...

**5**

votes

**0**answers

288 views

### Cardinal characteristics without choice

(I'm taking my definition of a cardinal characteristic from Blass' excellent article http://www.math.lsa.umich.edu/~ablass/need.pdf, which cites Vojtas/Fremlin/Miller; theirs is more general, but I'm ...

**5**

votes

**0**answers

148 views

### The (global) theory of Borel equivalence relations

What do we know about the complexity of the theory of Borel equivalence relations, with the Borel reducibility order $\leq_B$?
That is, let $\mathcal{B}$ be the set of all Borel equivalence ...

**5**

votes

**0**answers

253 views

### $\infty$-Borel Determinacy?

An $\infty$-Borel set is a set $X\subseteq\mathbb{R}$ which has an $\infty$-Borel code - a set $r$ coding the construction of $X$ via open sets, complementation, and well-ordered unions (see ...

**5**

votes

**0**answers

132 views

### A result of Steel on characterizing lightface pointclasses

In the article Projectively wellordered inner models, Steel proves the following theorem (4.12):
Theorem: Let $n < \omega$ and suppose $\mathcal{M}_n^{\sharp}$ exists. Let ...

**5**

votes

**0**answers

233 views

### Existence of an universally measurable pullback

Let $X,Y$ and $Z$ be standard Borel spaces:
topological spaces homeomorphic to Borel subsets of complete separable metric spaces.
Let $K\subseteq X\times Y$ be analytic. Assume that $K_x$ is not ...

**5**

votes

**0**answers

356 views

### Proof of “AD + every set of reals is Suslin” implies AD$_\mathbb{R}$

Could someone point me toward a proof that "ZF + AD + every set of reals is Suslin" (+ $\mathsf{DC}\_\mathbb{R}$?) implies $\mathsf{AD}\_\mathbb{R}$, either with a reference or a hint?
I am ...

**5**

votes

**0**answers

381 views

### Closure properties of familes of $G_\delta$ sets.

Given a family of sets $G\subset P(X)$, can one characterize by "closure properties" alone whether or not $G$ arises as the family of all $G_\delta$ for some topology on $X$? some Polish space ...

**5**

votes

**0**answers

229 views

### Infinity-Borel sets in ZFC

The notion of an $\infty$-Borel set of reals is useful in the study of AD. Under ZFC it becomes trivial: every set of reals is $\infty$-Borel. However, the notion of an $\infty$-Borel code is still ...

**5**

votes

**0**answers

635 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 ...

**4**

votes

**0**answers

62 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 ...

**4**

votes

**0**answers

118 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 $\Sigma_{2n+2}$ ...

**4**

votes

**0**answers

111 views

### Fixed Points of the Friedman Stanley Jump

Consider the situation of a pair $(X,E)$, where $X$ is a standard Borel space and $E$ is an invariant equivalence relation on $X$*. The Friedman-Stanley jump of this pair is an equivalence relation ...

**4**

votes

**0**answers

142 views

### Almost universal properties

Suppose that a small category $C$ has sets of objects and arrows that carry the structure of, say, Polish spaces, in some appropriately compatible way. For example $C$ might be an internal category in ...

**4**

votes

**0**answers

530 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. ...

**3**

votes

**0**answers

164 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 ...

**3**

votes

**0**answers

385 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: ...

**3**

votes

**0**answers

48 views

### If $x≤^∗y$ has the $\text{BP}$(in $X^2$) we can say that A is meager?

Let $X$ be a Polish space. Let $(I,<)$ be a wellordered set and $(A_i)_{i∈I}$ a family of sets meager in $X$. Let $A=⋃_{i∈I}A_i$. If consider the relation $x≤^∗y$ defined by:
$x,y∈A$ $\wedge$ (the ...

**3**

votes

**0**answers

228 views

### Uniformization/measurable selection theorems

Let $X,Y$ be measurable spaces and $F\subseteq X\times Y$. We say that $f:X\to Y$ is a uniformization map for $F$ if $(x,f(x))\in F$ for each $x\in \pi_X(F)$ where $\pi_X$ is the left projection map. ...

**3**

votes

**0**answers

236 views

### How many Dedekind-finite sets can $\mathbb{R}$ be partitioned into?

Building off Asaf Karagila's answer to my previous question (Can $\mathbb{R}$ be partitioned into dedekind-finite sets?) on partitioning $\mathbb{R}$ into strictly Dedekind-finite sets:
(1) What ...

**3**

votes

**0**answers

129 views

### Theory of (definable) ideals on a multi-dimensional countable set

I'm wondering if there are any results about definable ideals/filters on $\omega\times\omega$, $\omega\times\omega\times\omega$, etc.
To give a sense of the kind of results I might be looking for: ...

**3**

votes

**0**answers

266 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 ...

**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 ...

**2**

votes

**0**answers

212 views

### Is there a $\Sigma^0_3$-complete ideal on $\omega$?

In Kechris's book "Classical Descriputive set theory" Chapter 23 (Exercise 23.4), it claim that there is a $\Sigma^0_\xi$-complete ideal on $\omega$ for each $\xi\geq 3$.
There is a candidate ...

**2**

votes

**0**answers

304 views

### A query on how to climb inaccessibles in £

I am investigating to what extent extensions of the librationist property or set theory £ may support relative inaccessible sets; see Librationist Closures of the Paradoxes and Elements of ...

**2**

votes

**0**answers

128 views

### Classify spaces that make extension theorems hold

Recall a Polish space is a completely metrizable separable space.
Say a Polish space $Y$ is a terminal space if for any Polish space $X$ and any closed $C \subseteq X$, one can extend a continuous ...

**2**

votes

**0**answers

117 views

### Open games formed by pasting together infinitely many clopen games

Throughout, I think of games and their underlying trees as the same: so a "clopen game" and a "well-founded tree" mean the same thing.
Fix a sequence of clopen games $\lbrace T_i: i\in\omega\rbrace$. ...

**2**

votes

**0**answers

300 views

### Descriptive set theory on $\mathbb{R}^\mathbb{N}$

The short version of my question is, What is a good source for learning about descriptive set theory on the space $\mathbb{R}^\mathbb{N}$, under the product topology coming from the discrete topology ...