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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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 ...
2
votes
3answers
94 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
92 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
361 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 ...
5
votes
3answers
378 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
514 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
85 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
446 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
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 ...
2
votes
2answers
148 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
723 views

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

Does there exist set that contains all the cardinal numbers?
5
votes
2answers
814 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
132 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
881 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
102 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
713 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.
3
votes
3answers
95 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 ...
6
votes
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 $\mathbb{R}$. ...
2
votes
1answer
180 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
votes
1answer
218 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 ...
2
votes
2answers
170 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 $A\times A$
0
votes
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?
0
votes
0answers
44 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 ...
1
vote
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 ...
2
votes
1answer
284 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?
5
votes
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 ...
2
votes
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 ...
9
votes
1answer
378 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 ...
3
votes
2answers
158 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
votes
2answers
146 views

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

What is the cardinality of $\Bbb{N^N}$? my answer: $|\mathbb{R}|$ $=$$|2^\mathbb{N}|$ $\leqslant$ $|\mathbb{N}^\mathbb{N}|$ $\leqslant$ $|\mathbb{R}^\mathbb{N}|$ $=$ $|(2^\mathbb{N})^\mathbb{N}|$ $=$ ...
16
votes
1answer
436 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 ...
0
votes
2answers
107 views

Question on singular cardinals

This is a question concerning the lemma discussed in Question about a proof about singular cardinals. In this question, it is proved that if an infinite cardinal $\kappa$ is singular, then it can be ...
7
votes
1answer
220 views

Question about the order of a Stationary subset of $ \kappa$

Greets I'm trying to prove one part of exercise 8.14 of Jech's "Set Theory", namely that if $o(k)\geq k$, then $k$ is weakly inaccessible, where $\kappa$ is regular; $o(\kappa)$ is defined as ...
4
votes
1answer
283 views

Cardinality of the set of increasing real functions

Could you show me how to "calculate" the cardinality of the set of increasing (not necessarily strictly) functions $\ f: \mathbb{R} \rightarrow \mathbb{R}$ ?
0
votes
1answer
130 views

How many n-element subsets of real numbers are there

I was wondering if anyone could show me how to express the cardinality of all n-element subsets of real numbers.
0
votes
1answer
152 views

At least two circles meeting these cond. have nonempty intersection

Here is a problem I've been trying to solve for some time now. Maybe you could help me. We have two sets $\mathcal {S}$ is a family of circles in the plane such that for any $x \in \mathbb{R}$ there ...
0
votes
1answer
164 views

Mapping: each nonempty finite subset of $\Bbb R$ - sum of its elements

Could you give me a hint how to solve this problem? Let $ D:= \left\{E \subset \mathbb{R} \ | \ 0< \mathrm{card}(E)< + \infty \right\} $. $\phi\colon D\to\mathbb R$ defined by $\phi(E)\sum_{x ...
3
votes
1answer
112 views

Cardinalities of topologies in which not each open set is a union of regular open sets

Suppose, a topological space $(X, \mathscr{T})$ consists of a set $X$ with the cardinality $\kappa$, and a topology $\mathscr{T}$ in which it is not true that each open subset of $X$ can be written as ...
2
votes
1answer
161 views

Infitive distributive law in boolean valued models

I'm posting the problem 2.14 and 2.15 of the book "Set theory" of J.L. Bell. These problem are proposed after the forcing relation chapter and I'm new in this kind of stuff, so I have some little ...
3
votes
2answers
129 views

Computing $\kappa^{<\lambda}$, for cardinals $\kappa$ and $\lambda$

I'm trying to show that, for $\lambda$ an infinite cardinal and $\kappa$ any cardinal, that $$\kappa^{<\lambda} = \sup\{\kappa^\theta:\theta<\lambda\land\theta\text{ a cardinal}\},$$ where ...
8
votes
0answers
2k views

cardinality of set of all real continuous functions [duplicate]

Could somebody explain to me how to prove that the cardinality of all real continuous functions is $c$ ? The first problem is that I don't know how to show that each real continuous function $f: X ...
0
votes
3answers
348 views

Proof of equal cardinality $|\Bbb N \times\Bbb N \times\Bbb N| = |\Bbb N|$

How do I prove that the following sets have equal cardinality? $|\Bbb N \times\Bbb N \times\Bbb N| = |\Bbb N|$ ($|\Bbb N \times\Bbb N| = |\Bbb N|$ also for that matter) $|\Bbb Z \times\Bbb Z| = ...
6
votes
3answers
715 views

Easiest way to prove that $2^{\aleph_0} = c$

$\aleph_0$ is the cardinality of the set of natural numbers, $\aleph_0 = |N|$. $c$ is the cardinality of the continuum, i.e. the set of real numbers $c = |R|$. I know that $|P(A)| = 2^{|A|}$. This ...
5
votes
2answers
121 views

Weak cardinal powers and singular cardinals

Suppose $\kappa > \operatorname{cf}(\kappa)$. Show that: i) if $\kappa$ strong limit then $\kappa^{<\kappa} = \kappa^{\operatorname{cf}(\kappa)}$ ii) if $\kappa$ not strong limit then ...
18
votes
3answers
790 views

For any two sets $A,B$ , $|A|\leq|B|$ or $|B|\leq|A|$

Let $A,B$ be any two sets. I really think that the statement $|A|\leq|B|$ or $|B|\leq|A|$ is true. Formally: $$\forall A\forall B[\,|A|\leq|B| \lor\ |B|\leq|A|\,]$$ If this statement is true, ...