This tag is for questions about cardinals and related topics such as cardinal arithmetics, clubs, stationary sets, cofinality, and principles such as $\lozenge$. Do not confuse with [large-cardinals] which is a technical concept about strong axioms of infinity.

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General question: what happens if we replace the regularity stipulation in GCH with other conditions?

I went to bed last night pondering the following. We can formulate both a weak and a strong generalized continuum hypothesis. GCH0. If $\kappa$ is an infinite cardinal number, then $2^\kappa$ is ...
0
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2answers
20 views

Proving that $f$ is a bijection from $N$x$N$ to $N$.

I am having trouble with the following problem: $f: N\times N\rightarrow N$ and $f(i,j)=2^{i-1}(2j-1)$. Prove that $f$ is a bijection thus $N\times N$ and $N$ are numerically equivalent. Work: I ...
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3answers
27 views

Prove that $\{(a,b):a,b\in\mathbb N, a\geq b\}$ is denumerable.

If $S=\{(a,b):a,b\in\mathbb N, a\geq b\}$, how do I prove that $S$ is denumerable? Work: Since $S \subseteq\mathbb{N\times N}$ I know that $S$ is denumerable. But I don't know how to structure the ...
2
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2answers
53 views

Proving that (0,1) and [0,1] are numerically equivalent.

as the title suggests, I need help proving that the cardinality of $(0,1)$ and $[0,1]$ are the same. Here is my work: $f:[0,1] \rightarrow (0,1)$ Let $n\in N$ Let $A=\{\frac{1}{2}, \frac{1}{3}, ...
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2answers
32 views

cardinality with finite sets

$A,B,C$ are finite sets. Suppose $A\subseteq B \subseteq C$ and $\#A=\#C$. Prove that $\#A=\#B$ and $\#B=\#C$. Should I prove this by showing that there exist an element in $A$ that exist in $B$ and ...
1
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2answers
38 views

Cardinality of $\lim_{k\to\infty}\mathbb N^k$ vs. $\mathbb N^\infty$

My friend and I are having a disagreement over whether the number of terms in the following series is countable or uncountable: $$\sum_{i=1}^\infty a_i + \sum_{i=1}^\infty\sum_{j=1}^\infty a_{ij}+ ...
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0answers
15 views

If $X$ is a finite set of cardinality $n$, where $n$ exists in $P$, show that the following conditions on a function $f: X \to X$ are equivalent: [duplicate]

(a) $f$ is an injection (b) $f$ is a surjection (c) $f$ is a bijection I know that (c) implies (a) and (b) and (a) and (b) imply (c). I also have the following definition that I've been playing ...
0
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2answers
29 views

Cardinality of two sets cross-multiplied

Let $A$ and $B$ be sets. Prove that $ \#(A \times B) = \#(B \times A)$. What I have done: There exist an element $m$ in $A$ such that the element also exists in $B$. If $\#A = \#B$, then $\#B = ...
4
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2answers
40 views

Proving that $\mathrm{card}(2^{\mathbb{N}})=\mathrm{card}(\mathbb{N}^\mathbb{N})$

I'd like to prove that $\mathrm{card}(2^{\mathbb{N}})=\mathrm{card}(\mathbb{N}^\mathbb{N})$, I have the following 'sketch' but I'm not sure if this works. ...
0
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1answer
22 views

An alternative succinct proof needed for trivial cardinality fact

Let $|X|$ denote the cardinality of a set, i.e. the least ordinal $\alpha$ such that there is a bijection between X and $\alpha$. For any sets $X$ and $Y$ we write $X\preccurlyeq Y$ if the exists an ...
2
votes
1answer
78 views

What are interesting examples of existential proofs based on cardinality arguments?

Probably the most famous example of a proof, where consideration of cardinalities is used to show existence of some object, it the Cantor's proof that there exist transcendental numbers. What are ...
1
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1answer
66 views

Bigger infinity than real number infinity [duplicate]

Is there a bigger infinity than the infinity of cardinality of the real numbers $R$ ? i.e. is there a set to which real numbers can't be mapped one-one to ?
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3answers
52 views

Are the following sets countable?

I'm trying to determine if the following sets are countable: (a) $\mathbb{Z}^{[0,1]}, (b) [0,1]^{\mathbb{Z}}, (c) \mathbb{Z}^{\mathbb{Z}}$, (d) the set given by functions $f:\mathbb{Z}\to\mathbb{R}$ ...
1
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1answer
16 views

Cardinality of rational exponentiation orbit space

Let $X=(0,\infty)$ be the set of positive real numbers. Let $G=\mathbb{Q}\backslash\{0\}$ be the multiplicative group of rational numbers. $G$ acts freely on $X$ by exponentiation: $r\cdot x=x^r$ for ...
0
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1answer
23 views

Cardinal arithmetic basics

Let's say we have $\omega + \omega$. Since these sets are not disjoint we can replace them by disjoint sets of the same cardinality, namely $\omega \times \{0\}$ and $\omega \times \{1\}$. Then $\big ...
5
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1answer
66 views

Existence of a real uncountable $\aleph_{\alpha}$ without $\mathsf{AC}$

Set theory (Jech) $\text{p.}\;27:$ It is an open problem whether one can prove without the axiom of choice that there exists a regular uncountable $\aleph_{\alpha}\;($the informed guess is that ...
2
votes
1answer
33 views

Can we define ordinals such that the following sentences are independent of ZFC?

Can we explicitly define two ordinals $\alpha$ and $\beta$ in the language of $\{\in\}$ such that the following hold? ZFC proves that $\alpha$ and $\beta$ exist. ZFC proves that $\beth_\beta \neq ...
4
votes
1answer
30 views

If $2^{\kappa}<\lambda$, how many subsets of size $\kappa$ are there of a set of size $\lambda$.

Assume both cardinals are infinite. Also assume AC as needed. So, the obvious bound is that there are no more than $\lambda^\kappa\leq 2^\lambda$ of them. But it seems there should be an easy bound ...
1
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1answer
23 views

Does for a set of cardinals a finite subset exist such that for any cardinal in the set a larger cardinal in the subset exists?

I am writing an essay for which I need to prove that sufficiently many graphs of a certain type exist. Is it true that for any set of sets (or set of cardinals) $S$ a countable subset $C$ exists such ...
2
votes
2answers
108 views

A property of strong limit cardinal

Suppose $\lambda$ is a strong limit cardinal, i.e. $\forall \alpha<\lambda \ 2^\alpha<\lambda$, and the cofinality of $\lambda$: $cf(\lambda)=\omega$. How do we show that $2^\lambda \leq ...
0
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2answers
36 views

A question about the size of the set of all countably-infinite subsets of a countably-infinite set

Let $A$ be a countably-infinite set , then how do we prove that the power set of $A$ and the set of all countably-infinite subsets of $A$ have the same cardinality (i.e. that there is a bijection) ? ...
3
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2answers
84 views

Is there, for every set $X$, a set $Y$ for which $|Y| < |X|$ but $|\mathcal{P}(Y)| \geq |X|$?

As the title says, my question is: Is there, for every set $X$, a set $Y$ for which $|Y| < |X|$ but $|\mathcal{P}(Y)| \geq |X|$? I'm fairly certain this is true for finite sets but maybe ...
4
votes
4answers
212 views

Conclusion about cardinalty.

Assume that: $$\left| T \right| > {\aleph _0}$$ Why can't one assume immediately that: $$\left| T \right| \cdot \left| T \right| > \left| T \right| \cdot {\aleph _0}$$
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0answers
43 views

What is the cardinal number of the following?

What is the cardinal number of $\{f:\mathbb{R}\longrightarrow \mathbb{N}\ |\text{ $ f$ is an injective function}\}$?
4
votes
2answers
94 views

about the smallest $k$ that $V_k$ is a model of ZFC

Let $k$ to be the smallest ordinal that $V_k$ is a model of ZFC. I know that $k$ need not to be inaccessible cardinal,and $k$ has confinality $\omega$. Then how big is $k$? How to write down $k$ in ...
0
votes
1answer
35 views

What is the cardinality of the equivalence class

Consider this relation: $$R = \left\{ {\left\langle {f,g} \right\rangle \in {{\left\{ {0,1} \right\}}^N} \times {{\left\{ {0,1} \right\}}^N}|\exists k \in N\left| {\left\{ {i \in N|f(i) \ne g(i)} ...
0
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1answer
34 views

What is the cardinality of $M_2(\mathbb{R})$

What is the cardinality of $M_2(\mathbb{R})$, i.e the set of all 2 by 2 real matrices( $|M_2(\mathbb{R})|$)?
2
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1answer
59 views

Suppose that S and T each have cardinality c. Show that $S\cup T $ also has cardinality c.

I tried to use the Cantor-Bernstein Theorem. First, we have $S\subset S\cup T$, so that $\left | S \right |\leqslant \left | S\cup T\right | $. This implies $\left | S\cup T \right |\geqslant c$. But ...
20
votes
3answers
1k views

If the infinite cardinals aleph-null, aleph-two, etc. continue indefinitely, is there any meaning in the idea of aleph-aleph-null?

If the infinite cardinals aleph-null, aleph-two, etc. continue indefinitely, is there any meaning in the idea of aleph-aleph-null? Apologies if this isn't a sensible question, I really don't know too ...
1
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1answer
38 views

Cardinality of Orderings of $\mathbb{R}$

For a finite set $S$ there are $\vert S\vert!$ orderings of its elements. What is the cardinality of all orderings of $\mathbb{N}$? What would $$\vert \mathbb{N}\vert!$$ mean? Is it ...
0
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1answer
50 views

Is there always isomorphism between two sets that have the same cardinality?

Is there always isomorphism between two sets that has the same cardinality ? We only know that the two sets have the same cardinality. I tried to find a counter example but couldn't.
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0answers
13 views

Finding the cardinal of monotone increasing sequences of natural numbers [duplicate]

Find the cardinality of the set of all monotone increasing sequences of natural numbers. Well let's ignore the monotone increasing condition for a moment, then the cardinality of a set of all the ...
0
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4answers
30 views

Cardinality for all rational strongly increasing sequences

What is the cardinality for all rational strongly increasing sequences? Using diagonalization, I can show easily that for each list $f_n$ of sequnces, we can present a sequence which is not in ...
5
votes
4answers
102 views

Cardinality of Irrational Numbers

I know and I have proved more than once that the set of irrational numbers ($\mathbb{I}$) is uncountable, but now I'm given to solve this problem: Show that $|\mathbb{I}|=|\mathbb{R}|$, How can I ...
0
votes
1answer
57 views

Cardinality Of Borel Sets

I was trying to show that Borel $\sigma$ algebra is smaller than lebesgue measurable sets. I could come up with a proof for the cardinality of lebesgue measurable sets being $2^c$. Cardinality of ...
4
votes
2answers
110 views

The regularity of successor cardinal

I was looking at two different proofs of the fact that successor cardinals are regular. It struck me as odd that both proofs used AC. Looking at the concepts involved in defining cofinality I feel as ...
5
votes
3answers
214 views

Uncountable Cardinals without AC

I am doing an exercise, proving that without AC or Replacement that there are uncountable cardinals. As a point of reference I looked at the proof in Kunen's "The Foundations of Mathematics" that ...
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1answer
59 views

About alephs and beths

If $2^{\aleph_{0}} \ge \aleph_{\omega_1}$, show that $\beth_{\aleph_{\omega}} = 2^{\aleph_{0}}$ , and that $\beth_{\aleph_{\omega_1}} = 2^{\aleph_{1}}$ I don´t know how to start, can you give me a ...
2
votes
1answer
42 views

Cardinality of Cartesian Product of Uncountable Set with Countable Set

Is it true that if $I$ is an infinite set, then $I\times \mathbb{N}$ has the same cardinality as $I$? I believe it, but I have minimal background in set theory. My guess is that we can construct an ...
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1answer
54 views

Ordinality of a Set

What is the difference between Ordinal number and cardinal number of a set?....I have a confusion in understanding the difference between the two.Can anyone help me to understand these two things? ...
3
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1answer
309 views

A question about splitting sets

I've been looking into combinatorics and small cardinals, in particular, the splitting number $\mathfrak{s}$. By definition, a set $X \subseteq \omega$ splits an infinite set $Y \subseteq \omega$ if ...
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1answer
42 views

Are there ordinals other than the set of natural numbers which satisfy this property?

Let $\alpha$ be an ordinal. We say that $\alpha$ is good iff for every $\beta\in \alpha$, there exists $\gamma\in \alpha$ such that $|\scr{P}(\beta)|\leq |\gamma|$. Question: Is the set of natural ...
2
votes
1answer
45 views

Cofinality assuming GCH

There is this statement that GCH holds iff any pair of regular cardinals $\kappa,\lambda$ such that $\kappa<\lambda$ satisfy that $\lambda^\kappa = \lambda$. Assume we do have two such cardinals. ...
2
votes
1answer
28 views

Identity on singular strong limit cardinals

Let $\lambda$ be a singular strong limit cardinal. Prove that $2^\lambda = \lambda^{\mbox{cf}\lambda}$. It has been a while since I had to prove anything relating to cardinals, and I am not sure ...
4
votes
1answer
125 views

How much is ${\aleph_0}^{\aleph _ 0}$? [duplicate]

How much is ${\aleph_0}^{\aleph _ 0}$? On the left I can find ${2}^{\aleph_0}\le {\aleph_0}^{\aleph _ 0}$ but on the right I can not found someone that is $\le$. In general, how do I use ...
1
vote
1answer
26 views

Cardinality of arithmetic sequences

Let $S$ be the set of arithmetic sequences $(a_n)_n$ in $\mathbb{Z}$, i.e. there exists $d\in\mathbb{Z}$ such that $\forall n\in\mathbb{N}: a_{n+1} -a_n=d$. What is the cardinality of $S$? I ...
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votes
1answer
56 views

How many disjoint disks can be found in $\mathbb{R} \times \mathbb{R}$?

I know that the answer is $\mathbb{Q} \times \mathbb{Q}$ so the answer is $\aleph_0$ But why? Can't I find a $\mathbb{R} \times \mathbb{R}$ point in every disk?
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0answers
23 views

Infinite One-Time Pad

As you know, when used correctly, a one-time pad allows one to send a message, such that the only thing that can be found out about it is the maximum size (which is also the key length.) It is ...
1
vote
1answer
63 views

What is cardinal of set of all Cauchy sequences?

Here are basically two questions. The first, what is the cardinal of equivalent Cauchy sequences of rationals? I know it's $\beth_1$ because of the set is essentially real numbers. But I want to know ...
1
vote
1answer
88 views

Continuum Hypothesis $\iff ?$?

I have read that CH cannot be proved nor disproved within ZFC, and I was wondering: Which (If any) branches/fields of Mathematics are built upon CH being true? Are there any subjects built upon ...