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

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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 ...
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0answers
62 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 $\aleph_\alpha$ ...
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2answers
122 views

Can we conclude $\prod_{\kappa \in Crd, \kappa=1}^{\kappa<\aleph_\alpha}\kappa=2^{\aleph_\alpha}$ in ZFC?

In complex analysis, there is a function called Euler's Gamma function. Whenever given a positive integer $n+1$, it will return $n!=\prod_{i=1}^{i < n+1}i$. I'm not sure if there is similar ...
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1answer
70 views

Trouble understanding cardinality

Hi guys I am having trouble understanding cardinality. I am given this practice question. 1) Use Cantor-Schroder-Bernstein Theorem to prove that the intervals $(0,1)$ and $[0,1]$ have the same ...
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1answer
119 views

Product of a family of spaces of countable tightness

I recently learned the concept of cardinal functions and some of the definitions and theorems are not clear to me. How can we prove this theorem? Finite family of compact spaces of countable ...
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1answer
2k views

minimal infinite sigma algebra [duplicate]

Does there exist sigma algebra whose cardinality is countably infinite? If yes tell me some examples. If not how to show every infinite sigma algebra is uncountable?
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2answers
333 views

Is the class of all ordinals independent of set theory?

I am little confused with this. In any model of set theory (or is only in ZFC?), we start with an initial set, and then by transfinite recursion we build the universe of sets, which is a proper class. ...
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3answers
46 views

Small claim regarding addition of cardinals

I want to show that if $\alpha,\beta$ are cardinals such that $\alpha=\alpha+\beta$ and $0<\beta$ then $ \aleph_{0}\leq\alpha$ It should be fairly simple but for some reason I keep getting stuck.
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1answer
146 views

Books/Review material on infinite cardinality for undergrad

You may have noticed me using asking many questions on Infinite Cardinalities on this fine website. Although many of the answers to my questions here were very in-depth and amazing, I just can't help ...
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2answers
115 views

If CH assumed, can we prove this?

$$\aleph_2^{\aleph_0}=\aleph_2$$ Appreciate your help
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2answers
699 views

If $A$ and $B$ are denumerable sets, and $C$ is a finite set, then $A \cup B \cup C$ is denumerable

I have a statement here I wish to prove and I would love some help on it :) If $A$ and $B$ are denumerable sets, and $C$ is a finite set, then $A \cup B \cup C$ is denumerable Here is my thoughts!...
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1answer
145 views

About cardinalities of almost disjoint systems of functions

Let us call a system $\{f_i; i\in I\}$ of functions $f_i\colon\omega\to\omega$ almost disjoint if the set $\{n\in\omega; f_i(n)=f_j(n)\}$ is finite whenever $i\ne j$. (I am not entirely sure whether ...
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1answer
1k views

Proving every infinite set is a subset of some denumerable set and vice versa

I have 2 sets of statements that I wish to prove and I believe they are very closely related. I can prove one of them and the other I'm not so sure! 1: Every infinite set has a denumerable subset ...
3
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1answer
66 views

“big” Hausdorff space with dense subspace of given cardinality

In a topology course we proved the following proposition: Let $A$ be an infinite set. Then there exists a Hausdorff space $X$ of cardinality $|\mathfrak{P}(\mathfrak{P}(A))|$ which contains a ...
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3answers
2k views

Let $A$ be any uncountable set, and let $B$ be a countable subset of $A$. Prove that the cardinality of $A = A - B $

I am going over my professors answer to the following problem and to be honest I am quite confused :/ Help would be greatly appreciated! Let $A$ be any uncountable set, and let $B$ be a countable ...
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3answers
2k views

Cardinality of a set A is strictly less than the cardinality of the power set of A

I am trying to prove the following statement but have trouble comprehending/going forward with some parts! Here is the statement: If $A$ is any set, then $|A|$ $<$ $|P(A)|$ Here is what I ...
3
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3answers
111 views

$P(A)$ and $2^A$ are numerically equivalent

Whilst reading some notes on the cardinality of infinite sets, I got to this question which has been bugging me for a while. Help would be greatly appreciated! For every nonempty set A, the sets $...
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1answer
209 views

It is possible to generalize the “real” line to be able to embed $\omega_1$ or any uncountable ordinal into a finite segment of it?

This question is motivated from a previous question, but is in itself independent of it. So, I understand that it is not possible to embed $\omega_1$ or any uncountable ordinal into the real line, ...
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1answer
125 views

Question about cardinality of some set of functions

The question in its original form deals with the problem of deciding whether the set $T$ of all irrational numbers in the set $[0,1]$ such that they have only digits $0$ and $1$ in their decimal ...
3
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1answer
82 views

How to show this space $X$ is countably compact, first countable?

Consider the subspace $X$ of $(2^\omega)^+$, i.e., the smallest cardinal greater then $2^\omega$, equipped with the ordered topology consisting of all ordinals of countable cofinality. How to ...
3
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4answers
779 views

Bijection between $\mathbb R^\mathbb N$ and $\mathbb R$ [duplicate]

$\mathbb R^\mathbb N$ is the set of all functions from the naturals to the reals. I have to prove that $\mathbb R^\mathbb N$ has the same cardinality as $\mathbb R$. I found an injective function ...
2
votes
2answers
51 views

length of a normal not cofinal sequence

Let $\kappa$ an infinite cardinal. Does every normal sequence $\langle\alpha_\xi\rangle \subseteq \kappa$ with $\sup \alpha_\xi<\kappa$ is of length $<cf\kappa$ ? I think yes but I have some ...
4
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1answer
79 views

GCH implies that $2^{<\kappa}=\kappa$

If GCH holds, then $ 2^{<\kappa}=\kappa$ for all $\kappa$ It is true that $2^{<\kappa}=\sup_{\delta <\kappa}(2^\delta)$ some explain this for me. Thanks in advance
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4answers
573 views

Looking for a problem where one could use a cardinality argument to find a solution.

I would like to find an exercise of the type: Find some $x$ in $A\setminus B$. Solution: since $A$ is uncountable and $B$ is countable such $x$ exists...
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5answers
463 views

cardinality of the set of $ \varphi: \mathbb N \to \mathbb N$ such that $\varphi$ is an increasing sequence

I know that the set of functions $ f:\mathbb N \to \mathbb N$ is uncountable, but what if we consider only $f$ such that $f$ is increasing? I want to know if this set is countable D: and also the case ...
2
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1answer
191 views

The class of cardinal numbers is well ordered

I'm looking for a proof that the class of cardinal numbers is well ordered under the order relation $|A|\leq |B| \Leftrightarrow$ exists an injection $f:A \to B$. In fact, I've found a very beautiful ...
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1answer
52 views

X ≼ Y ≼ Z and |X| = |Z|. Prove |Y|=|Z|.

$\qquad\qquad\qquad\qquad\qquad X \preceq Y \preceq Z$ and $|X| = |Z|$. Prove that $|Y|=|Z|$. Just started on cardinalities. Not sure about this one. Am I right if I do something along the lines of:...
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1answer
49 views

CCC and point separating weight

CCC means countable chain condition; A cover $\cal A$ of a set $E$ is separating if for each $p\in E$, $\bigcap \{A: A \in \mathcal{A}, p\in A\}=\{p\}.$ The point separating weight of $X$, denoted $...
6
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1answer
219 views

How to prove that a regular cardinal cannot be expressed as a union of sets with less cardinality?

My question is about the following: Using the Axiom of Choice show that: If $\kappa\ge\omega$ is a regular cardinal, $\gamma\le\kappa$, and $\langle A_\alpha\mid\alpha\lt\gamma\rangle$ is a ...
3
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4answers
296 views

If A is a denumerable set, and there exists a surjective function from A to B, then B is denumerable

I am having some trouble solving the following homework question and some help would be greatly appreciated!! Q: Prove that if $A$ is a denumerable set, and there exists a surjective function from $A$...
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2answers
267 views

Proving that if $A$ is a non-empty set, then $|A| ≤ |A \times A|$

I just need some help with this problem. Let $A$ be an non-empty set. Prove that $|A| \leq |A \times A|$. $A$ may or may not be infinite! Intuitively, this statement makes sense. $A \times A$ ...
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5answers
817 views

Why hasn't GCH become a standard axiom of ZFC?

I've never seen a text that includes GCH in the ZFC axioms. I presume this means that GCH has not achieved widespread acceptance. This seems surprising to me, given that: The cardinal numbers ...
5
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1answer
137 views

What indexes do the subgroups of $\mathrm{GL}_n(\Bbb C)$ have?

Let $B_n\subset\mathrm{GL}_n(\Bbb C)$ be the group of invertible upper-triangular matrices. What is the index $[\mathrm{GL}_n(\Bbb C):B_n]?$ (By index I mean the cardinality of a coset space.) In ...
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2answers
583 views

Set theory, property of addition of natural numbers in the cardinal way

Consider the set of natural numbers $\mathbb N$. On this set we define an operation '+', as follows: for all $n,m \in \mathbb N$ we put $n+m$ to be the unique natural number $t \in \mathbb N$ such ...
4
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1answer
61 views

Inequality in cardinal function: $|X|\le 2^{s(X)\psi(X)}$

How to prove that $|X|\le 2^{s(X)\psi(X)}$ by using the Erdős-Rado theorem when $s(X)=\psi(X)=\omega$? $s(X)=\sup \{ |D|: D \subset X, D \text{ is discrete} \} + \omega $ $\psi(X)= \sup\{\psi(p,X): ...
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1answer
144 views

Instance of Continuum Hypothesis implying cardinal inequality

I'm currently trying to solve Exercise 5.27 of Jech's Set Theory (3rd Millennium ed.), viz: If $2^{\aleph_1}=\aleph_2$, then $\aleph_{\omega}^{\aleph_0} \ne \aleph_{\omega_1}$. The presumption ...
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1answer
164 views

Why should the union of such a family of sets have cardinality $\aleph_1?$

Let $\mathcal A$ be a family of infinite countable sets which is linearly ordered by inclusion and such that $\bigcup\mathcal A$ is uncountable. I have to prove that $|\bigcup\mathcal A|=\aleph_1.$ I ...
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votes
3answers
329 views

The real line has cardinality at most $\aleph_2$, but transfinite ordinal space has arbitrarily high cardinality: what is wrong?

In the context of supertasks, people and mathematicians are comfortable with the idea of transfinite ordinal time, that is, that time can be divided into an arbitrarily high number of steps. In most ...
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2answers
545 views

How many total order relations on a set $A$?

Let's define a set $T_A$ which is the set of all total order relations on $A$. This set is a subset of the set of all $2$-adic relations on $A$: $$T_A \subset \mathcal P(A^2) $$ 1-Which is the ...
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votes
4answers
447 views

Are there countably or uncountably many infinite subsets of the positive even integers?

Let $S$ be the set of all infinite subsets of $\mathbb N$ such that $S$ consists only of even numbers. Is $S$ countable or uncountable? I know that set $F$ of all finite subsets of $\mathbb N$ is ...
11
votes
1answer
235 views

Is the cardinality of uncountable $G_{\delta}$ set of $\mathbb{R}$ equals the cardinality of the continuum?

It is known that closed sets of $\mathbb{R}$ satisfies continuum hypothesis, that is, every closed subset of $\mathbb{R}$ is either countable or of the cardinality of the continuum. Is the ...
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2answers
327 views

Can proper classes also have cardinality?

In some set theories such as ZF+GAC, in which GAC is global axiom of choice, the Von Neumann universe $V$ bijects to $Ord$, the class of ordinals. It suggests us that proper classes may also have ...
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1answer
56 views

A question on a set theoretic theorem

Erdős-Rado theorem: Let $\kappa$ be an infinite cardinal. Let $E$ be a set with $|E|>2^\kappa$, and suppose $[E]^2=\bigcup_{\alpha<\kappa}P_\alpha$. Then there exists $\alpha<\kappa$ and a ...
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1answer
166 views

ultrafilter $\kappa$-complete

Let $A$ an infinite set, $D$ an ultrafilter on $A$ and $\kappa$ an infinite cardinal. I want to show the following : $D$ is $\kappa$-complete iff $\forall\tau<\kappa$ and $\forall$ partition $\{X_\...
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2answers
220 views

Countable or uncountable

(1) $C$ is the set of all circles $C(z,r)$ with $z\in\mathbb{Q}\times\mathbb{Q}$ and $r\in\mathbb{Q}^+$. What is the cardinality of $C$? (2) Let $S$ be the set of all sequences $X=\{X_n\}_{n=1}^\...
5
votes
1answer
142 views

Differences in worlds with and without $\aleph_0<|S|<2^{\aleph_0}$

Paul Cohen told us that whether or not there is $S$ with \begin{equation} \aleph_0<|S|<2^{\aleph_0} \end{equation} cannot be decided within ZFC, and hence it is reasonable to work in two ...
3
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1answer
105 views

A question on the power set

Let $\mathcal{P}_\kappa(E)$ is the collection of all subsets of $E$ of cardinality $\le \kappa$, and $[E]^\kappa=\{A: A\subset E, |A|=\kappa\}.$ Then $|\mathcal{P}_\kappa(E)|=|E|^\kappa$ or only $|\...
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vote
1answer
135 views

Axiom of choice , Hartogs ordinals, well-ordering principle

I'm trying to prove the following: If it holds that if for any two sets $A$ and $B$, $A$ can be injected into $B$ or $B$ can be injected into $A$, then every set can be well-ordered (axiom of choice ...
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0answers
73 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 $\sup(\operatorname{rg}(f))=\...
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2answers
84 views

If $A$ is a set, then $\mathrm{card}(P(A)) = 2^{\mathrm{card} A}$.

Can anyone help me please (since I don't know how to work with Maps)? I looked online, but those proofs don't make any sense to me.