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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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
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}|$ $\leqslant$ $|\mathbb{N}^\mathbb{N}|$ $\leqslant$ $|\mathbb{R}^\mathbb{N}|$ $=$ $|(2^\mathbb{N})^\mathbb{N}|$ $=$ ...
16
votes
1answer
437 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
286 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
130 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
352 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
724 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
796 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, ...
2
votes
1answer
70 views

Possible typo in Just/Weese's set theory

In Just Weese on page 197 there are the following corollaries: Regarding Corollary 24: Is this a typo and should say "$CON(ZF) \not\rightarrow CON(ZF + \exists \text{ "a strongly ...
1
vote
2answers
151 views

Number of Vertices of Graphs

So, I was looking at some graph theoretical stuff, more specifically Topological Graph Theory, and I had a question about the definition of graphs: is there usually a condition in the definition ...
2
votes
1answer
146 views

Assuming $(GCH)$, strongly inaccessible and weakly inaccessible coincide

My book says "... If $GCH$ holds, then the notions of strongly inaccessible and weakly inaccessible cardinals coincide, ..." In $ZFC$ I can prove this. But the paragraph from which I have excerpted ...
1
vote
2answers
181 views

Defining strong limit cardinals in $ZF$

I do not understand the following passage/footnote in the book I am currently reading: An initial ordinal $\lambda$ is called a strong limit cardinal if $2^\kappa < \lambda$ for every $\kappa ...
1
vote
1answer
148 views

Proving Axiom Schema of Replacement holds in $H_\lambda$

I think I proved the following, can you tell me if my proof is correct? Exercise 23: Show that if $\lambda$ is a regular infinite cardinal then $\langle H_\lambda , \overline{\in} \rangle$ satisfies ...
3
votes
2answers
110 views

Possible typo in proof of Bukovský-Hechler

In the proof of the following theorem: Theorem 29 (Bukovský-Hechler): Let $\kappa, \lambda$ be infinite cardinals such that $\mathrm{cf}(\kappa) \le \lambda$ and $\mathrm{cf}(\kappa) < ...
2
votes
0answers
92 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 ...
5
votes
2answers
229 views

Cardinal arithmetic gone wrong?

I am trying to calculate $\kappa^\lambda = \aleph_{\omega_1}^{\aleph_0}$. I know that if $\kappa$ is a limit cardinal and $0 < \lambda < \mathrm{cf}(\kappa)$ then $\kappa^{\lambda} = ...
6
votes
2answers
221 views

Is it viable to ask in an infinite set about the Cardinality?

Can you ask given an infinite set about its cardinality? Does an infinite set have a cardinality? So, for example, what would be the cardinality of $+\infty$?
2
votes
2answers
98 views

Cardinal sums over successor ordinals are equal?

Can you tell me if the following claim and subsequent proof are correct? Thanks. Claim: If $\alpha = \delta + 1$ is an infinite successor ordinal then $\sum_{\xi < \alpha } \kappa_\xi = \sum_{\xi ...
3
votes
1answer
115 views

Question about a proof about singular cardinals

The following is a lemma in Just/Weese on page 179: Lemma 17: Let $\kappa$ be an infinite cardinal. Then $\kappa$ is singular iff there exist an $\alpha < \kappa$ and a set of cardinals ...
3
votes
3answers
358 views

Defining cardinals without choice

According to Wikipedia if we assume AC we define a cardinals number to be an ordinal that is not in bijection with any smaller ordinal. Without AC, one takes the cardinality of a set $X$ to be the ...
0
votes
1answer
59 views

Deducing $|B^A|+|B^A|=|B^A|$ from $|A|+|A|=|A|$,

How attacking this question? Show that if $A$ and $B$ are sets such that $A$ is infinite, $|A|+|A|=|A|$, and $|B|\geq 2$, then $|B^A|+|B^A|=|B^A|$
0
votes
2answers
102 views

Cardinality calculation

How to simplify the following: $$2^{\aleph_0}(\aleph_0+\aleph_0)^{2^{\aleph_0}}$$ Thank you for every help.
5
votes
2answers
118 views

Question about proof of Hessenberg: $\kappa \cdot \lambda = \lambda$

The following is a theorem: (Hessenberg) Let $1 \le \kappa \le \lambda$ where $\lambda$ is an infinite cardinal. Then $\kappa \cdot \lambda = \lambda$. The proof in the book proceeds by transfinite ...
1
vote
1answer
79 views

Constructing a bijection between $\xi$ and $\xi + 1$

I did the following exercise, can you tell me if I have it right, thank you (Just/Weese p 176): Show that $|\xi + 1|$ is either finite or equal to $|\xi|$. (here $\xi$ is an ordinal) By ...
2
votes
1answer
150 views

Proving properties of enumeration of infinite cardinals

I am doing the following exercise from Just/Weese: where $F$ is defined as follows: (a) I'm not sure whether the following passes as "convince myself": Apparently $F$ is defined for all ...
4
votes
2answers
89 views

About a proof of “$\bigcup A$ is a limit cardinal”

Assume that if $A$ is a set of cardinals such that $A$ contains no largest element and assume that we have shown that $\bigcup A$ is a cardinal. Now we want to show that $\bigcup A$ is a limit ...
7
votes
1answer
161 views

Bijection between $\mathbb{Z}\times\mathbb{Z}\times\dots$ and $\mathbb{R}$

Is there a bijection between $\mathbb{Z}\times\mathbb{Z}\times\dots$ for countably infinitely many $\mathbb{Z}$'s and $\mathbb{R}$? That is, is $\mathbb{Z}\times\mathbb{Z}\times\dots$, repeated ...
4
votes
7answers
992 views

Proving the uncountability of $[a,b]$ and $(a,b)$

I am trying to prove that $[a,b]$ and $(a,b)$ are uncountable for $a,b\in \mathbb{R}$. I looked up Rudin and I am not too inclined to read the chapter on topology, for his proof involves perfect ...
14
votes
2answers
496 views

What's the difference between saying that there is no cardinal between $\aleph_0$ and $\aleph_1$ as opposed to saying that…

... $\aleph_1$ is the immediate successor of $\aleph_0$? I was reading the wiki article on $\aleph_1$ where a distinction is made between the two. If there's isn't a cardinal between $\aleph_1$ and ...
2
votes
1answer
108 views

Cardinality of all possible partitions of a infinite set

I suppose this problem should be a commonplace, but I only find this one, in which notations are kind of idiosyncratic, along with a glaring defect. My question is where I can find a more formal ...
5
votes
1answer
98 views

Why $\kappa^{<\kappa}=2^{<\kappa}$, if $\kappa$ is a regular and limit cardinal?

On Page 60, Set Theory Jech(2006) (Show that)if $\kappa$ is regular and limit, then $\kappa^{<\kappa}=2^{<\kappa}$. It's not difficult to show that $\kappa^{<\kappa}\geq2^{<\kappa}$. ...
2
votes
1answer
104 views

Is $\kappa^\lambda=2^\lambda$($2 \le \kappa<\lambda$,$\lambda$ infinite) valid in set models of ZF?

Let $2 \le \kappa<\lambda$(both cardinal numbers), in which $\lambda$ is infinite. Then these formula as follows hold where in ZFC: $\lambda+\kappa=\lambda$ $\lambda\cdot\kappa=\lambda$ ...
2
votes
2answers
61 views

Does $\kappa^\lambda=\kappa$ imply $\mu^\lambda=\mu$ for all $\mu>\kappa$, given $\kappa$, $\mu$, $\lambda$ are cardinals?

This problem is originated from the experience that I was trying to prove 5.19 and 5.20 on Page 60 of Set Theory, Jech(2006). It seems to be right with AC, but I don't know how to prove it.
1
vote
1answer
55 views

$A+\alpha\sim A$ when $\omega\le\alpha<h(A)$

I have been considering about $A+\alpha\sim A$ when $\omega\le\alpha<h(A)$, in which $h(A)$ is the Hartogs number of $A$. If there is $\alpha$ s.t. $\omega\le\alpha<h(A)$, we can get $\alpha ...
1
vote
2answers
145 views

Is there any Dedekind-infinite set can be split to two smaller Dedekind-infinite sets?

I have been considering about $A+\alpha\sim A$ when $\omega\le\alpha<h(A)$, in which $h(A)$ is the Hartogs number of $A$. If there is $\alpha$ s.t. $\omega\le\alpha<h(A)$, we can get $\alpha ...
2
votes
1answer
154 views

question on formulations of Generalized Continuum Hypothesis and Singular Cardinal Hypothesis

I hope this is not a silly question(well, not too silly, I hope). After all, a relevent question at a deeper level is already out there, even though it seems the solution is missing. Why not ...
1
vote
1answer
2k views

Does $\mathbb R^2$ contain more numbers than $\mathbb R^1$? [duplicate]

Possible Duplicate: bijective map from $\mathbb{R}^3\rightarrow \mathbb{R}$ Do the real numbers and the complex numbers have the same cardinality? Does $\mathbb R^2$ contain more numbers ...
4
votes
1answer
156 views

Is $\aleph_0$ the minimum infinite cardinal number in $ZF$?

$\aleph_0$ is the least infinite cardinal number in ZFC. However, without AC, not every set is well-ordered. So is it consistent that a set is infinite but not $\ge \aleph_0$? In other words, is it ...
0
votes
2answers
46 views

Can a set(in ZFC) be well-ordered with any order type equipotent to it?

Let $D$ be a set with cardinality $\aleph_\alpha$, and give an ordinal number $\beta$ between $\omega_\alpha$ and $\omega_{\alpha+1}$, can $D$ be well-ordered with order type $\beta$?
4
votes
1answer
153 views

Is this theory $\kappa$-categorical when $\kappa$ is infinite and not $\le \aleph_0$?

Let $\mathcal L$ be a language with only a binary predicate $E$, and let $T$ be a theory of structures in which $E$ is an equivalence relation which partitions the structure into two infinite ...
3
votes
1answer
86 views

Why $2^\kappa=\kappa^{\operatorname{cf}{\kappa}}$, if $\kappa$ is a strong limit cardinal?

On Page 58, Set Theory, Thomas Jech(2006) states the following fact without details. Another fact worth mentioning is: If $\kappa$ is a strong limit cardinal, ...