Questions tagged [cardinal-characteristics]
For questions about various cardinal invariants, cardinal characteristics of the continuum and related topics.
95 questions
7
votes
0
answers
151
views
Cardinal characteristics and $\mathfrak{c} < \aleph_\omega$
Let $\mathsf{R}$ denote some finitely many relations about finitely many cardinal characteristics (e.g. $\mathfrak{a} \leq \mathfrak{s}$, $\mathfrak{a} < \mathfrak{d} = \mathfrak{r}$, $\mathfrak{b} ...
- 1,442
8
votes
1
answer
246
views
Is $\operatorname{non}(\mathcal{M}) < \mathfrak{a}$ consistent?
Let $\operatorname{non}(\mathcal{M})$ be the least cardinality of a non-meagre subset of the reals. Let $\mathfrak{a}$ be the least cardinality of an infinite maximal almost disjoint family (i.e. $\...
- 1,442
8
votes
1
answer
385
views
Is "every infinite set of strictly subnumerous sets is supernumerous to its union" equivalent to AC?
Is the following sentence equivalent to $\sf AC$ over the rest of axioms of $\sf ZF$?
For each infinite set $X$: if for all $y \in X$ we have $|y| < |X|$, then $| \bigcup X|\leq |X| $?
Note: ...
- 11.3k
4
votes
0
answers
150
views
Does this rule imply axiom of choice?
if $\kappa; \lambda$ are infinite Scott cardinals, then: $$2^\kappa = 2^\lambda \leftrightarrow
\kappa \leq \lambda < 2^\kappa \lor \lambda \leq \kappa < 2^\lambda $$
Would adding the ...
- 11.3k
6
votes
2
answers
470
views
Does Well-Ordered Interval Power Set "WOIPS" principle , prove $\sf AC$ in $\sf ZFA$?
Does $\sf ZFA + WOIPS$ prove $\sf AC$?
Where $\sf WOIPS$ is phrased as: for every infinite set $X$ the set of intervening cardinals between $|X|$ and $|\mathcal P(X)|$ is well-ordered.
In $\sf ZF$, I ...
- 11.3k
3
votes
0
answers
154
views
How the cardinalities of $\mathcal H^*_x$ and $\mathcal P(x)$ compare?
Working in $\sf ZF$, regarding an arbitrary Dedekind infinite set $x$, what's the relationship between the cardinality of $\mathcal P(x)$ and $\mathcal H^*_x$? The latter is the set of all sets ...
- 11.3k
2
votes
1
answer
240
views
Can there be a minimal remote cardinal?
Working in $\sf Z- Reg.$ we can have sets bigger than every well-founded set, let's label such sets as "remote". Now, suppose we'll add the axiom of existence of transitive closures, and the ...
- 11.3k
3
votes
2
answers
622
views
Can uncountable sets be proved to exist in this variant of ZFC with definability restrictions?
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 ...
- 11.3k
11
votes
0
answers
172
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 ...
- 18.6k
8
votes
1
answer
351
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 ...
- 20.5k
3
votes
1
answer
160
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 ...
- 505
4
votes
1
answer
221
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 ...
- 20.5k
5
votes
1
answer
177
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 ...
- 1,151
4
votes
1
answer
239
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 \...
- 505
8
votes
1
answer
291
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})...
- 2,286
4
votes
0
answers
237
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 ...
- 2,206
1
vote
0
answers
130
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 ...
- 11.3k
5
votes
0
answers
109
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 ...
- 1,882
6
votes
1
answer
162
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}$, ...
- 505
0
votes
0
answers
57
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(...
- 11.3k
5
votes
1
answer
370
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 ...
- 1,101
14
votes
1
answer
581
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 ...
- 3,011
5
votes
1
answer
215
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-...
- 505
12
votes
1
answer
363
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 $\...
- 2,206
9
votes
0
answers
252
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 ...
- 20.5k
9
votes
0
answers
243
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 $\...
- 20.5k
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$?...
- 11.3k
6
votes
1
answer
191
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 ...
- 42k
7
votes
0
answers
138
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 ...
- 42k
3
votes
0
answers
85
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 ...
- 11.3k
2
votes
1
answer
109
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 ...
- 48.9k
3
votes
0
answers
80
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 ...
- 42k
1
vote
1
answer
193
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 ...
- 11.3k
4
votes
1
answer
151
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\...
- 20.5k
9
votes
2
answers
347
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 ...
- 524
3
votes
0
answers
122
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 ...
- 4,413
1
vote
1
answer
151
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 ...
- 11.3k
2
votes
0
answers
82
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 ...
- 42k
2
votes
1
answer
155
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 $\...
- 42k
5
votes
1
answer
199
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 ...
- 42k
5
votes
1
answer
155
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
$$\...
- 42k
3
votes
1
answer
163
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}...
- 5,409
5
votes
1
answer
320
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{\...
- 5,409
16
votes
2
answers
649
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 ...
- 296
11
votes
1
answer
711
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 ...
- 236k
4
votes
0
answers
293
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 $F[A]\in\mathbb{B}$...
- 20.5k
6
votes
1
answer
258
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 $X\in\mathbb{A}$...
- 20.5k
1
vote
0
answers
88
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 ...
- 621
9
votes
0
answers
174
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 $\operatorname{non}(\mathcal B)$) ...
- 1,882
4
votes
0
answers
222
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 ...
- 13.4k