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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Cardinal of the set of real functions

We know that the cardinal of natural numbers is $\aleph_0$, and the cardinal of real numbers is $\mathfrak c$. Is it correct that the cardinal of real functions is $2^{\mathfrak c}$?
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28 views

The Cardinality of infinite series of natural numbers?

Given an infinite sequence $a_1,a_2,a_3,...$,and the map $F(a_1,a_2,a_3...) = {p_1}^{-a_1}{p_2}^{-a_2}{p_3}^{-a_3}...$ Where $p_i$ is the ith prime (chosen by the axiom of choice). Why isn't this ...
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1answer
36 views

Cardinality of $F\times\Bbb N$

Suppose $F$ is an infinite set (that is $\#F\geq\#\mathbb N$). Various sources I have consulted claim that $$\# F=\# (F\times\mathbb N)$$ without proof (# denotes cardinality). I guess that this is so ...
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1answer
90 views

continuum and aleph one

We have symbols of cardinal numbers. The most known are aleph zero and continuum. Somewhere I've noticed the sequence of cardinal numbers as aleph zero, aleph one, aleph two... where $\aleph_n$ = ...
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1answer
124 views

Are there any infinites not from a powerset of the natural numbers?

With the cardinality of the natural numbers as $|\mathbb{N}| = \aleph_0$ and its powerset as $|\mathcal{P}(\mathbb{N})| = 2^{\aleph_0}$, the continuum hypothesis and the axiom of choice says that ...
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2answers
96 views

Cardinality of Integers, Positive Integers, and Rational Numbers all equal $\aleph_0$

Prove that $|\mathbb{Z}|=|\mathbb{Z}^+|=|\mathbb{Q}|=\aleph_0$ I am to use cardinal addition and multiplication to reduce this to finding an injection $\mathbb{Q}^+ \to ...
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3answers
48 views

Altering an Infinite Set does not change cardinality

Let X be an infinite set. Show that adding or subtracting a single point does not change its cardinality. I have a plan but need help writing the actual proof. I need to show that it doesn't matter ...
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2answers
60 views

What's the diference between $A<\infty$ and $A<\aleph_0$?

In my topology class the teacher gave some examples of topologies, and I'm trying to prove that they really are topologies. If $X$ is a set then: $\mathcal C=\{A:\# (X-A)<\infty\}$ is a topology ...
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1answer
95 views

Cardinality of the set of permutations of a set $ A $

I've some trouble calculating the cardinality of the set of the permutations of a given set $ A $. For notational purpose let $ k = |A|$ and define $ P_A = \{ f : A \to A | f \text{ is a bijective ...
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2answers
128 views

Why do “Set of even Integers” and “Set of all Integers” have same cardinality? [duplicate]

Despite "Set of even Integers" and "Set of all Integers" are infinite sets, we can see that 3 is member of only one of them. Only one example is enough to say that both can't have same cardinality ...
2
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1answer
194 views

Prove: if $A$ is infinite set and $B$ is finite set then $\left| {A - B} \right| = \left| A \right|$

Prove: if $A$ is infinite set and $B$ is finite set then $\left| {A - B} \right| = \left| A \right|$ Well, one way to show it, is to find an injective function, for both directions. First, ...
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1answer
53 views

Prove: $2^{\aleph_0}\not=\aleph_{\epsilon_0}$

Prove: $2^{\aleph_0}\not=\aleph_{\epsilon_0}$ I would like a hint for this problem
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2answers
22 views

What $P(E)\sim (E \to \{ 0,1\} )$ mean?

i was asked to prove: $P(E)\sim (E \to \{ 0,1\} )$. the left-hand side is the set of all subsets of $E$. (Right?) What about the right-hand side? Thanks.
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1answer
32 views

Proof Check: The cardinality of the set of all binary series with an infinite amount of 0's and 1's:

Label the set of all binary series with an infinite amount of 0's and 1's as $C$. It's easy to prove that the set (labeled $A$) of all binary series with a finite number of 1's is countable. I can ...
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1answer
22 views

Given two cardinalities, $m$ and $n$, how many solutions does $n*x = m$ have? How about $n+x=m$?

It looks to me like in the case where both the cardinalites are finite, there exists one solution for the first equation and one solution for the second as long as $m>=n$. Otherwise there's none. ...
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1answer
46 views

Subset of an infinite set with same cardinality

Let $A$ be an infinite set. Show that there is a subset $B\subseteq A$ such that $|B|=|A-B|=|A|$. I've tried using Zorn's lemma with $P(A)$ and $\subset$ and got nowhere. I would like a hint.
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1answer
44 views

Set of Cardinals

Let $A$ be a set of cardinals. Prove that there is a cardinal that that is greater than every cardinal in $A$. Assume that there isn't such a cardinal. Then for any cardinal $x$ there is $y\in A$ such ...
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2answers
71 views

equivalence classes and cardinality

I need to prove that every equivalence class created by the equivalnce relation $\sim$ on $\mathbb{R}$, that is defined by: $a\sim b \Leftrightarrow (a-b) \in \mathbb{Q}$, is $\aleph_0$. Furthermore, ...
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1answer
69 views

$\ell^p(I)$ space and a dense set of this space

Let $I$ be an infinite set and $1\leq p<\infty$. Show that $\ell^p(I)$ has a dense set of the same cardinality as $I$. For this I put $$X=\{(x_i); x_i\in \Bbb C , x_i=0 \text{ for all but ...
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1answer
46 views

Find the cardinal of the set of all infinite sequences of $0,1,-1$ such that each sequence contains each digit at least once - Check my answer

As the title says, we are asked to find the cardinal of the set of all infinite sequences made from the digits $0,1,-1$ such that each sequence contains each digit at least once. My answer I solved ...
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1answer
98 views

$\aleph$ function fixed points below a weakly inaccessible cardinal are a club set

I am throwing yet another one of my solutions out here for the internets to debug and for future set-theory students. Let $\aleph_\delta$ a weakly inaccessible cardinal. Prove that $A =\{\alpha ...
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2answers
2k views

What's “the catch” in this question?

I am solving old exam questions and I came across this question: Let $\langle A_n \mid n < \omega\rangle$ disjoint sets such that $\bigcup_{n < \omega}A_n = \mathbb{R}$. Prove that there ...
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1answer
46 views

Supremum of a set of cardinalities.

Let $A$ be a set of cardinalities. Does $A$ have a supremum among all cardinalities. How about infimum?
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2answers
102 views

Some questions about elementary set theory (cardinal and ordinal numbers)

I have three questions about elementary set teory and i don't figure out how to solve them: 1)Let $X$ a subset of the cardinal number $2^{\aleph_0}$ (seen as an initial ordinal). Is true or false ...
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0answers
92 views

$\mathfrak b \leq \mathfrak r$. Is this proof correct?

I am trying to prove that $\mathfrak b \leq \mathfrak r$ where $\mathfrak r$ is the minimal cardinality of a reaping family of sets. A family $\mathcal R \subseteq [\omega]^\omega$ of sets $\mathcal ...
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1answer
78 views

How to show that $\mathfrak s \leq \mathfrak d$

I am trying to understand why $\mathfrak s \leq \mathfrak d$. Can anyone state a proof of it? I have a proof , which I don't understand yet. My question regarding that proof is here below: At the ...
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1answer
77 views

A verification for a proof that $\omega_1 \leq \mathfrak s$

I am trying to prove that $\omega_1 \leq \mathfrak s$ where $\mathfrak s$ is the splitting number which is the smallest cardinality of any splitting family. This statement was left as an ...
4
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1answer
81 views

How to prove that $\omega_1 \leq \mathfrak p$

I am trying to understand the stated above Theorem. $\mathfrak p$ is the smallest cardinality of any family $\mathcal F \subseteq [\omega]^\omega$, which has the strong finite intersection property, ...
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2answers
128 views

How many functions $f^m(n) = n$ over $\mathbb{N}$?

I got a task that i have problem with. I have to find how many functions are there that satisfies $$f^m(n) = n$$ for some $m > 0$. So, i came up with an idea. How many functions there are for ...
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2answers
69 views

The cardinal characteristic $\mathfrak d$

I am reading a chapter in a book of Andreas Blass which is called: "Combinatorial Cardinal Characteristics of the Continuum". In there, the cardinal characteristic $\mathfrak d$ is defined as folows: ...
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1answer
133 views

Injective or constant function on measurable cardinal

I'm working on the following question: Let $\kappa$ regular cardinal, and $f:\kappa \rightarrow \kappa$. Prove that $\{\alpha < \kappa\mid f|_\alpha : \alpha \rightarrow \alpha\}$ is a ...
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3answers
46 views

Function composition giving the same value

Let $A = \{f: \mathbb N \rightarrow \mathbb N$ | $\forall n\in \mathbb N \ \ \exists p \ge 1 \ \ f^p(n)=n$} What is $\overline{\overline{A}}$?
3
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1answer
33 views

Question about cardinals.

I have heard that $2^{\omega_1} = \omega_1^{\omega_1}$. Is that true? Why is that? I have tried to find a bijection between the set of all subsets of $\omega_1$ and the set of all functions $\alpha: ...
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2answers
177 views

Ordinals that satisfy $\alpha = \aleph_\alpha$ with cofinality $\kappa$

I am trying to solve the following question: Prove that for every regular cardinal, $\kappa \gt \aleph_0$, there is a exists an $\alpha$ with cofinality $\kappa$ such that $\alpha = ...
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1answer
44 views

What is the defenition of $\mathcal{c}$ and $\aleph_1$ if we assume ZFC without CH.

I am reading an intro to a chapter of Andreas Blass called "Combinatorial Cardinal Characteristics of the Continuum" and I am getting a bit confused. When I studied "Discrete Mathematics", it was ...
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1answer
63 views

Prove that for every $ \quad n\in\mathbb{N},\quad \mathbb{R}^{n} = \mathfrak{c}$?

I was thinking about induction like: Base: $$\#\mathbb{R}^{1} = \#\mathbb{R} = \mathfrak{c}$$ And for $n+1$ $$\#\mathbb{R}^{n+1} = \#\mathbb{R}^{n}\mathbb{R} ...
2
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1answer
68 views

chain A s.t. $|X|<|A|\leq |P(X)|$ [duplicate]

Can we prove that there exists at least one chain $A$ in P(X), where X is a non-empty set (finite or infinite), s.t. $ |X|<|A|\leq |P(X)|$? If you can't solve it, ideas/possible directions are ...
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1answer
137 views

What would a world where $\mathsf{CH}$ is false look like?

My question is a little more specific than the title may lead to believe. In the article The set-theoretic multiverse (J.D. Hamkins), the author writes the following: [...] the continuum is ...
2
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1answer
141 views

Every club of $\kappa$ in $M[G]$ contains a club in $M$.

I'm trying to solve exercise (H1) of chapter VII on Kunen's Introduction to Independence Proofs and I would like some hint. I would prefer a hint instead of the full solution :) Assume in M that ...
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2answers
163 views

Cantor's infinities, and the cardinality of reals vs. complex

Cantor devised 1:1 mappings to prove that the set of integers was the same cardinality as positive integers, odd, etc. And he proved that reals are infinitely more dense. As I recall he called the ...
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3answers
146 views

Cardinality of all sequences of non-negative integers with finite number of non-zero terms. (NBHM 2012)

Consider the set $S$ of all sequences of non-negative integers with finite number of non-zero terms. Is the set $S$ countable or not? What is the cardinality of the set $S$ if it is not countable? ...
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1answer
68 views

Proving that $|\{B\subseteq S: |B|<\infty \}|=|S|$ [duplicate]

I've some elementary set theory problem that I came across with: Let $S\subseteq\mathbb{R}$ be infinite set, and let $A=\{B\subseteq S: |B|<\infty \}$. I'm interested in showing that cardinality ...
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3answers
101 views

countable or not countable

Good evening everyone; Can you tell me whether are they countable or not ? $$ 2^\mathbb{R}\\ 2^\mathbb{Z}\\ 2^\mathbb{N} $$ where $\mathbb{R}$ is the set of real numbers. $\mathbb{Z}$ is set of ...
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2answers
94 views

What is the meaning of $n\in \aleph$

Using mathematical induction, prove that, for each $n\in \aleph$ $$n<3^n$$ Dear all, what is the meaning of "$n\in \aleph$" . How to substitute it and prove that? please give me one step ...
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1answer
71 views

$(k^{<\lambda})^{<\lambda}=k^{<\lambda}$ if $\lambda$ is regular

I'm in need of some help... Why does $(k^{<\lambda})^{<\lambda}=k^{<\lambda}$ if $\lambda$ is regular? I can't see why... Any hints?
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1answer
28 views

Trying to understand an equality between sets

Consider the following equivalence class: $${[\mathbb{N}]_s} = \{ A \in P(\mathbb{Z}) : |A| = |A \cup \mathbb{N}| \wedge |\mathbb{N}| = |A \cup \mathbb{N}|\} $$ So, $A$ must be infinite set with the ...
2
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1answer
150 views

Largest infinite cardinal used in a proof

I've heard before that Knuth holds the record for the largest constant used in a mathematical proof. I was wondering what is the largest cardinal ever explicitly considered in set theory. I presume ...
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1answer
118 views

Bijection between countable ordinals and reals

The set of all countable ordinals is $\omega_1$, which has a cardinality of $\aleph_1$. When accepting the continuum hypothesis, $2^{\aleph_0} = \aleph_1$, so a bijection between countable ordinals ...
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1answer
71 views

Cardinality of cantor set $K$

Is the Cantor function bijective from $[0,1]$ to Cantor set $K$? As $K$ is uncountable I think cardinality of $K$ must be $\mathfrak c$ as $K$ is a subset of $[0,1]$. But I am surprised whether there ...
2
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1answer
84 views

prove that $\Bbb{Z \times ((0,1]\cap Q)}$ and $\Bbb Q$ have the same cardinality

I have to prove that $\Bbb{Z \times ((0,1]\cap Q)}$ and $\Bbb Q$ have the same cardinality. I think I have bijective function($f(\langle a,b\rangle)=a-b$) between thse sets, but I don't know how to ...