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.

learn more… | top users | synonyms

0
votes
1answer
77 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.
0
votes
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 ...
0
votes
4answers
47 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
173 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
95 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
124 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
232 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 ...
1
vote
1answer
71 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
175 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 ...
1
vote
1answer
102 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
votes
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
votes
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
votes
1answer
47 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
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
148 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
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 ...
-2
votes
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?
0
votes
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
vote
1answer
80 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
99 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 ...
0
votes
2answers
63 views

How do you solve for the cardinality of a power set of some complex set? (i.e. $|\mathcal P(A^n)|$ , $|\mathcal P(A\cup B)|$ )

Suppose $A$ is some set such that $A = \{a_1,a_2,\dotsb,a_n\}$. We know that $|A|=n$. We know that $\mathcal P(A)= 2^n$. Now let $A^n$ denote the cartesian product of a set A with itself n times. ...
1
vote
1answer
63 views

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 $2^{\mathfrak c}$?
0
votes
2answers
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 ...
2
votes
1answer
35 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 ...
1
vote
1answer
86 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 $\aleph_n$ = ...
1
vote
1answer
123 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 ...
0
votes
2answers
86 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 ...
1
vote
3answers
43 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 ...
1
vote
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 ...
3
votes
1answer
87 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 ...
0
votes
2answers
105 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
votes
1answer
192 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, ...
1
vote
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
1
vote
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.
0
votes
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 ...
1
vote
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. ...
0
votes
1answer
42 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.
2
votes
1answer
39 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 ...
1
vote
2answers
69 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, ...
1
vote
1answer
68 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 ...
0
votes
1answer
45 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
votes
1answer
97 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 ...
10
votes
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
votes
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?
0
votes
2answers
95 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
votes
0answers
90 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
votes
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 ...
5
votes
1answer
76 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
votes
1answer
79 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, ...
3
votes
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 ...