16
votes
1answer
403 views

Three old questions on the Sacks forcing

I came across the two following Qs in 1970. Find reals $a,b$ such that $a$ is Sacks over $L[b]$ and vice versa $b$ is Sacks over $L[a]$. Note that a Sacks $\times$ Sacks generic pair definitely does ...
6
votes
2answers
413 views

Can we define an “empirically generic” real number?

Summary: My question, in a nutshell, is how we should intuitively imagine a generic real number (as opposed to a random one), and whether we can construct numbers which empirically behave like generic ...
10
votes
1answer
426 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 "Turing ...
8
votes
2answers
311 views

When is $A$ “$L$-ish” whenever $B$ is “$L$-ish”?

My question is about a kind of relative constructibility in set theory. Fix a countable transitive model $W\models ZFC$ which is much bigger than $L^W$. There is a natural way within $W$ to compare ...
8
votes
1answer
267 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
2answers
229 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 $p\in P$ ...
-1
votes
1answer
112 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 ...
10
votes
2answers
231 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 ...
4
votes
0answers
183 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 ...
14
votes
2answers
466 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
1answer
207 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. ...
5
votes
0answers
113 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 ...
4
votes
1answer
217 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 ...
5
votes
1answer
215 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, ...
9
votes
1answer
188 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 ...
2
votes
0answers
172 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 ...
-1
votes
1answer
154 views

Algebra generated by a tree [Edit] [closed]

Suppose that $(T,\leq)$ is a partially ordered set, we say $T$ is a tree* if for every $i\in T$, $\{s: s\in T, s\leq t\}$ is a well-founded chain. What I need to know is: Can the algebra ...
8
votes
1answer
249 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 ...
9
votes
1answer
200 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 ...
3
votes
0answers
177 views

Strong measure zero sets and selection principles

A set of reals $X$ is strong measure zero if for any sequence of positive real numbers $ ( \epsilon_n ) _{n \in \omega }$ there is a sequence of open intervals $ ( a_n ) _{n \in \omega }$ which ...
6
votes
1answer
383 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 ...
5
votes
1answer
265 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 ...
9
votes
1answer
244 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 ...
6
votes
1answer
196 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 ...
9
votes
1answer
155 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 ...
11
votes
0answers
201 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
1answer
263 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 ...
9
votes
1answer
288 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
2answers
230 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
1answer
223 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?
16
votes
1answer
461 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 ...
10
votes
1answer
247 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 ...
4
votes
1answer
212 views

From universal measurability to measurability

Let $(\Omega,\Sigma)$ be a measurable space and $K$ be a compact metrizable space endowed with its Borel $\sigma$-algebra $\mathcal{B}(K)$. Let $A\subseteq\Omega\times K$ be universally ...
10
votes
1answer
403 views

Restrictions of null/meager ideal

Let I denote the null (resp. meager) ideal on reals. Is it consistent that for any pair of non null (resp. meager) sets A and B, there is a null (resp. meager) preserving bijection between A and B? In ...
7
votes
1answer
452 views

How long can it take to generate a $\sigma$-algebra?

I want to know if there is a $\sigma$-algebra such that for every countable ordinal $\alpha$ the $\sigma$-algebra can be generated in more than $\alpha$ steps but less than $\omega_{1}$ steps. Given ...
4
votes
1answer
301 views

Models of Determinacy

Today we have that $L(\mathbb{R}) \models AD$ (assuming there are $\omega$ many Woodin cardinals and a measurable above them all). I was wondering what other models of $AD$ might look like and if it ...
4
votes
1answer
375 views

Definition of HYP in $L_{\omega_1^{CK}}[a]$?

The structure $L_{\omega_1^{CK}}$ consists of only HYP sets (I believe) and HYP in this structure is the same as the actual hyperaritmetic sets. Now if I move to the structure $L_{\omega_1^{CK}}[a]$ ...
6
votes
0answers
167 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 ...
2
votes
0answers
251 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 ...
10
votes
3answers
882 views

Games that never begin

Games that never end play a major role in descriptive set theory. See for example Kechris' GTM. Question: Does there exist a literature concerning games that never begin? I have in mind two ...
5
votes
0answers
332 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
275 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 $x_0$ ...
10
votes
1answer
240 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
469 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
0answers
311 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
214 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
123 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
199 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 ...
9
votes
0answers
464 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 ...
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 $\mathbb{R}$, ...