This tag is for questions about cardinals and related topics such as cardinal arithmetics, clubs, stationary sets, cofinality, and principles such as $\lozenge$. Do not confuse with [large-cardinals] which is a technical concept about strong axioms of infinity.

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2
votes
3answers
58 views

Countability of a set: only 2 options?

So I know sets can be countable (bijection between set and $\mathbb{N}$, finite) or uncountable. Is there another option or are all sets either or?
1
vote
2answers
37 views

cardinality with finite sets

$A,B,C$ are finite sets. Suppose $A\subseteq B \subseteq C$ and $\#A=\#C$. Prove that $\#A=\#B$ and $\#B=\#C$. Should I prove this by showing that there exist an element in $A$ that exist in $B$ and ...
1
vote
1answer
64 views

Cartesian product of large sets

For a non-empty set $A$ let $A'$ denote the Cartesian product of $A$ with itself taken denumerably many times. Now given a set $S$ whose cardinality is strictly greater than the cardinality of ...
0
votes
1answer
22 views

An alternative succinct proof needed for trivial cardinality fact

Let $|X|$ denote the cardinality of a set, i.e. the least ordinal $\alpha$ such that there is a bijection between X and $\alpha$. For any sets $X$ and $Y$ we write $X\preccurlyeq Y$ if the exists an ...
0
votes
1answer
23 views

Cardinal arithmetic basics

Let's say we have $\omega + \omega$. Since these sets are not disjoint we can replace them by disjoint sets of the same cardinality, namely $\omega \times \{0\}$ and $\omega \times \{1\}$. Then $\big ...
0
votes
1answer
41 views

If $\gamma >0,$ then $ \alpha < \beta $ implies that $\gamma \ . \alpha<\gamma \ . \beta$ and that $\alpha \ . \ \gamma \leq \beta \ . \ \gamma.$

Show that if $\gamma >0,$ then $ \alpha < \beta $ implies that $\gamma \ . \alpha<\gamma \ . \beta$ and that $\alpha \ . \ \gamma \leq \beta \ . \ \gamma.$ (All operations are cardinal ...
0
votes
1answer
56 views

Cardinals and $2^\aleph$

The cardinal of $\{e^{ax}\,\, |\,\, a \in \mathbb{R}\}$ is $\aleph$, What is the cardinal of the group of all functions that are linear combinations of ones from the first group? If it's not ...
-2
votes
1answer
56 views

How many disjoint disks can be found in $\mathbb{R} \times \mathbb{R}$?

I know that the answer is $\mathbb{Q} \times \mathbb{Q}$ so the answer is $\aleph_0$ But why? Can't I find a $\mathbb{R} \times \mathbb{R}$ point in every disk?
11
votes
0answers
362 views

Formulations of Singular Cardinals Hypothesis

I have stumbled on a few different formulations of the Singular Cardinals Hypothesis. The most common are: SCH1: $\quad 2^{cf(\kappa)}<\kappa \ \Longrightarrow \ \kappa^{cf(\kappa)}=\kappa^+$ for ...
6
votes
0answers
107 views

AC iff $P(\delta)$ can be well-ordered for all $\delta\in{\bf On}$

I'm trying to do the following exercise: Exercise. Assume ZF. Then AC is equivalent to the statement that $P(\delta)$ can be well-ordered for all $\delta\in{\bf On}$. I'm struggling with the ...
6
votes
0answers
219 views

About singular $\beth_{\alpha}$ for limit ordinals $\alpha$

Why are there cofinitely many regular cardinals of type $\beth_{\beta}$ below any given singular $\beth_{\alpha}$, if $\alpha$ is a limit ordinal?
6
votes
0answers
126 views

Is there an upper bound to the number of rings that can be obtained from a semigroup with zero by defining an additive operation?

Let $\mathscr S$ be the class of all semigroups with zero. For $(S,\times,0)\in\mathscr S,$ I want to count additive operations $+$ on $S$ such that $(S,+,\times,0)$ is a ring (possibly without ...
5
votes
0answers
156 views

Is there a simple example of a ring that satifies the DCC on two-sided ideals, but doesn't satisfy the ACC on two-sided ideals?

It follows from the Hopkins–Levitzki theorem that if a ring satisfies the DCC on left ideals, then it also satisfies the ACC on left ideals. I've been trying to find a counterexample to the following ...
4
votes
0answers
64 views
+50

What happens if we replace “regularity” in GCH with other conditions?

We can formulate both a weak and a strong generalized continuum hypothesis. GCH0. If $\kappa$ is an infinite cardinal number, then $2^\kappa$ is regular. GCH1. If $\kappa$ is an infinite cardinal ...
4
votes
0answers
80 views

Solovay Theorem

Solovay's Theorem states If $\kappa$ is regular uncountable, then any stationary set in $\kappa$ can be partitioned into $\kappa$-many pairwise disjoint stationary sets. For regular uncountable ...
4
votes
0answers
76 views

Is $X$ pseudocompact

The following example with a little modified from the handbook of set theoretic topology, Page 574: Let $\kappa$ be any cardinal for which there exists a family $\{H_\alpha: \alpha < \kappa\}$ ...
4
votes
0answers
37 views

Cardinal characteristics — generalisation?

I'm just reading about what is called cardinal characteristics of the continuum. For example there are the bounding number $\mathfrak b$ and the dominating number $\mathfrak d$ etc. which are defined ...
4
votes
0answers
172 views

Property of Galvin-Hajnal rank

In what follows, $\|\Phi\|_I$ means the Galvin-Hajnal rank of a function $\Phi:\kappa \to Ord$ with respect to a $\omega_1$-complete ideal $I$ of $\kappa$. Lemma 2.2.5 of Introduction to Cardinal ...
3
votes
0answers
65 views

Which Field Would You Use to Represent a Group Larger than $\aleph _1$?

I understand that in representation theory we try to understand a group $G$ by studying the homomorphisms $\rho\ \colon G \to $ GL$(V)$ where $V$ is a vector space over some field. I believe complex ...
2
votes
0answers
71 views

$\mathfrak b \leq \mathfrak r$. Is this proof correct?

I am trying to prove that $\mathfrak b \leq \mathfrak r$ where $\mathfrak r$ is the minimal cardinality of a reaping family of sets. A family $\mathcal R \subseteq [\omega]^\omega$ of sets $\mathcal ...
2
votes
0answers
24 views

(S is a proper class wrt model M) implies ((|T|=|S| under model N) implies (T is a proper class wrt M)). Why?

If I understand correctly (which is far from guaranteed, so a reply telling me that it is rubbish will be better than no reply at all): Suppose we have two models N and M such that N is a ...
2
votes
0answers
52 views

A question on semi-stratifiable spaces

A space $(X,\tau)$ is called semi-stratifiable if there exists a function $g:\omega\times X\to\tau$ such that: (i) for any point $x$ of $X$ holds $\{x\}=\bigcap_{n\in\omega} g(n,x)$; (ii) for any ...
2
votes
0answers
90 views

Can any large cardinal axiom be reached by a recursive sequence of unbounded sequences and fixed points?

The lowest large cardinals can be seen as a simple recursive rule that defines an unbounded sequence, then defines a fixed point of the sequence, then an unbounded sequence of fixed points, a fixed ...
1
vote
0answers
61 views

stationary subset of $cf(\kappa)$

Let $\kappa$ a infinite cardinal with uncountable cofinality and $S\subset\kappa$. We can find a normal function $f$ from $\operatorname{cf}(\kappa)$ on $\kappa$ such that ...
1
vote
0answers
79 views

Closing a subcategory under finite colimits by transfinite induction

Let $\mathcal{C}$ be a locally small category with all finite colimits, and let $\mathcal{A}$ be a small full subcategory. I wish to prove the following: Proposition. There exists a full subcategory ...
0
votes
0answers
43 views

What is the cardinal number of the following?

What is the cardinal number of $\{f:\mathbb{R}\longrightarrow \mathbb{N}\ |\text{ $ f$ is an injective function}\}$?
0
votes
0answers
23 views

Infinite One-Time Pad

As you know, when used correctly, a one-time pad allows one to send a message, such that the only thing that can be found out about it is the maximum size (which is also the key length.) It is ...
0
votes
0answers
27 views

cardinality of finite subset

I have the following question: Prove that a set A has the same cardinality of a subset of a Set B, if and only if exists an injective function A to B. I find it hard to prove it because I can easily ...
0
votes
0answers
49 views

Is this sentence OK?

I'm starting to write a paper. This is the sentence which I want to put first in the paper. It is well known that diagonal properties are useful in estimating certain cardinal invariants of a ...
0
votes
0answers
51 views

Can we conclude that $|\prod_{\xi \in Ord, \xi=1}^{\xi<\alpha}\xi|=2^{|\alpha|}$ for all infinite ordinal $\alpha$ in $\mathsf{ZFC}$?

According to this question, $\prod_{\kappa \in Crd, \kappa=1}^{\kappa<\aleph_\alpha}\kappa$, the cardinal product of all positive cardinals smaller than a given infinite cardinal $\aleph_\alpha$ ...
0
votes
0answers
69 views

A quick question about proof of Bukovský-Hechler

The following is an exercise in Just/Weese (page 179), Question 1: can you tell me if I got it right? Thank you! Question 2: Shouldn't it be equality rather than less equals in $\mu = \sum_{\alpha ...
0
votes
0answers
146 views

Enumeration of subsets

I want to show that if $\{X_\beta : \beta<\lambda^+\}$ is an enumeration of $[\lambda\times\lambda^+]^{\leq\lambda}$ then, for all $i<\lambda$, the set $\{(X_\beta)_i:\beta<\lambda^+\}$ is ...