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
134 views

ZFC + not-CH, can a set with cardinality between $\aleph_0$ and $2^{\aleph_0}$ be defined?

Continuum hypothesis states, there is no set with cardinality between the integers and the reals. There is a milestone result, that CH is independent from ZFC. That means, both of ZFC + CH, and ZFC + ...
1
vote
1answer
43 views

How to determine cardinality of an infinite set using Aleph numbers?

So I was reading a little bit about cardinal infinities, and I thought it was pretty interesting. However I wanted to know a little bit more about how to use them. For example, how would I determine ...
3
votes
1answer
54 views

Powers of $\mathfrak{c}^+$

Denote by $\mathfrak{c}^+$ the cardinal successor of continuum. Can we prove in $\mathsf{ZFC}$ that $(\mathfrak{c}^+)^{\aleph_0} = \mathfrak{c}^+$? I guess not. Of course this question is ...
2
votes
1answer
24 views

If $f\colon\kappa\rightarrow\kappa$, then the set of all $\alpha < \kappa$ such that $f(\xi) < \alpha$ for all $\xi < \alpha$ is closed and unbounded.

If $f\colon\kappa\rightarrow \kappa$, then the set of all $\alpha < \kappa$ such that $f(\xi) < \alpha$ for all $\xi < \alpha$ is closed and unbounded. This is from Jech's book (page 103) so ...
1
vote
0answers
28 views

Ordinals that are not cardinals [duplicate]

I am reading Jech's set theory and he defines a cardinal number as an ordinal $\alpha$ (a cardinal) if $|\alpha| \neq |\beta|$ for all $\beta < \alpha$, and he says that all infinite cardinals are ...
0
votes
1answer
23 views

bijective, one-to-one, and number of elements

How does one reconcile the following (seemingly) contradiction in using "number of elements" argument? In the "range" [0,1] in R there are more points than in N, to be shown as "take the inverse of ...
0
votes
2answers
33 views

Cardinality of the set of all complex sequences converging to zero.

I was asked to show that the set of all complex sequences converging to zero has the same cardinality as the set [0,1]. This is the only hole in a proof that I am working on. I need to show there ...
5
votes
1answer
97 views

A club-guessing exercise

I came across this club-guessing exercise on Cardinal Arithmetic by Abraham and Magidor in the Handbook of Set Theory. Let $\kappa, \lambda$ be regular cardinals $\kappa^{++}<\lambda$ and let ...
1
vote
1answer
46 views

Question on Komjáth's “three clouds may cover the plane”

I am reading a wonderful paper by Komjáth, "Three clouds may cover the plane," and am having difficulty proving that certain sets are countable. Assume CH (the continuum hypothesis) holds. Let ...
0
votes
0answers
59 views

Vector spaces over Z/2Z,countably many predicates added,invariant

Suppose we have a class of vector spaces over $Z/2Z$ with predicates $P_n,n\in N$ added for independent subspaces of co-dimension 2. (BTW, independent in which sense,$P_n$ is independent with ...
3
votes
1answer
40 views

Does any set admit a total order? [duplicate]

Is it true that any set $P$ can be endowed with a total order $"\leq" \subseteq P\times P$?
1
vote
2answers
37 views

Prove there is either a chain or an antichain of infinite cardinal.

Let $K$ be an set of infinite cardinal, $X$. Let $(K,\le)$ be a partially ordered set. Prove there is either a chain $C$ such that $|C|=X$ or there is an antichain $A$ such that $|A|=X$. I guess I ...
0
votes
1answer
41 views

Prove $A$ is either finite or countable.

Let $A$ be a set and let $f:A \to \Bbb{N}$ be an injection. Prove $A$ is either finite or countable. How can one prove such a thing? Since this is so obvious it gets me in a place where I don't know ...
1
vote
1answer
21 views

Find $|f^{-1}(\emptyset)|$ where $f: P(\Bbb{R})\to P(\Bbb{N})$, $f(X)=X\cap \Bbb{N}$.

Let $f: P(\Bbb{R})\to P(\Bbb{N})$, $f(X)=X\cap \Bbb{N}$. Find $|f^{-1}(\emptyset)|$. Prove that $|f^{-1}(\emptyset)|=|f^{-1}(\Bbb{N})|$. I am having a difficulty solving 2., but this is what I ...
0
votes
0answers
11 views

CH is preserved under a $Fn(\kappa,\lambda)$ forcing? (Kunen IV.7.10)

The following is Exercise IV.7.10 in the 2013 edition of Kunen's "Set Theory": Let $M$ be a countable, transitive model for ZFC. In $M$, let $\aleph_0 \le \kappa < \lambda$ be cardinals and ...
0
votes
0answers
34 views

Is it possible to create a bijection between all pairs of reals and a real? [duplicate]

The title basically says it all. Is it possible to associate with each pair of reals, another unique real? I guess you could say I'm looking for functions of two real arguments that return a ...
1
vote
2answers
51 views

How many infinite subsets of N are there anyway? [duplicate]

I was reading 2 proofs one that the powerset of $ N$ has a higher cardinality than $N$ two a proof that the cardinality of the set of all finite subsets of $N$ has the same cardinality than $N$ ...
0
votes
1answer
74 views

Can Aleph Numbers be multiplied?

i.e., does it make sense to say something like $(2 * \aleph_0) > \aleph_0$ ? The original question I was thinking about is: if A = $\mathbb{Z}$ and B = {the set of even integers} is it correct to ...
0
votes
1answer
34 views

What is the cardinality of a set of all finite subsets of $\Bbb{N}$? [duplicate]

I'm looking for cardinality of $P_{fin}(\Bbb{N})=\{x|x\subset\Bbb{N}$ and $x$ finite$\}$. I was told in my classes that it's $\aleph_0$, but how to prove it?
1
vote
1answer
40 views

Showing that the class of all sets of a particular cardinality is not a set.

How to show that the class of all sets of a particular cardinality ,say $h$ is not a set. My argument: I assume that I've shown the following lemma. Lemma: If $X$ is an infinite set of cardinality ...
0
votes
1answer
47 views

Cardinality of the set of all (real) discontinuous functions

This is a question from the book Introduction to Set Theory (Hrbacek and Jech), chapter 5, question 2.6. (Show that) The cardinality of the set of all discontinuous functions is ...
2
votes
1answer
37 views

Cardinality of sets: $|A|\le|B|\Rightarrow(|A\cup B|=|B|\land|A\times B|=|B|)$

My book of mathematical logic states the facts that, if we call $|X|$ the cardinality of set $X$, then, for any two sets $A,B$ such that $|A|\le|B|$, $$|A\cup B|=|B|\quad\text{ and }\quad|A\times ...
1
vote
1answer
30 views

Proving that if $A\leq_c B$ then $\chi(A)\leq_o \chi(B)$

By Hartog's Theorem we knoe that for every set $A$ There is a definite operation $\chi(A)$ which associates with each set $A$, a well ordered set $\chi(A) = (h(A),{\leq_{\chi(A)}}),$ such that $h(A) ...
0
votes
0answers
39 views

Connection beween closure property in forcing and preservation of $H_{\kappa}$

I would like to know about the relation between closure property of forcing notions and preservation of hierarchy of hereditary small sets, $\langle H_\lambda \mid \lambda\in \mathrm{Card}\rangle$, at ...
4
votes
2answers
175 views

Understanding proof of Hartogs’ Theorem on Set Theory

I'm trying to understand the proof of the Hartogs’ Theorem on page 100 of this book. My especific question is: If we have for each set $A$ $$\mathrm{WO}(A)=\{ (U,\leq_{U}) \, | \, U\subseteq A \, ...
1
vote
2answers
56 views

Injections from all ordinals into a set $X$

We are working in $\mathsf{ZF}$. Let $X$ be a set. Let $A$ be the class of all injections $f: \alpha \to X$ for arbitrary ordinals $\alpha$. I am quite sure that, in fact, $A$ is a set, since if ...
1
vote
0answers
24 views

Connectedness ( cardinality and connectedness) [duplicate]

$(X,d)$ metric space and $A\subset X$ and $A$ is connected. $$ \text{Card}(A) > 2 \implies \text{Card}(A) \geq \text{Card}(\mathbb{R}).$$ How do I prove it ?Waiting for your help?
0
votes
2answers
29 views

what is the cardinality of powerset of a union set?

Is there exist something like P(X+Y) (P STANDS FOR POWERSET)? I am confuse because power set is the set of all subset of Cartesian product, and X+Y wont give Cartesian product but (x,0) U (y,1), and ...
2
votes
1answer
23 views

Find the maximum number of a continuous function

Lets define a function $z:\mathbb{R}^\mathbb{R}\to\mathcal P(\mathbb R)$ that gives you the set of zeros of any $\mathbb R ^\mathbb R$ function. Now, we define a set $S=\{z(f):f\in\mathbb R ^\mathbb ...
4
votes
2answers
47 views

$\aleph_0 \aleph_1 =\aleph_1$? But I don’t know any way to prove or disprove it

What is the value of $\aleph_0 \aleph_1$? Clearly $\aleph_0\le \aleph_1$ implies $\aleph_0=\aleph_0\aleph_0\le \aleph_1 \aleph_0$ and again $\aleph_0 \aleph_1\le \aleph_1 \aleph_1=\aleph_1$. But ...
3
votes
1answer
45 views

I can't find the mistake in this argument (Cofinality and König's theorem)

I have some trouble explaining this apparent contradiction: we know that given $k>\aleph_0$ an infinite cardinal, $\mu=cof(k)$ is the minimum cardinal such that $k=\sum_{i\in \mu} k_i$ where ...
0
votes
1answer
29 views

If $\alpha$, $\beta$ are finite cardinals such that both are greater than $1$ and $\gamma$ is an infinite cardinal then $\alpha^\gamma=\beta^\gamma$.

Can someone suggest a rigorous proof of the following: If $\alpha$, $\beta$ are finite cardinals such that both are greater than $1$ and $\gamma$ is an infinite cardinal then $\alpha^\gamma = ...
3
votes
1answer
61 views

Cofinality of $2^{\aleph_\omega}$

Is the following statement correct: $\operatorname{cf}(2^{\aleph_\omega})=\aleph_0$? It appears in the "Jech" book. Wikipedia however states that $\operatorname{cf}(\aleph_\omega)=\aleph_0$. The ...
2
votes
1answer
64 views

Is it true that $({\aleph_1})^{\aleph_0} = {\aleph_1}$?

Is it true that $({\aleph_1})^{\aleph_0} = {\aleph_1}$? My scenario is as follows: The cardinal number of $\mathbb R$ is $|\mathbb R|={\aleph_1}$ and the cardinal number of the Cartesian product of ...
0
votes
1answer
17 views

closed unbounded set,regular cardinals,cofinality

Given two regular cardinals $\lambda>\mu$, why this set is a closed unbounded set in $\lambda$? {$\alpha$ | cf($\alpha$)=$\mu$ , $\alpha<\lambda$}
9
votes
1answer
108 views

Are injections harder to find than surjections?

Given two finite sets $A$ and $B$ with $|A|<|B|$ There are more functions from $B$ to $A$ than from $A$ to $B$ except when $|A|=1$ or $|A|=2,|B|=3,4$. See here for proof. It is also true there are ...
3
votes
3answers
390 views

Are there fields and ordered fields of every infinite cardinality?

On the top of my head, I cannot think of any fields of cardinality more than that of the reals. (It is known that the process of algebraic closure does not increase the cardinality of an infinite ...
0
votes
0answers
18 views

Mapping vectors to real numbers [duplicate]

Does there exist an invertible mapping that takes n-dimensional vectors (RN) to real numbers (R). Any countable set can be mapped in this way to another countable set. This is the sense in which ...
1
vote
1answer
61 views

Notation about cardinals

Does a cardinal $\mathcal{k}$ such that $2^\mathcal{k}=k^+$ have any special name? I never encountered any name for this property, but I think it is possible they have one.
0
votes
1answer
20 views

Prove that if A~B then Sym(A)~Sym(B).

I tried to prove it with sets. Really, truly clumsy. I know |A|=|B|. Can I simply conclude that |A|!=|B|! => Sym(A)~Sym(B)?? (Sym(A) for a set A is the set of all bijections from A to A.)
0
votes
0answers
26 views

Prove $α · β ≤ α · γ$ and $α^ β ≤ α^ γ$ for any three cardinals, where $ β ≤ γ$.

This is what I did: a. let $|A|=α, |B|=β, |C|=γ. |B|≤|C|$ and therefore there is an injection $f: B \to C$, such that $f(b)=c$ for some $b \in B, c \in C.$ Upgrading this function to ...
0
votes
3answers
84 views

Prove that $2^\aleph+\aleph$ equals $2^\aleph$.

How do I show this elegantly? I can't seem to find the right sets for it... Maybe there are some substantial laws I could use? I would appreciate your help... What I said is: $2^\mathfrak c$ is for ...
8
votes
0answers
98 views

Do an infinite set and its double have the same cardinality? [duplicate]

My question was inspired by this answer. Suppose $A$ is an infinite set. Does its double, $A\times\{0,1\}$, always have the same cardinality? In my head I quickly spotted a simple proof that the ...
0
votes
1answer
24 views

Transfinite fixed points of a function

Let the function $F\colon On \rightarrow On$ be defined by the following recursion: $F(0) = \aleph_0$ $F(\alpha+1) = 2^{F(\alpha)}$ (cardinal exponentiation) $F(\lambda) = \sup\{F(\alpha): \alpha ...
0
votes
4answers
26 views

Why we can conclude immediately that $x \in B$, if $(x, y) \in A \times B = B \times A$

The following statements are part of a proof involving cartesian products, specifically involving this theorem: $A \times B = B \times A \iff$ either $A = \emptyset$, $B= \emptyset$, or $A = B$ ...
0
votes
2answers
61 views

Showing the set of functions $\{0, 1\} \to \mathbb{N}$ is countably infinite.

I'm doing a question it asked me to show that $\mathbb{N} \times \mathbb{N}$ was countably infinite but I am stuck on the following part of the question: deduce that the set of all functions $f : ...
1
vote
3answers
67 views

Power of sets - $\{0,1\}^\mathbb{N} \simeq \mathbb{N}^\mathbb{N}$ [duplicate]

I've got a problem with prove about cardinality of sets. How can I prove that $\lbrace 0,1 \rbrace^\mathbb{N} \simeq \mathbb{N}^\mathbb{N}$?
2
votes
1answer
82 views

Does $\aleph_0\cdot\kappa=\kappa$ for every $\kappa\ge\aleph_0$ hold in ZF?

It is easy to show that for any (Dedekind) infinite cardinal $\kappa$ we have $\aleph_0+\kappa=\kappa$. Definition of an infinite cardinal is a cardinal such that $\aleph_0\le\kappa$. (I believe ...
1
vote
2answers
60 views

Cardinality of the countably infinite product of a two-point set $\{0,1\}$?

I'm so confused about cardinalities of some sets. What is the countable infinite product of a two points set $\{0,1\}$? Does it have the same cardinality as the real number $\mathbb R$? Or is the ...
4
votes
1answer
70 views

Prove that $\forall\alpha\geq\omega$, $|L_\alpha|=|\alpha|$ without AC

Without using Axiom of Choice, prove that $$\forall\alpha\geq\omega,~~|L_\alpha|=|\alpha|,$$ in which $\alpha$ is an ordinals, $\omega$ is the set of natural numbers, $L_\alpha$ is the $\alpha$-th ...