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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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.
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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 ...
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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.
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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 $...
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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
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1answer
186 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 $\kappa$, ...
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3answers
213 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.
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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
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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
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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
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1answer
88 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
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1answer
119 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 ...
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2answers
735 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
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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 ...
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3answers
95 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
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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
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3answers
363 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.
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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?
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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
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3answers
385 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.
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2answers
581 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
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1answer
189 views

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

Is there any neat way to calculate the value of $\aleph_1^{\aleph_0}$?
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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 ...
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3answers
449 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 ...
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3answers
105 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 ...
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2answers
150 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 ...
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2answers
765 views

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

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

Why are infinite cardinals limit ordinals?

My book states this as obvious, but it isn't so trivial to me. thanks
5
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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
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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, ...
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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 $G_{\omega}(\omega)$ ...
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1answer
906 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. ...
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2answers
103 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 ...
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2answers
723 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.
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3answers
96 views

Are $\prod_{i\in I}{X_i^2}$ and $(\prod_{i\in I}{X_i})^2$ the same?

We have: $$\prod_{i\in I}{X_i}=\left\{f:I\to\bigcup_{i\in I}{X_i}~\Big|~ (\forall i\in I)\big(f(i)\in X_i\big)\right\}$$ Is it true: $$\left|\prod_{i\in I}{X_i^2}\right|=\left|\left(\prod_{i\in I}{...
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5answers
3k views

Show that open segment $(a,b)$, close segment $[a,b]$ have the same cardinality as $\mathbb{R}$

a) Show that any open segment $(a,b)$ with $a<b$ has the same cardinality as $\mathbb{R}$. b) Show that any closed segment $[a,b]$ with $a<b$ has the same cardinality as $\mathbb{R}$. ...
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1answer
189 views

Regular cardinals and unions

If a cardinal $\kappa$ is regular then it cannot be written as a union of fewer than $\kappa$ sets, each of size less than $\kappa$. This seems to be a very useful characterization. I have seen a ...
2
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1answer
220 views

Proof of one-one correspondence

a. Show that every infinite set can be put into a bijection with a proper subset of itself. b. Show that the initial segment determined by $n$ cannot be put into a bijection with the initial ...
2
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2answers
174 views

Do there exist bijections between the following sets?

Let $A$ be an infinite set. Do there exist bijections between the following sets? $A$ and $A\setminus B$ where $B$ is a finite subset $A$ and $A\times \{1, 2, \dots, n\}$ $A$ and $A\times A$
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4answers
185 views

Bijective function between $\mathbb R$ and $\mathbb R^\mathbb R$

Let $F=\{\text{all functions}\ f:\mathbb{R} \rightarrow \mathbb{R}\}$. Then $ \nexists$ a bijection $\alpha: \mathbb{R}\rightarrow F$. Why is this the case? I do not know why?
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0answers
45 views

What is the cardinality of the set of transfinite cardinals? [duplicate]

Possible Duplicate: Cardinality of all cardinalities What is the cardinality of the set of transfinite cardinals? The generalized continuum hypothesis ($2^{\aleph_a} = \aleph_{a+1}$) seems to ...
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1answer
58 views

Uncountable models for a language $L_Q$

$L$ is first-order language with identity and $L_Q$ a language obtained by adding to $L$ the quantifier $Q$. Definition of $Q$: If $P$ is a formula and $x$ a variable, $QxP$ is a formula of $L_Q$....
2
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1answer
287 views

Cardinality of the set of clopen subsets of a topological space

Is there some way to find the cardinality of set of all clopen subsets of a topological space, say, Cantor space, Baire space?
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2answers
4k views

Cardinality of $\mathbb{R}$ and $\mathbb{R}^2$

I am working on this exercise for an introductory Real Analysis course: Show that |$\mathbb{R}$| = |$\mathbb{R}^2$|. I know that $\mathbb{R}$ is uncountable. I also know that two sets $A$ and $B$...
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1answer
71 views

Countability of “center” points of line segments in complement of Cantor set

So, start with the set [0,1] of the real line. Remove the middle third, and keep removing the middle thirds of the remaining line segments as usual when making the Cantor set. Each time you remove a ...
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1answer
398 views

A “reverse” diagonal argument?

Cantor's diagonal argument can be used to show that a set $S$ is always smaller than its power set $\wp(S)$. The proof works by showing that no function $f : S \rightarrow \wp(S)$ can be surjective ...
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2answers
159 views

Why isn't this function $f:\mathbb N \to \mathcal P(\mathbb N)$ a surjection?

Let $f:\mathbb N \to \mathcal P(\mathbb N)$ be a funtion which maps to each odd natural number an unitary set from $P(\mathbb N)$. Then, we map the even, non-four multiples with the sets formed by two ...
2
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2answers
148 views

What is the cardinality of $\Bbb{N^N}$?

What is the cardinality of $\Bbb{N^N}$? my answer: $|\mathbb{R}|$ $=$$|2^\mathbb{N}|$ $\leqslant$ $|\mathbb{N}^\mathbb{N}|$ $\leqslant$ $|\mathbb{R}^\mathbb{N}|$ $=$ $|(2^\mathbb{N})^\mathbb{N}|$ $=$ ...
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1answer
442 views

Cardinality of a locally compact Hausdorff space without isolated points

I am interested in the following result: Theorem. A locally compact Hausdorff topological space $X$ without isolated points has at least cardinality $\mathfrak{c}$. To prove it, one can find two ...