The tag has no wiki summary.

learn more… | top users | synonyms

4
votes
0answers
139 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 ...
5
votes
0answers
339 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 ...
7
votes
1answer
284 views

Determinacy and definable ultrafilters

It is a simple consequence of AD that there are no non-principal ultrafilters on $\omega$: for $U$ an ultrafilter on $\omega$, consider the game $G_U$ where players I and II play natural numbers $x_0$ ...
11
votes
1answer
249 views

Which forcings preserve (some) determinacy?

The question is exactly as in the title. I'm interested in general in all questions of the form "which forcings preserve property P?" for any P, but determinacy assumptions occupy a special place in ...
8
votes
0answers
481 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 ...
5
votes
1answer
427 views

A question about Q?

Let A=$\{a_n : n\in \omega \}\subset 2^{\omega\times\omega}$ be nonempty countable without isolated points (i.e. homeomorphic to $\mathbb{Q}$), and satisfy $ \forall n\in \omega \exists^\infty ...
5
votes
0answers
328 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 ...
3
votes
1answer
220 views

Sets of reals and absoluteness

Schoenfield's absoluteness states that if $\phi$ is $\Sigma^1_2$ then $V\models \phi$ iff $L\models \phi$. The set of reals in $L$ is $\Sigma^1_2$ and it is the largest countable $\Sigma^1_2$ set of ...
3
votes
0answers
126 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: ...
5
votes
0answers
204 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 ...
10
votes
0answers
476 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
1answer
445 views

Constructing an injective reduction of equivalence relations

[Metastuff: I asked this question in a slightly different way on mathSE last week, and it didn't go anywhere, which is why I am asking here. I added the DST tag because it's basically a problem about ...
5
votes
1answer
202 views

Companion of the pointclass of inductive sets

This question is about the notion of a companion for a Spector class, as defined in Moschovakis's book Elementary Induction on Abstract Structures. I am interested in Spector classes on $\mathbb{R}$, ...
3
votes
1answer
217 views

sigma-algebra generated by OD sets

Assume $V=L(\mathbb{R})$ and the Axiom of Determinacy. Is every set of reals generated by ordinal-definable sets of reals under the operations of countable union and intersection? The class of sets ...
6
votes
2answers
313 views

Weakly homogeneous trees under AD

If AD$_\mathbb{R}$ holds and $\kappa < \Theta$ then every tree $T$ on $\kappa$ is weakly homogeneous (Martin–Woodin, "Weakly homogeneous trees.") I recall hearing that the hypothesis can be ...
5
votes
0answers
324 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 ...
3
votes
1answer
209 views

Wadge Determinacy of some related classes.

Let $A, B \subset \omega^\omega$. The Wadge Game $(A, B)$ is played as followed: Player I plays $a_0 \in \omega$. Then Player II plays $b_0 \in \omega$. Then Player I choose $a_1 \in \omega$; ...
0
votes
1answer
177 views

Trees on $\omega$

I am going to give a construction of a tree on $\omega$ that at first appears as though it is well founded. However, this tree cannot be well-founded because, using the rank function on finite ...
7
votes
0answers
141 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 ...
5
votes
3answers
671 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) ...
3
votes
1answer
236 views

Indeterminacy of long games

Hello, all, Several months ago I sat in on a seminar on AD+, which was incredibly wonderful even though I could barely follow it at all. AD+ is a technical variant of AD, the axiom of determinacy, ...
4
votes
2answers
267 views

Is the Turing equivalence relation the orbit equiv. relation of the action of a countable group?

The Turing equivalence relation on $\cal P(\mathbb{N})$ is defined by $A\equiv_T B$ iff $A\leq_T B$ and $B\leq_T A$. This is a countable Borel equivalence relation on the polish space $\cal ...
11
votes
1answer
437 views

How badly does compactness fail in $\mathcal{L}_{\omega_1\omega}$?

I would like to get a better idea of how badly compactness fails in $\mathcal{L}_{\omega_1\omega}$. Let $\Gamma$ be an arbitrary set of sentences from $\mathcal{L}_{\omega_1\omega}$. Let the ...
8
votes
2answers
700 views

Good source for Effective Descriptive Set Theory

I just finished a first course in Descriptive Set Theory using Kechris' "Classical Descriptive Set Theory" and was hoping to find a good source for learning some of the Effective DST. Kechris doesn't ...
14
votes
2answers
842 views

Decomposing $\mathbf{\Pi}^1_1$ sets into closed sets

It is well known that every $\mathbf{\Pi}^1_1$-set is a union of $\aleph_1$-many Borel sets. I wonder whether it can be improved under certain reasonable set theory axioms assumption. For example, ...
2
votes
0answers
135 views

Unbounded Class of Orbit Equivalence Relations

In their paper titled "The Classification of Hypersmooth Borel Equivalence Relations" Alexander Kechris and Alain Louveau quote the following (Theorem 5.2 in the article) as "Harrington, unpublished": ...
5
votes
0answers
568 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 ...
27
votes
0answers
990 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 ...
5
votes
1answer
541 views

Countable admissible ordinals

Jensen claimed that for any finite increasing sequence countable admissible ordinals $\omega= \alpha_0<\alpha_1\cdots <\alpha_n$, there is a real $x$ so that, for each $m\leq n$, $\alpha_m$ is ...
0
votes
0answers
201 views

Universally measurable sets and the perfect set property

Is it true that all universally measurable sets (say on $[0,1]$ ) have the perfect set property? I am not an expert in this at all and the answer may be known, but I was not able to find it. I ...
6
votes
2answers
288 views

Are Vitali-type nonmeasurable sets determinate?

Here, by a Vitali set, I mean the following. Call $f_1,f_2:\omega\rightarrow 2$ tail-equivalent if {$n| f_1(n)\not=f_2(n)$}$<\infty$. Vitali sets (existence via AC) contain one such $f$ from ...
5
votes
1answer
278 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
231 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 ...
47
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
410 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
377 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
316 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
478 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
1k 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
290 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
318 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
506 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
628 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
327 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
309 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
447 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
576 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
232 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
991 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
537 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?