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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2answers
109 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
91 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
228 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
216 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
97 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
114 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
347 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
117 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
149 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
978 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
493 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
143 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
153 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
153 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, ...
3
votes
1answer
109 views

Cardinal Exponentiation $\lim_{\alpha\to\kappa} \alpha^\lambda$

On Page 57 of Jech's Set Theory, Lemma 5.19 If $\kappa$ is a limit cardinal, and $\lambda \geq \operatorname{cf}{\kappa}$, then $\kappa^\lambda = (\lim_{\alpha\to\kappa} ...
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votes
3answers
201 views

Cardinality of the Union is less than the cardinality of the Cartesian product

Suppose I have two collections of sets, $\{A_i\}_{i \in I}$ and $\{B_i\}_{i \in I}$. It is given in the problem that $$\mathrm{card}(A_i) < \mathrm{card}(B_i) \text{ for all }i \in I $$ I want to ...
14
votes
4answers
6k views

Do the real numbers and the complex numbers have the same cardinality?

So it's easy to show that the rationals and the integers have the same size, using everyone's favorite spiral-around-the-grid. Can the approach be extended to say that the set of complex numbers has ...
6
votes
1answer
230 views

Why continuum function isn't strictly increasing?

Is there any example that for cardinal numbers $\kappa < \lambda$, we have $2^\kappa = 2^\lambda$? My guess is that it only depends on whether GCH holds. Is it true?
3
votes
3answers
119 views

How to understand $\operatorname{cf}(2^{\aleph_0}) > \aleph_0$

As a corollary of König's theorem, we have $\operatorname{cf}(2^{\aleph_0}) > \aleph_0$ . On the other hand, we have $\operatorname{cf}(\aleph_\omega) = \aleph_0$. Why the logic in the latter ...
3
votes
2answers
319 views

The cofinality of $\aleph_{\omega\cdot9+3}$

I am studying for a test and I was able to find the cofinality 3 of the 4 ones given, but am having a lot of trouble with the 4th. the 3 first ones are: ...
3
votes
1answer
84 views

The product of the finite non-zero ordinals, and the product of the finite non-zero cardinals.

I am trying to study for a test I have on Set Theory, and after being given the practice test, I am having some big problems with simple things. One question I am having a lot of problems with is: ...
2
votes
2answers
118 views

Prove that $S$ is countably infinite

Suppose we have the set $S\subset\mathbb{ N \times N}$ where $\mathbb N$ is the set of positive integers $\{1, 2,\ldots\}$ with the property: $S = \{\langle m, n \rangle\mid m \leq n\}$. Suppose ...
3
votes
3answers
297 views

Why the principle of counting does not match with our common sense

Principle of counting says that "the number of odd integers, which is the same as the number of even integers, is also the same as the number of integers overall." This does not match with my ...
4
votes
1answer
465 views

Counterexamples to the continuum hypothesis

Assume the continuum hypothesis is false, and add that as an axiom to ZF set theory. How many cardinalities are between the rationals and the reals in this case? Only one? Infinitely many? Countably ...
7
votes
2answers
435 views

A question about the cardinality of the set of all the bijections from $M$ to itself

$M$ is a set.We denote the set of all the bijections from $M$ to itself by $K$.Is the cardinality of $K$ larger than the cardinality of $M$?
1
vote
1answer
122 views

Cardinality natural numbers

Is the cardinality of the natural numbers a natural number? $|\mathbb{N}| \in \mathbb{N}\text{ or } |\mathbb{N}| \notin \mathbb{N}$, that is the question.
0
votes
1answer
519 views

Infinite Dedekind Finite sets

I realized that i have used argument below many times before and I'm not sure if it is true. Let $A=\{n\in \omega|\Phi(n)\}$. Then $A\preceq \aleph_0$. (i)Suppose $A$ is dedekind-infinite and find ...
1
vote
4answers
144 views

Which kind product of non-zero number non-zero cardinal numbers yields zero?

Let $I$ be a non-empty set. $\kappa_i$ is non-zero cardinal number for all $i \in I$. If without AC, then $\prod_{i \in I}\kappa_i=0$ seems can be true(despite I still cannot believe it). But what ...
7
votes
1answer
586 views

Can an infinite cardinal number be a sum of two smaller cardinal number?

Let $\kappa$ be an infinite cardinal number. My question is whether there are $\lambda$ and $\mu$ such that both $<\kappa$ but $\lambda+\mu=\kappa$? If AC holds, then the answer is definitely ...
6
votes
1answer
219 views

Exercise on partially ordered sets from Kunen's *Set Theory*

This problem is Exercise (F4) from Chapter VII of Kunen's text. Apparently, it is a result of Tarski. It goes as follows: Let $ \mathbb{P} $ be a partial order. Define $ \text{c.c.}(\mathbb{P}) $ ...
0
votes
4answers
375 views

Comparing the cardinality of sets

An exercise is the following: Compare the cardinality of the following sets: The class of all real numbers $\mathbb{R} =: A$ The class of all polynomials $\mathbb{R}[X] =: B$ The class of all real ...
4
votes
1answer
83 views

A cardinal $\kappa$ such that $2^{\lambda}<\kappa$ for all $\lambda<\kappa$ is regular?

A cardinal $\kappa$ such that $2^{\lambda}<\kappa$ for all $\lambda<\kappa$ is regular? I would appreciate very much an answer
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vote
2answers
353 views

Cardinality of Sets and Infinite Sets

The following are homework questions I would like assistance on. I will do what I can to work on these problems; any feedback is helpful. In the following problems, S is an infinite set (we do not ...
4
votes
4answers
711 views

Cardinality of Vitali sets: countably or uncountably infinite?

I am a bit confused about the cardinality of the Vitali sets. Just a quick background on what I gather about their construction so far: We divide the real interval $[0,1]$ into an uncountable number ...
0
votes
2answers
3k views

Definition of denumerable (countable) set

When we say that a set $S$ is denumerable, that is, there is a bijection $S \to \omega$, do we mean that there exists such a bijection or do we mean that we have one and are talking about a pair ...
2
votes
4answers
766 views

What's the definition of $\omega$?

This is a follow up on a comment to one of my previous questions. What's the definition of $\omega$? Are the following equivalent definition of $\omega$: $\omega$ is the initial ordinal of ...