# 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.

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### If $f=h\circ g$, then there's a measurable function $\tilde h$ such that $f=\tilde h\circ g$

Let $X,Y,Z$ be three standard measurable spaces and $f:X\to Z$ and $g:X\to Y$ two measurable functions. Suppose that there's a function $h:Y\to Z$ such that $f=h\circ g$. How can I show that there's a ...
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### Periodicity in the cumulative hierarchy

Under Reinhardt cardinals in ZF, the cumulative hierarchy exhibits a periodicity in that for large enough $λ$, certain properties of $V_λ$ depend on whether $λ$ is even vs odd. See Periodicity in the ...
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### Examples of concrete games to apply Borel determinacy to

I'm teaching a course on various mathematical aspects of games, and I'd like to find some examples to illustrate Borel determinacy. Open or closed determinacy is easy to motivate because it proves ...
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### Sufficient condition for the graph of a measurable map to be measurable

Let $f:X \to Y$ be measurable map between measurable spaces w.r.t. to their corresponding $\sigma$-algebras $\Sigma_X$ and $\Sigma_Y$, resp. If $(X,\Sigma_X)$ is a standard Borel space can we always ...
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### Consistency of definability beyond P(Ord) in ZF

Is it consistent with ZF that the satisfaction relation of $L(P(Ord))$ is $Δ^V_2$ definable? More generally, is it consistent with ZF that there is a $Δ^V_2$ formula (taking $α$ as a parameter) that ...
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### Uniformization and functions on Turing degrees

Assuming Martin's Conjecture on functions between Turing degrees, is AD + DC consistent with existence of an $f:\mathcal{D}_t → \mathcal{D}_t$ of rank $Θ$ ? $\mathcal{D}_t$ is the set of Turing ...
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### meaning of "the singletons of $\Gamma$ are a basis for $\Gamma$"

In a set of notes for the Cabal Seminar a theorem is attributed to Moschovakis which includes the statement that for certain pointclasses $\Gamma$ the singletons of $\Gamma$ are a basis for $\Gamma$. ...
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### The cardinality of projections of subsets of the Hilbert cube by inner products

I have three related questions. Question 1: Is there a subset $X$ of the Hilbert cube $[0,1]^{\Bbb N}$ of cardinality continuum, such that for each sequence $a\in [0,1]^{\Bbb N}$ with $\sum a_n$ ...
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### Is there a Bernstein subset $X$ of $\mathbb{R}$ such that no continuous map $f : X → [0,1]$ is surjective?

Is there a Bernstein subset $X$ of $\mathbb{R}$ such that no continuous map $f : X → [0,1]$ is surjective ?
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### Proof of: No rapid filter is Lebesgue measurable

I'm studying the following theorem in (Schindler, 2014: Set Theory Exploring Independence and Truth), p. 178-180: Theorem 9.16 (Mokobodzki) No rapid filter F $\subset$ ${}^\omega 2$ is Lebesgue ...
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### Is every compact, sober, second-countable space the image of $2^\omega$?

As a bonus, is every compact, $T_0$, second-countable space the image of $2^\omega \times \omega$? As a further bonus, can we strengthen "image" to "quotient"? My motivation for ...
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### Does every (Abelian) Polish group have a nontrivial locally compact subgroup?

The question is pretty much in the title, suppose that $G$ is an (Abelian) nontrivial Polish group, must $G$ have a nontrivial locally compact (in the induced topology, hence necessarily closed) ...
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### The Hausdorff dimension of the set of reals of inner models

Suppose that both $M$ and $N$ are models of $ZFC$ with $M\subseteq N$ so that $M$ is definable in $N$. Question Can $(\mathbb{R})^M$ have Hausdorff dimension strictly between $0$ and $1$ in $N$? How ...
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### Follow up question: Shelah's "Can you take Solovay's inaccessible away?"

In this answer to the question " Shelah's "Can you take Solovay's inaccessible away?" " the following is stated: Assume that $\aleph_1$ is not inaccessible in $L$, hence a ...
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### Erdős–Sierpiński duality in locally compact Polish groups (e.g. $\mathbb{R}^n$)

Erdős–Sierpiński mapping for a locally compact Polish group $G$ is a bijection $f$ from $G$ to $G$ such that $A$ is a null set in $G$ with respect to the Haar measure if and only if $f(A)$ is a meager ...
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### A variation on pinned equivalence relations

Recall (see e.g. Zapletal, Pinned equivalence relations) that a Borel equivalence relation $E$ on $\omega^\omega$ is pinned iff for every forcing $\mathbb{P}$ and every $\mathbb{P}$-name $\nu$ we have ...
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### How much Dependent Choice is provable in $Z_2$? And what about Projective Determinacy?

So, second order arithmetic, $Z_2$, is capable of proving quite a few things. One thing which would be of use is dependent choice for $\mathbb{R}$. Basically, dependent choice on $\mathbb{R}$ says ...
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### Existence of Borel uniformization for coanalytic set with non-$K_\sigma$ sections

Suppose that $X$ is a Polish (or standard Borel) space and $\omega^\omega$ is the Baire space of all natural number sequences. My question is: If $A\subseteq X\times \omega^\omega$ is a coanalytic set ...
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### Is every real number in [0,1] a product of three (or more) Cantor set's numbers?

It is well known that every number $x$ in the unit interval $[0,1]$ is the arithmetic mean of two elements of the (triadic) Cantor set $C$. The way to see it I like the most: the Cantor set is the ...
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### Approximating evalutation maps at open sets over invariant measures

Let $G$ be a group acting by homeomorphisms on a compact metrizable space, say $X$; let's denote by $\alpha:G\to\mathrm{Homeo}(X)$ the action, $g\mapsto\alpha_g$, and consider the weak-$^*$ compact ...
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### Can the set of parafinite congruences be descriptive-set-theoretically complicated?

Fix an algebra $\mathfrak{A}$ with underlying set $\mathbb{N}$ and finite language $\Sigma$. The set of congruences on $\mathfrak{A}$ is a closed subset $C_\mathfrak{A}$ of $2^\mathbb{N}$ (with the ...
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### Two dimensional perfect sets

Consider the following family of sets  \begin{align*} \mathcal{F} = \{X\subseteq [0,1]\times [0,1] \mid \ &X \text{ is closed and }\\& \forall x \in \pi_0 (X) (\{y \in [0,1] \mid (x,y) \in ...
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### What is an "open Baire set"?

In Measures Which Agree on Balls by Hoffmann-Jørgensen, it is stated that if $\varphi$ is a Baire function (which I presume means a pointwise limit of continuous functions), then $\{a<\varphi\}$ is ...
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### "Compactness length" of Baire space

Intuitively, my question is: how many times do we have to mod out by an closed equivalence relation with all classes compact in order to collapse Baire space $\omega^\omega$ to a singleton? In more ...
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### Can convergence in distribution necessarily be realised by almost-sure convergence?

Let $X$ be a Polish space. Let $(\mu_n)_{n \in \mathbb{N} \cup \{\infty\}}$ be a family of Borel probability measures $\mu_n$ on $X$ such that $\mu_n \to \mu_\infty$ weakly as $n \to \infty$. For each ...
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### The most powerful inner model and a $\Delta^2_1$ well-ordering of the reals

With the current research, it seems that we are in a position to get extremely powerful absoluteness theorems (like $\Sigma^2_0$-absoluteness, $\Sigma^2_1$-absoluteness, $\Sigma^2_2$, $\diamondsuit_G$,...
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### Inner model for KP and a Well-Ordering of the Reals

It is well known that Gödel proved the following theorem: $\mathsf{ZFC + V=L}$ has a $\mathit{\Delta}^1_2$-good well-ordering of $\mathbb{R}$. (Gödel, Addison) So: Is there an inner model for KP/Z/....
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### How much of second-order arithmetic do you need for $\mathbf{\Sigma}^1_1$-determinacy to give you countable transitive models of $\mathsf{ZFC}$?

This is in some sense a follow-up to this question. The answer there says that over $\mathsf{Z}_2$ (second-order arithmetic), (boldface) $\mathbf{\Sigma}^1_1$-determinacy is enough to entail the ...
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### Borel sets in Vietoris topology

Let $\mathcal{K} = \mathcal{K}(\mathbb{N}^{\mathbb{N}})$ be the set of all non-empty compact subsets of the Baire space $\mathbb{N}^\mathbb{N}$ equipped with the Vietoris topology. Let $G$ be a Borel ...
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### How much determinacy do you need for second order arithmetic to be as strong as ZFC?

From Wikipedia (I couldn't find the original source): $\text{ZFC} + \{\text{there are$n$Woodin cardinals:$n$is a natural number}\}$ is conservative over $\text{Z}_2$ with projective determinacy. ...
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