# Questions tagged [cardinal-characteristics]

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

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### 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 ...

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### 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 $\mathfrak{b} = \mathfrak{d}$, ...

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### 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(...

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### 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 ...

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### 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 ...

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### 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-...

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### 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 $\...

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### 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 ...

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### 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 $\...

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### 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 $\sf ZFC$?...

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### 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 ...

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### 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 ...

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### 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 ...

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### 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 ...

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### 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 ...

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### 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 ...

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### 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\...

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### 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 ...

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### 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 ...

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### 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 ...

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### 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 ...

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### 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 $\...

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### 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 ...

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### 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
$$\...

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### 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}...

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### 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{\...

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### 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 ...

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### 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 ...

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### 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 $F[A]\in\mathbb{B}$...

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### 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 $X\in\mathbb{A}$...

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### 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 ...

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### 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 $\operatorname{non}(\mathcal B)$) ...

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### 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 ...

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### 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: ...

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### 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 ...

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### 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 ...

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### Upper bound for constructibility orders of elements of the continuum

In a constructible universe (ZFC + V=L), is there any known upper bound for the constructibility orders of all elements of the continuum, i.e. some separately described ordinal $\alpha$ such that we ...

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### Is there a condensation from $\aleph_1^{\aleph_0}$ onto a metrizable compact space?

Is there a condensation (continuous bijective mapping) from $D^{\aleph_0}$ onto a metrizable compact space ?
$D$ - discrete space of cardinality $\aleph_1$.
CH implies it is a positive answer. In ...

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### Cardinality of the closure of subset of a dense subset

Is the following true? Given a first countable Hausdorff space $X$ and a dense subset $Y\subset X$ which is initially $2^\omega$-initially Lindelöf in $X$. Let $F$ be a subset of $Y$ such that $\vert ...

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### Selecting an almost disjoint family in a given family of sets

A family $\mathcal A$ of infinite subsets of $\omega$ is called almost disjoint if for any distinct sets $A,B\in\mathcal A$ the intersection $A\cap B$ is finite.
Let $\mathfrak a'$ be the largest ...

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### What is the smallest cardinality of a base of an ultrafilter on $\omega$ related to an almost disjoint family of cardinality $\mathfrak c$?

Let $(A_\alpha)_{\alpha\in\mathfrak c}$ be an almost disjoint family of infinite subsets of $\omega$. The almost disjointness of the family means that $A_\alpha\cap A_\beta$ is finite for any ordinals ...

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### What is the smallest cardinality of a maximal ultrafamily of infinite subsets of $\omega$?

A family $\mathcal U$ of infinite subsets of $\omega$ is called an ultrafamily if for any sets $U,V\in\mathcal U$ one of the sets $U\setminus V$, $U\cap V$ or $V\setminus U$ is finite.
By the ...

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### Cardinal numbers and the Bolzano-Weierstrass theorem

Let $\kappa$ be a cardinal number, define $\textsf{M}(\kappa)$ and $\textsf{BW}(\kappa)$ as follows:
$\textsf{M}(\kappa)$ : For every sequence $(f_{n}:\kappa\to \mathbb{R})_{n\in\mathbb{N}}$ of real-...

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### Influence of cardinal characteristics on nonstandard analysis?

As I understand, nonstandard analysis usually proceeds by taking a ultrapower of the universe by some nonprincipal ultrafilter on $\mathbb N$. There are continuum many “integers” of this model, but ...

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### Strong chains of uncountable functions and cardinal characteristics

A family of functions $\langle f_\alpha:\alpha<\kappa\rangle$ from $\omega_1$ to $\omega_1$ is called a strong chain if $\alpha<\beta<\kappa\Longrightarrow \{\xi<\omega_1: f_\beta(\xi)\leq ...

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### Is $\mathfrak j_{2:1}=\mathfrak{j}_{2:2}$ in ZFC?

A function $f:\omega\to\omega$ is called
$\bullet$ 2-to-1 if $|f^{-1}(y)|\le 2$ for any $y\in\omega$;
$\bullet$ almost injective if the set $\{y\in \omega:|f^{-1}(y)|>1\}$ is finite.
Let us ...

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### Calculate the $\downarrow$, $\downarrow\uparrow$ and $\uparrow\downarrow$ cofinalities of the poset of nontrivial finitary partitions of $\omega$

Let $(P,\le)$ be a poset. For a point $x\in P$ let
$${\downarrow}x=\{p\in P:p\le x\}\quad\text{and}\quad{\uparrow}x=\{p\in P:x\le p\}$$be the lower and upper sets of the point $x$, and for a subset $...

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### Pseudo-intersections, splitting families, and ultrafilters

Suppose $U$ is a non-principal ultrafilter on $\omega$, and let us define $\tau(U)$ to be the minimum cardinality of a family $\mathcal{X}\subseteq U$ such that $\mathcal{X}$ does not have an infinite ...

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### Bounding and dominating numbers for preordering $\leq^*$ on bijections $f:\omega\to\omega$

For functions $f, g:\omega\to\omega$ we write $f \leq^* g$ if $\{x\in\omega: f(x)> g(x)\}$ is finite.
Let $S_\omega$ denote the collection of bijections $\varphi:\omega\to\omega$ Similarly to the ...

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### Bounding and domination numbers for relation $\leq$ modulo $\omega$-nullsets

We say that $A\subseteq \omega$ is a nullset if $$\lim\sup_{n\to \infty} \frac{|A\cap n|}{n+1} = 0.$$
Let $\omega^\omega$ denote the set of functions $f:\omega\to\omega$. We define a pre-ordering ...