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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1answer
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Preserve Cardinals and Adding No Bounded Subsets

In Chapter 15 at the bottom of page 228 of $\textit{Set Theory}$ by Jech, he writes that if $\kappa$ is a cardinal in $V$ and if $\kappa$ has no new bounded subsets in $V[G]$, then $\kappa$ remains a ...
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1answer
61 views

How many ways can we totally order a set $X$ (but not necessarily the whole set $X$, perhaps just a subset) up to isomorphism?

Here's Attempt #2 at asking this question. Suppose $X$ is a set (not necessarily finite), and let $T$ denote the collection of all totally ordered sets $(A,\leq)$ such that $A \subseteq X.$ Now let ...
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3answers
53 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 ...
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2answers
79 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
42 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 ...
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2answers
46 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 ...
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2answers
54 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
55 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. ...
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1answer
23 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
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1answer
119 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 ...
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1answer
82 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
60 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
21 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
31 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 ...
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1answer
95 views

Existence of a regular 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
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1answer
44 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
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1answer
36 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 ...
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1answer
26 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
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2answers
117 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 ...
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2answers
38 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) ? ...
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2answers
91 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
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4answers
217 views

Conclusion about cardinalty.

Assume that: $$\left| T \right| > {\aleph _0}$$ Why can't one assume immediately that: $$\left| T \right| \cdot \left| T \right| > \left| T \right| \cdot {\aleph _0}$$
4
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2answers
127 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
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1answer
37 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
40 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
61 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 ...
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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 ...
3
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1answer
70 views

In $ \mathsf{ZFC} $, is it true that $ \text{cf}(\kappa) < \text{cf}(2^{\kappa}) $ for all cardinals $ \kappa $?

Question. In $ \mathsf{ZFC} $, is it true that $ \text{cf}(\kappa) < \text{cf}(2^{\kappa}) $ for all cardinals $ \kappa $? I am particularly interested in the case when $ \kappa = \mathfrak{c} ...
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1answer
43 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
73 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
14 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 ...
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4answers
44 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
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4answers
147 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
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1answer
90 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
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2answers
123 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
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3answers
228 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
70 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
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1answer
115 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
94 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? ...
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1answer
314 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 ...
2
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1answer
47 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
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1answer
46 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
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1answer
29 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
144 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 ...
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1answer
30 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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1answer
57 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
26 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
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1answer
74 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
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1answer
98 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 ...