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.

learn more… | top users | synonyms (1)

3
votes
2answers
592 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
votes
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): ...
7
votes
1answer
145 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 ...
7
votes
1answer
165 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 ...
4
votes
3answers
333 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 ...
1
vote
2answers
555 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 ...
2
votes
4answers
454 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
239 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 ...
9
votes
2answers
337 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 ...
2
votes
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 ...
1
vote
1answer
168 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_\...
0
votes
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
votes
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 $|\...
1
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 ...
1
vote
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))=\...
1
vote
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.
2
votes
1answer
113 views

Two Questions on “Set Theory” [closed]

Q1 Prove that the set $\mathbb{R}^+$ of the positive reals can be written as the union of two non-empty sets, say $A$ & $B$ , both these set are closed unnder addition. Q2 $\aleph_\omega ...
2
votes
1answer
167 views

How to prove that every infinite cardinal $Z$ is equal the countable sum of sets of size $Z$?

Any infinite cardinal $Z$ can be expressed as a countable union of disjoint sets, each of them has the same size $Z$. Any help will be appreciated.
1
vote
1answer
73 views

Operations with cardinals in transcendence base proof

I want to prove the theorem that two transcendence bases for a transcendental field extension have the same cardinality. I've come with this situation : if $A$ and $B$ are two transcendence bases for $...
1
vote
3answers
97 views

How can I prove that $|A| + |B| = |A\cup B| + |A\cap B|$?

If $A$ and $B$ are sets then $|A| + |B| = |A\cup B| + |A\cap B|$.
5
votes
1answer
187 views

Orders of subgroups of Infinite Profinite Groups

This question is in some sense equivalent to my question here. A proof would answer that question in the case when the base field is perfect. Let $G$ be a profinite group of cardinality $\kappa$, ...
0
votes
3answers
215 views

Prove if $A$ is infinite and $B$ is finite, then card($A \cup B$) = card($A$).

Can anyone provide me with function and detail instruction Please.
9
votes
3answers
279 views

Existence of a sequence that has every element of $\mathbb N$ infinite number of times

I was wondering if a sequence that has every element of $\mathbb N$ infinite number of times exists ($\mathbb N$ includes $0$). It feels like it should, but I just have a few doubts. Like, assume ...
6
votes
1answer
218 views

How to explain that $\Bbb{R}$ is not countable to a non-mathematician

What is the best way to explain that $\Bbb{R}$ is not countable assuming that the audience is formed of people who are not mathematicians? I ask this because these days I'm in a debate with someone ...
2
votes
1answer
114 views

A question on almost disjoint collection

Here is a theorem: Let $E$ be an uncountable infinite set. Then there is a collection $\mathcal{A}$ of subsets of $E$ such that $|\mathcal{A}|=|E|^\omega$, $|A|=\omega$ for each $A\in \mathcal{A}$,...
6
votes
1answer
89 views

How to show $\kappa^{cf(\kappa)}>\kappa$?

For $\kappa \geq \omega$, which is a cardinal, then how to show $\kappa^{cf(\kappa)}>\kappa$? My idea: When $\kappa=\aleph_\omega$, then $cf(\kappa)=\omega$, is $\kappa^\omega>\kappa$? It seems ...
7
votes
1answer
120 views

$V=L[A]$ implies GCH for $A\subset \aleph_1$

On page 14 of the introduction to Vol. II of Gödel's collected works: By a slightly more difficult argument one can show that GCH continues to hold if $V=L[a]$ and $a\subseteq\aleph_1$. Does ...
4
votes
2answers
745 views

Applications of cardinal numbers

I know basic things about cardinality (I'm only in High School) like that since $\mathbb{Q}$ is countable, its cardinality is $\aleph_0$. Also that the cardinality of $\mathbb{R}$ is $2^{\aleph_0}$. ...
4
votes
1answer
84 views

Clarification of a proof in Herrlich

In Herrlich on page 5 he gives a proof of $\textbf{AC} \implies \textbf{WOT}$: He does not give a definition of cardinality $|X|$ before this proof and I searched the index for a definition but ...
2
votes
3answers
96 views

Cardinals of set operations without AC

Given info: $|A|=\mathfrak{c}$ , $|B|=\aleph_0$ in ZF (no axiom of choice). Prove: $|A\cup B|=\mathfrak{c}$ If $B \subset A\implies|A \backslash B|=\mathfrak{c}$? I have found several places ...
2
votes
1answer
93 views

Cardinality of a set containing subsets of $\omega_{1}$

Consider the set $ \{ X \subseteq \omega_{1} \ | \text{ such that } |X| = \aleph_{0} \} $ I know $\omega$ is in this set. But then I thought about it and realized that {2,3,4,... } was also in this ...
6
votes
3answers
364 views

What is the first cardinal number which is grearter than continuum?

What is the first cardinal number which is grearter than continuum? We denote it by ? Thanks very much.
0
votes
2answers
61 views

Question about power of sets

If two sets are finite and they have the same power, can we say that the two sets are equivalent? Is every finite set countable?
1
vote
3answers
50 views

Dimension of a space

I'm reading a book about Hilbert spaces, and in chapter 1 (which is supposed to be a revision of linear algebra), there's a problem I can't solve. I read the solution, which is in the book, and I don'...
5
votes
3answers
390 views

Question about Generalized Continuum Hypothesis

I wonder how the Generalized Continuum Hypothesis reveal that $A\times A$ is equivalent to $A$? $A$ is any infinite set.
3
votes
2answers
604 views

How to define a injective and surjective function from $\mathbb{Z}$ to $\mathbb{N}$?

I want to show that $|\mathbb{Z}|=|\mathbb{N}|$. FWIW, I think again that I must define a injective and surjective function from $\mathbb{Z}$ to $\mathbb{N}$. But how? Is there any proof as to how ...
5
votes
1answer
189 views

What is the value of $\aleph_1^{\aleph_0}$?

Is there any neat way to calculate the value of $\aleph_1^{\aleph_0}$?
0
votes
1answer
86 views

Strong limit cardinal - power set operation

Strong limit cardinal is defined as some cardinal that cannot be reached by power set operation - but $2^{\aleph_0}$, strong limit cardinal, can be reached by power set of $\aleph_0$! So there must be ...
4
votes
3answers
452 views

Proving that for infinite $\kappa$, $|[\kappa]^\lambda|=\kappa^\lambda$

Initially assume ZFC. Let $\binom{\kappa}{\lambda}$ denotes $\left|[\kappa]^{\lambda}\right|$ where $[\kappa]^{\lambda}$ is the collection of all subsets of $\kappa$ with cardinality $\lambda$. That ...
0
votes
3answers
106 views

Countability of the continuum

I googled the word countability of continuum and the first result (Ok, second to this thread!) was from Arxiv. I was wondering how valid this argument is. I would also appreciate any additional ...
2
votes
2answers
151 views

Forcing Question about a sequence of functions from $\aleph_{0}$ into $2 = \{0, 1\}$

I'm trying to go through Timothy Chow's A Beginner's Guide to Forcing (arxiv pdf). My particular question is about the first paragraph of Section $5$ (on p. $7$ of the above pdf). First, my vague ...
0
votes
2answers
785 views

Does there exist a set of all cardinals? [duplicate]

Does there exist set that contains all the cardinal numbers?
5
votes
2answers
854 views

Why are infinite cardinals limit ordinals?

My book states this as obvious, but it isn't so trivial to me. thanks
5
votes
1answer
135 views

Is $\alpha^{\beta}=2^{\beta}$ for all infinite cardinalities $\alpha, \beta$?

Is $\alpha^{\beta}=2^{\beta}$ for all infinite cardinalities $\alpha, \beta$? I was wondering about this since I've encountered examples of times where this holds, but I can't seem to prove it myself ...
1
vote
1answer
193 views

countable sets help

In each item below, you may rely on the earlier items. A real number r is called algebraic if there exists a polynomial $P(x) = a_nx_n + \cdots + a_2x_2 + a_1x + a_0$ whose coefficients $a_0, \ldots, ...
2
votes
1answer
64 views

Cardinality of strategy space of $G_{\omega}(\mathbb{R})$ up to an equivalence relation

Suppose, in $G_{\omega}(\mathbb{R})$, a player's two strategies are equivalent, if, for any strategy of his opponent, the outcome incurred are the same. It can be shown that in $G_{\omega}(\omega)$ ...
14
votes
1answer
919 views

Do you need the Axiom of Choice to accept Cantor's Diagonal Proof?

Math people: It is my understanding that set theorists/logicians compare cardinalities of sets using injections rather than surjections. Wikipedia defines countable sets in terms of injections. ...
2
votes
2answers
107 views

Addition of cardinalities

"It is impossible to define addition of cardinalities since the resulting operation is not well-defined" The above is the true and false question and what i think the statement above is false and my ...
1
vote
2answers
725 views

Cardinality of all the functions from $\mathbb N$ to $\{0,1\}$.

Is it true to say that: $$|\{0,1\}^\mathbb N| = |\{0,1\}|^{|\mathbb N|} = 2^{\aleph_0}=\aleph$$ As I know the right part of the equation is true, but I don't know if the equations to it are allowed.