Questions tagged [cardinal-characteristics]

For questions about various cardinal invariants, cardinal characteristics of the continuum and related topics.

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If parameters in ZFC are restricted to definable sets, can existence of uncountable sets be proven?

If we restrict all parameters in the set building axioms of $\sf ZFC$ to definable sets, would the celebrated Cantor's theorem still apply? Can existence of uncountable sets be proven at all? The set ...
10 votes
0 answers
152 views

Can the nowhere dense sets be more complicated than the meager sets?

Suppose $X$ is a completely metrizable space with no isolated points. Let $\mathcal{ND}_X$ denote the ideal of nowhere dense subsets of $X$, and let $\mathcal{M}_X$ denote the ideal of meager subsets ...
8 votes
1 answer
337 views

"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 ...
3 votes
1 answer
153 views

Is there any class of ideals for which $\mathfrak{b}(\mathcal I)\neq\mathfrak{b}$?

Consider the Baire space $\mathbb{N}^\mathbb{N}$. A natural pre-order $\leq^*$ on $\mathbb{N}^\mathbb{N}$ is defined by $f\leq^*g$ if and only if $f(n)\leq g(n)$ for all but finitely many $n$. A ...
24 votes
2 answers
2k views

Short proof of $\frak p=t$

It is known for a while now that $\frak p=t$, as a result of Malliaris-Shelah. The original paper draws from model theoretic methods. I've heard rumors that there was a proof which was purely set ...
4 votes
1 answer
215 views

Comparing bornologies for cardinal characteristics via Borel maps

This question is "take 2" of this older one, following a suggestion of Francois Dorais. Consider the following bornologies $\mathbb{D},\mathbb{E}$ on the set $\mathcal{N}$ of all functions ...
4 votes
0 answers
289 views

How does "spreading-with-determinacy" compare with Cichon's diagram?

For $\mathbb{A},\mathbb{B}\subseteq\mathcal{P}(\omega^\omega)$, say that $\mathbb{A}$ spreads onto $\mathbb{B}$ iff there is some $F:\omega^\omega\rightarrow\omega^\omega$ such that $F[A]\in\mathbb{B}$...
5 votes
1 answer
163 views

Complemented subspaces of a dual Banach space

Let $\kappa$ be an infinite cardinal number and by $\mathcal{B}(\kappa)$ denote the class of all Banach spaces of density $\kappa$. My question reads as follows: Does there exist $\kappa$ for which ...
4 votes
1 answer
235 views

Cofinal rectangles in poset

Suppose $(P, <)$ is a poset of cofinality $\aleph_2$ and additivity (least cardinality of an unbounded subset) $\aleph_1$. Can we conclude the existence of a cofinal subset of order-type $\omega_1 \...
7 votes
1 answer
234 views

$\operatorname{cof}(\mathcal{L}) = \aleph_1$ and $2^{\aleph_0} = \aleph_3$

Here $\operatorname{cof}(\mathcal{L})$ refers to the largest cardinal characteristics of the continuum in Cichon's diagram. My question is: Is the theory $\mathsf{ZFC}+ \operatorname{cof}(\mathcal{L})...
4 votes
0 answers
204 views

Where can I find information about this concept of 'dual ideals'?

I have come across the following notion of (what I am calling) dual ideals, and I am looking for any work in which this notion has been considered, and particularly anything about transferring ...
1 vote
1 answer
134 views

Thinning directed sets ${\frak P}$ of partitions of $\omega$ with no ${\frak P}$-discrete subsets

This question branches from Taras Banakh's recent question on a cardinal characteristic connected to families of partitions that are directed in the ordering of partition refinement. A partition $\...
1 vote
0 answers
124 views

How much choice we can get from this partition principle?

For every set $X$ there cannot be a partition on $X$ of a larger size than the set $\iota``X$ of all singleton subsets of $X$. Formally: $$\begin{align} \forall X \forall P: P \text { is a partition ...
5 votes
0 answers
102 views

Comparing Mathias forcing notions relative to various filters

Let $\mathcal F$ be a (non-principle, non trivial, ...) filter on $\omega$. The Mathias Forcing relative to $\mathcal F$ is the forcing notion $\mathbb M(\mathcal F)$ consisting of pairs $(s, X)$ with ...
0 votes
0 answers
49 views

Does parallelism of cardinal comparison between sets and their power sets, enact a form of choice? [duplicate]

Let $ * $ range over cardinal relations $ \{<,<>\}$; if we add the following axiom to $\sf ZF$, would that prove a known form of choice? Parallelism: $ |x| * |y| \leftrightarrow |\mathcal P(...
6 votes
1 answer
135 views

Cofinal trees in $({}^\omega \omega , \leq^\ast )$

So, I know that the existence of a scale (that is, a linear cofinal set in $({}^\omega \omega , \leq^\ast )$, where $\leq^\ast $ is eventual domination, is equivalent to $\mathfrak{b} = \mathfrak{d}$, ...
5 votes
1 answer
364 views

Is it true that $\mathit{MA}(\omega_1)$ iff $\omega_1<\mathfrak{p}$?

Recall that $\mathfrak{p}=\min\{|F|: F$ is a subfamily of $[\omega]^{\omega}$ with the sfip which has no infinite pseudo-intersection $\}$. The cardinal $\mathfrak{q}_0$ defined as the smallest ...
14 votes
1 answer
555 views

How “disconnected” can a continuum be?

A continuum is a compact connected metrizable topological space. Given a cardinal $\kappa$, a topological space $X$ is called $\kappa$-connected if it is not possible to write $X$ as the disjoint ...
5 votes
1 answer
204 views

Cofinal well-founded subset in mod finite order

The mod finite order on ${}^\omega \omega$ is defined as $f \leq^\ast g$ if and only if $f(n) \leq g(n)$ except for finitely many $n \in \omega$. My question is: can we always extract a cofinal well-...
9 votes
0 answers
243 views

Another determinacy-related cardinal characteristic

This question is a kind of "dual" to an earlier one of mine. Although I don't know a reference for this, it's easy to show the following result: Suppose $G$ is a game in which neither ...
12 votes
1 answer
341 views

Can we force $\mathfrak{r}<\mathfrak{s}$?

Are there models of ZFC in which $\mathfrak{r}$ is strictly less than $\mathfrak{s}$? I've not been able to find any forcings that end up with this result. Here $\mathfrak{r}$ is the reaping number $\...
9 votes
0 answers
234 views

Is this cardinal characteristic trivial? (Number of strategies needed to guarantee at least one win)

(Previously asked at MSE.) Let the determinacy number, $\mathfrak{g}$ (for "game"), be the smallest cardinal such that for every (two-player, perfect-information, length-$\omega$) game on $\...
3 votes
0 answers
91 views

Can we have a model of ZFC for any increasing cardinality function over its ranks?

Per Löwenheim–Skolem theorems, is it a result that we can have a model of $\sf ZFC$ with any increasing cardinality function over its ranks? Formally, is the following scheme consistent with $\sf ZFC$?...
6 votes
1 answer
186 views

Steinhaus number of a group

$\newcommand\Sn{\mathit{Sn}}$A subset $A$ of a group $X$ is called algebraic if $A=\{x\in X: a_0xa_1x\dotsm xa_n=1\}$ for some elements $a_0,a_1,\dotsc,a_n\in X$. Let $\mathcal A_X$ be the family of ...
7 votes
0 answers
137 views

The smallest cardinality of a cover of a group by algebraic sets

$\DeclareMathOperator\cov{cov}$A subset $A$ of a semigroup $X$ is called algebraic if $$A=\{x\in X: a_0xa_1x...xa_n=b\}$$ for some $b\in X$ and $a_0,a_1,...,a_n \in X^1=X\cup \{1\}$. The smallest ...
2 votes
0 answers
83 views

How do the finite partitioning principle compare with infinite Dedekindian addition?

In a posting to MathStackExchange, what I've labeld as the finite partitioning principle was answered to be implied by the principle that for every infinite set $X$, $X+X=X$, what I'd label as the ...
2 votes
1 answer
107 views

Minimal cardinality of non-bipartite sub-family of $[\omega]^\omega$

Let $[\omega]^\omega$ the collection of infinite subsets of $\omega$. We say that $E\subseteq [\omega]^\omega$ is bipartite if there is $d\subseteq \omega$ such that for all $e\in E$ the intersections ...
3 votes
0 answers
76 views

Is $\mathfrak q_0$ equal to the smallest cardinality of a second-countable $T_1$-space which is not a $Q$-space?

A topological space $X$ is a $Q$-space if every subset of $X$ is of type $G_\delta$. The smallest cardinality of a metrizable separable space which is not a $Q$-space is denoted by $\mathfrak q_0$ and ...
0 votes
1 answer
166 views

At which large cardinal, the theory of the minimal transitive model of ZFC starts proving its absence?

Let's take the minimal transitive model of $\sf ZFC$ which, I came to know, is some minimal $L_\kappa$ for a countable $\kappa$, that models $\sf ZFC$, and since its minimal so no subset of it can be ...
2 votes
0 answers
115 views

Cardinal characteristics of the ideal of $\sigma$-continuity of the Pawlikowski function

A function $f:X\to Y$ between topological spaces is called $\sigma$-continuous if there exists a countable cover $\mathcal C$ of $X$ such that for every $C\in\mathcal C$ the restriction $f{\...
4 votes
1 answer
148 views

Comparing bornologies for domination/escaping

Consider the following bornologies $\mathbb{D},\mathbb{E}$ on the set $\mathcal{N}$ of all functions from $\mathbb{N}$ to $\mathbb{N}$: $\mathbb{D}=\{A: \exists f\in\mathcal{N}\forall g\in A\exists m\...
5 votes
0 answers
185 views

What are the known large cardinal axioms for which weaker and stronger set theories "catch up"?

I will clarify what I mean by the title in the following four ways: For which cardinals $\kappa$ do we have that ZFC-(Powerset axiom)+$\exists\kappa$ is equiconsistent with ZFC? If that is not ...
3 votes
0 answers
121 views

A space with independent tightness

Recall that the tightness of a topological space $X$ is defined as the least cardinal $\kappa$ such that for every non-closed subset $A$ of $X$ and every point $x \in \overline{A} \setminus A$, there ...
1 vote
1 answer
145 views

Is existence of a cardinal that witness non-failure of GCH everywhere everyway, a theorem of ZF?

In an earlier positing to $\mathcal MO$, it appears that the answer to if the $\sf GCH$ can fail everywhere in every way is to the negative, this is the case in $\sf ZFC$, however it also appears that ...
2 votes
0 answers
80 views

A convex version of the small uncountable cardinal $\mathfrak b$

Let us recall that $\mathfrak b$ is the smallest cardinality of a subset of $\omega^\omega$, which cannot be covered by countably many compact subsets of $\omega^\omega$. The definition of $\mathfrak ...
2 votes
1 answer
143 views

Game versions of the tower number $\mathfrak t$

Let us recall that $\mathfrak t$ is the smallest cardinal $\kappa$ for which there exists a family $(T_\alpha)_{\alpha\in\kappa}$ of infinite subsets of $\omega$ such that $\bullet$ for any ordinals $\...
5 votes
1 answer
193 views

The cardinal characteristic $\mathfrak r_{(X,f)}$ of a dynamical system

I am interested in a "dynamical" modification of the cardinals $\mathfrak r$ and $\mathfrak r_\sigma$, well-known in the theory of cardinal characteristics of the continuum. For a compact ...
13 votes
2 answers
983 views

Consistency results separating three cardinal characteristics simultaneously

(For information on cardinal characteristics of the continuum aka cardinal invariants see Joel David Hamkins' MO answer here; Andreas Blass's handbook article is an excellent reference.) Problem 2.3 ...
5 votes
1 answer
152 views

Can the Boolean group $C_2^\omega$ be covered by less than $\mathfrak b$ nowhere dense subgroups?

Let $\mathrm{cov}_H(C_2^\omega)$ be the smallest cardinality of a cover of the Boolean group $C_2^\omega=(\mathbb Z/2\mathbb Z)^\omega$ by closed subgroups of infinite index. It can be shown that $$\...
3 votes
1 answer
158 views

Bounds for a covering number of the circle group $\mathbb T$ by some its small subgroups

$\newcommand{\w}{\omega}\newcommand{\A}{\mathcal A}\newcommand{\F}{\mathcal F}\newcommand{\I}{\mathcal I}\newcommand{\J}{\mathcal J}\newcommand{\M}{\mathcal M}\newcommand{\N}{\mathcal N}\newcommand{\x}...
5 votes
1 answer
310 views

Bounds for a small cardinal

$\newcommand{\w}{\omega}\newcommand{\F}{\mathcal F}\newcommand{\I}{\mathcal I}\newcommand{\J}{\mathcal J}\newcommand{\M}{\mathcal M}\newcommand{\N}{\mathcal N}\newcommand{\x}{\mathfrak x}\newcommand{\...
16 votes
2 answers
637 views

Diagonalizing against $\omega_1$-sequences of functions mod finite

The following statement is a direct consequence of the Continuum Hypothesis: There exists a sequence $\langle f_\alpha:\omega_1\rightarrow\omega_1 ~ \vert ~ \alpha<\omega_1\rangle$ of functions ...
11 votes
1 answer
698 views

Can we separate the almost-disjointness sunflower numbers?

This question concerns a new cardinal characteristic of the continuum that arose out of issues in my answer to the question, Sunflowers in maximal almost disjoint families. A family $\cal A$ of ...
6 votes
1 answer
253 views

Do escaping sets "uniformly" cover dominating sets under determinacy?

For $\mathbb{A},\mathbb{B}\subseteq\mathcal{P}(\omega^\omega)$, say $\mathbb{A}$ spreads onto $\mathbb{B}$ iff there is some $F:\omega^\omega\rightarrow\omega^\omega$ such that for all $X\in\mathbb{A}$...
1 vote
0 answers
86 views

Are there results on cardinal function using o-tightness?

Recall that a space $X$ has countable $o$-tightness, if for every family $\mathcal U$ of open sets of $X$ and for each $x \in X$ with $x \in \overline{\bigcup \mathcal U}$, there exists a countable ...
7 votes
0 answers
152 views

A model where the uniformity of the meager ideal is strictly below the almost disjointness number

I'm looking for a model satisfying the inequality described in the title. Recall that the uniformity of the meager ideal, denoted $\operatorname{non}(\mathcal M)$ (or $\operatorname{non}(\mathcal B)$) ...
4 votes
0 answers
203 views

The list reaping number?

My question is inspired by a question of Dominic van der Zypen. Let $[\omega]^\omega$ denote the set of all infinite subsets of $\omega$. The reaping number, denoted by $\mathfrak r$, is the minimum ...
3 votes
1 answer
184 views

A sequence of cardinal characteristics constructed with hypergraph coloring

Let $[\omega]^\omega$ denote the collection of infinite subsets of $\omega$. A hypergraph $H=(V,E)$ consists of a set $V$ and a collection of subsets $E \subseteq {\cal P}(V)$. A coloring is a map $c: ...
2 votes
0 answers
187 views

What is the smallest number of nowhere dense affine subsets covering a topological group?

$\DeclareMathOperator\cov{cov}\newcommand\A{\text A}$A subset $A$ of a group $G$ is called affine if $A=xHy$ for some subgroup $H\subseteq G$ and some $x,y\in G$. Given a non-discrete topological ...
5 votes
0 answers
140 views

Two cardinal characteristics of the continuum, related to the Bohr topology on integers

For a subset $A\subseteq\mathbb T$ of the unit circle $\mathbb T=\{z\in\mathbb C:|z|=1\}$, let $\tau_A$ be the smallest topology on the additive group of integers $\mathbb Z$ such that for every $z\in ...