Questions tagged [descriptive-set-theory]
Descriptive Set Theory is the study of definable subsets of Polish spaces, where definable is taken to mean from the Borel or projective hierarchies. Other topics include infinite games and determinacy, definable equivalence relations and Borel reductions between them, Polish groups, and effective descriptive set theory.
653
questions
14
votes
2
answers
1k
views
Does Turing determinacy imply full determinacy?
The axiom of Turing determinacy is a weakening of the full axiom of determinacy, $AD$, in which only games with payoff sets which are $\equiv_T$-invariant are demanded to be determined.
In "...
14
votes
3
answers
764
views
When is $A$ "$L$-ish" whenever $B$ is "$L$-ish"?
My question is about a variant of the usual notion of relative constructibility, $\le_c$ (which an earlier version of this question confusingly denoted "$\le_L$"), in set theory.
Fix a ...
9
votes
1
answer
560
views
Is $\ell^\infty$ Polishable?
Consider $\ell^\infty$ as a subspace of the Polish space $\mathbb{R}^\omega$. It is easy to check that $\ell^\infty$ is not Polish in the subspace topology, as it is countable union of the compact ...
7
votes
1
answer
248
views
Is the set of measurable maps with countable range Borel?
Let $(X,\mu)$ be a standard probability space, and $(Y,\tau)$ an uncountable Polish space. Then the set $L^0(X,\mu,Y)$ of measurable maps from $X$ to $Y$ identified up to measure 0 is Polish w.r.t. ...
15
votes
1
answer
992
views
Higher recursion theory and reverse mathematics: What is to $\Pi^1_1$-$CA_0$ as $RCA_0$ is to $ACA_0$?
There is an extremely rich and well-understood analogy between "recursively enumerable" and "$\Pi^1_1$" – indeed, this is the starting point of metarecursion theory, and $\alpha$-...
7
votes
3
answers
636
views
Cofinality of a $\sigma$-ideal of $\mathbb{R}$
The cofinality of a partially ordered set $\left( P,\leq \right)$, written $cof(P)$, is the smallest cardinality of a subset $T$ of $P$ that is [EDIT: cofinal] in $P$, i.e. for every element $p\in P$ ...
10
votes
1
answer
444
views
Haar measurable sets and quotient maps
Let $G$ be a locally compact Hausdorff group with a Haar measure $\mu$, let $H$ be a closed normal subgroup of $G$, and let $q: G \to G/H$ be the quotient homomorphism. Let $\nu$ be a Haar measure ...
-2
votes
1
answer
176
views
($^{\omega}2$,<) is not well-order. [closed]
Let < be a lexicographic order on $^{\omega}2$ or in other words given distinct functions $f,g$ from $\omega$ to 2, let $f<g$ if and only if $f(n)=0$ and $g(n)=1$, where $n$ is the lease $m<\...
12
votes
2
answers
650
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 ...
6
votes
0
answers
399
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 http://en....
19
votes
4
answers
1k
views
Pathological behavior of Borel sets?
Usually in set theory, Borel sets are much more nicely behaved than arbitrary sets of reals. One reason for this is Borel determinacy, which immediately yields measurability, Baireness, and the ...
5
votes
1
answer
609
views
$\omega$ universally Baire sets, tree representations
I've recently encountered the notion of a universally Baire set, and I've tried to look at the paper by Feng, Magidor and Woodin where this notion is studied. There are several points that confuse me.
...
6
votes
0
answers
171
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 $\delta=\delta^1_{n+2}$....
5
votes
1
answer
407
views
Perfect set property implies $\omega_1$ is a limit cardinal in $L$
Specker proved in 1957 that if in $V$ every set of real numbers has the perfect set property, than in $L$, $\omega_1^V$ is actually a limit cardinal.
The original proof is in German, and I've been ...
4
votes
1
answer
99
views
Measurability of $\{ x \in X ; H_0 x \subset A \}$
Let $H$ be some Polish group and $X$ some standard Borel space. Assume that $H$ acts measurably on $X$, i.e. $(h,x) \mapsto hx$ is Borel. Let $H_0 \subset H$ and $A \subset X$ be some Borel sets. Is ...
6
votes
1
answer
289
views
A model of Krivine
In a paper by J.-L. Krivine, Modèles de ZF+AC dans lesquels tout ensemble de réels définissable en termes d'ordinaux est mesurable-Lebesgue [C. R. Acad. Sci. Paris Sér. A-B 269 (1969), A549–A552, ...
5
votes
3
answers
655
views
Does every separated measurable space embed into a power of $\{0,1\}$?
Let $(X,\Sigma)$ be a measurable space of arbitrary cardinality. I would like to understand under which conditions this space is isomorphic to a measurable subset of $\{0,1\}^\kappa$ for some cardinal ...
9
votes
1
answer
293
views
Obtaining a lightface pointclass from a boldface one
Define a pointclass to be:
boldface inductive-like if it is $\mathbb{R}$-parameterized, has the scale property, and is closed under $\wedge$, $\vee$, $\forall^\mathbb{R}$, $\exists^\mathbb{R}$, and ...
5
votes
0
answers
432
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 ...
9
votes
1
answer
610
views
Can $\mathbb{R}$ be partitioned into dedekind-finite sets?
Assuming $ZF$ itself is consistent, it is consistent that there are sets $D$ which are infinite but cannot be placed in bijection with any of their proper subsets; such sets are called "strictly ...
5
votes
2
answers
476
views
Uncountable atomless subalgebras of the Boolean algebra of all Jordan measurable sets in [0,1]
Definition: Suppose $\mathcal A$ is
the Boolean algebra of all Jordan measurable sets in $I=[0,1]$ (i.e $\mathcal A=\{A\subseteq I: \mu(\partial(A))=0\}$, where $\mu$ is the Lebesgue measure and $\...
15
votes
1
answer
582
views
Does there exist an uncountable separable metric space $X$ such that every subset of $X$ is a Borel set?
Is it consistent with ZFC that there exists an uncountable separable metric space $X$ such that every subset of $X$ is a Borel set?
If the continuum hypothesis holds, or more generally $2^{\aleph_{0}}...
0
votes
1
answer
133
views
Existence of a map connecting two marginals of a product measure
Let $X$ and $\bar X$ be two standard Borel spaces, and let $A\subseteq X\times\bar X$ be an analytic subset of the product space. Let $P$ be any probability measure such that $P(A) = 1$, and denote by ...
5
votes
1
answer
972
views
sets without perfect subset in a non-separable completely metrizable space
Suppose $X$ is a completely metrizable (but not separable) space. Suppose $D$ is a Borel (actually $F_{\sigma}$) subset of $X$. Is there any logical relation between the following statements?
[1] $D$...
11
votes
1
answer
739
views
Is there a suitably generalized Baire property for topological spaces of arbitrary cardinalities?
Is there some suitable generalization to the notion of Baire property for topological spaces of arbitrary cardinalities which satisfies the following condition:
The meager sets are sets which are ...
7
votes
1
answer
588
views
Failure of Shoenfield's Absoluteness
Shoenfield's absoluteness states that if $M \subseteq N$ are models of $ZF$ and $M \supseteq \omega_1^N$, then every $\Sigma^1_2$ formula with parameters in $M$ is absolute between $M$ and $N$. In ...
1
vote
0
answers
472
views
Comparing two metrics on the space of infinite sequences and relating open and closed sets
Let $X = \{ 0, 1 \}$ and $X^{\mathbb N_0} = \{ x_0 x_1 x_2 \ldots : x_i \in X \}$ be the space of all infinite sequences, then a metric could be defined on it
$$
d(u,v) := \frac{1}{2^r} \mbox{ with } ...
2
votes
1
answer
192
views
Particular neighborhoods of analytical sets
Let $X$ be a standard Borel space: a topological space isomorphic to a Borel subset of a complete separable metric space. Denote by $\mathcal P(X)$ the set of all Borel probability measures over $X$ ...
9
votes
1
answer
422
views
Are there trees for $(\Sigma^2_1)^{\text{uB}}$?
If there is a proper class of Woodin cardinals, then Woodin showed (using stationary towers) that $(\Sigma^2_1)^{\text{uB}}$ statements are generically absolute, where $\text{uB}$ denotes the ...
22
votes
2
answers
1k
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 ...
8
votes
2
answers
380
views
Natural $\Pi^1_2$ (or worse) classes of structures?
(To clarify, my interest is mainly lightface, that is, $\Pi^1_2$ instead of $\bf \Pi^1_2$, although it doesn't particularly matter.)
This is just an idle curiosity. In logic, I find myself frequently ...
17
votes
4
answers
1k
views
Continuity on a measure one set versus measure one set of points of continuity
In short: If $f$ is continuous on a measure one set, is there a function $g=f$ a.e. such that a.e. point is a point of continuity of $g$?
Now more carefully, with some notation: Suppose $(X, d_X)$ ...
7
votes
1
answer
349
views
Is the consistency of $\mathcal{L}_{\infty\omega}$-sentences absolute?
The question is exactly that of the title. Suppose $\varphi\in V$ is an $\mathcal{L}_{\infty\omega}$-sentence, and $W$ is an inner model of $V$ such that $\varphi\in W$. Is the statement
$\varphi$ ...
3
votes
1
answer
227
views
Product of Topological Measure Spaces
Def. A Radon measure $\mu$ on a compact Hausdorff space $X$ is uniformly regular if there is a countable family $\mathcal{A}$ of compact $G_\delta$-subsets of $X$ such that for every open set $U\...
9
votes
1
answer
224
views
n odd: $\bf\Delta^1_n$ wadge degrees are $< \bf\delta^1_{n+1}$
My adviser is out of town and there is a comment in the Van Wesep paper "wadge degrees and descriptive set theory" that I can't figure out.
Work in ZF+AD throughout.
As stated in the title, the ...
12
votes
2
answers
591
views
Partition $\Bbb{R}$ into a family of sets each one homeomorphic to the Cantor set
It is known that there is no (nontrivial) partition of $\Bbb{R}$ into a countable number of closed set. But is there a partition of $\Bbb{R}$ into sets, each one homeomorphic to the cantor ternary set?...
3
votes
1
answer
1k
views
Different Metrics for Baire Space and their induced Topologies
The Baire-Space is the set of all infinite sequences of integers, i.e.
$$
\mathcal N = \omega^{\omega}.
$$
On this space usually the following metric is given
$$
d(\alpha, \beta) = \left\{ \begin{...
11
votes
0
answers
329
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 > \...
9
votes
1
answer
407
views
Universal $(\Sigma^2_1)^{\text{Hom}_{\mathord{<}\lambda}}$ set
Does anyone know of a reference for the fact that if $\lambda$ is a limit of Woodin cardinals, then the pointclass $(\Sigma^2_1)^{\text{Hom}_{\mathord{<}\lambda}}$ is $\omega$-parameterized? By ...
11
votes
1
answer
425
views
Concerning Silver's result
Jack Silver proved that if $x$ is a real so that every $x$-admissible ordinal is a cardinal in $L$, then $0^{\sharp}$ exists.
I wonder whether various weaker or stronger versions of Silver's result ...
5
votes
2
answers
428
views
Does the boldface class $\Delta^1_2$ have the uniformization property? (assuming $V=L$)
DISCLAIMER: All pointclasses considered here are boldface.
Most of the time, when doing descriptive set theory, we want the projective sets to "behave well;" for example, maybe we don't want there to ...
7
votes
1
answer
337
views
Analytic uniformization
Suppose I am given a subset of $2^\omega\times\omega^\omega$ of some bounded Borel rank. Can I get an analytic uniformization of this set?
5
votes
0
answers
345
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 ...
15
votes
2
answers
3k
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 ...
6
votes
2
answers
309
views
Borel kernel over an analytic set implies existence of a Borel map
Let $X$ and $Y$ be standard Borel spaces, and let $A\subseteq X\times Y$ be an analytic set with a full projection on $X$: that is $\pi_X(A) = X$. Suppose that there exists a Borel-measurable kernel $\...
3
votes
1
answer
142
views
Maps that are a.e. equal have almost the same graphs
Let $X$ and $Y$ be two measurable spaces, and let $p$ be a probability measure on $X\times Y$. Denote by $p_X$ the marginal of $p$ on $X$, that is an image of $p$ under projection on $X$. Consider two ...
5
votes
1
answer
446
views
Universally measurable map coincides a.e. with a Borel map
Let $X$ be a standard Borel space: that is, a topological space equivalent to a Borel subset of $\Bbb R$. It is known that for any probability measure $p$ on $X$ and any universally measurable set $A\...
16
votes
1
answer
670
views
Question about product topology
Suppose $S\subset\mathbb{R}$ is dense without interior point, and for every open interval $I,J\subset\mathbb{R}$, $I\cap S$ is homeomorphic to $J\cap S$.
Is $S\times S$ homeomorphic to $S$?
By Luzin ...
12
votes
1
answer
747
views
Consistency strength of projective determinacy (PD)
Let PD stand for projective determinacy, and consider the two claims:
(1) For each n=1,2,..., Con(ZFC+PD) implies Con(ZFC + there are n Woodin cardinals)
(2) Con(ZFC+PD) implies Con(ZFC + there are ...
6
votes
1
answer
275
views
Is the ideal of compact operators strongly Borel?
Let $H$ be a separable infinite dimensional Hilbert space. Denote by $\mathcal{B}(H)$ the space of bounded operators on $H$, and $\mathcal{K}(H)$ the ideal of compact operators. When endowed with the ...