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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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
40 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
58 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
75 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
83 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 ...
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1answer
171 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
37 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
36 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
64 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
67 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
44 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 ...
2
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1answer
95 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 ...
3
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1answer
43 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
90 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 ...
3
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0answers
87 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 ...
4
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1answer
73 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 ...
5
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1answer
74 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
77 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
127 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 ...
3
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2answers
64 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: ...
5
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1answer
118 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
32 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: ...
4
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2answers
168 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 = ...
2
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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 ...
3
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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
65 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
136 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
137 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
151 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
120 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
65 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
91 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 ...
3
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2answers
93 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 ...
3
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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
26 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
133 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
111 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
62 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
80 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 ...
2
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0answers
24 views

(S is a proper class wrt model M) implies ((|T|=|S| under model N) implies (T is a proper class wrt M)). Why?

If I understand correctly (which is far from guaranteed, so a reply telling me that it is rubbish will be better than no reply at all): Suppose we have two models N and M such that N is a ...
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1answer
75 views

Russell's definition of finite cardinals

whether the thought had been previously adumbrated, perhaps confusedly, i know not, but the name of Bertrand Russell has become associated with the assertion that: the number $2$ is the set of all ...
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2answers
90 views

Proof of non-existence of non-principal $\kappa$-complete ultrafilter

Let $\lambda$ be a cardinal. I would like to prove that for all cardinals $\lambda < \kappa \leq 2^\lambda$, there can't be a $\kappa$-complete non-principal ultrafilter on $\kappa$. Here is my ...
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1answer
60 views

Cardinality of the set of differentiable functions

Is the cardinality of $$X = \{f: \Bbb R \to \Bbb R \;|\; f \text{ is differentiable everywhere}\}$$ the same as $\Bbb R$? How to prove it?
2
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1answer
103 views

Godel's pairing function and proving c = c*c for aleph cardinals

I have a few questions about Godel's pairing function and proving that c = c * c for aleph cardinals. Mostly, though, I'm concerned that most of the proofs I've seen are erroneous, and this concerns ...