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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1answer
64 views

What is ordinal expression of $\infty$? [closed]

$\infty$ - is cardinal expression? ?Origin of a line ray is ordinal expression of $\infty$, if distance from the $\infty$ to the $0$ origin of line ray?
1
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1answer
36 views

proving $|X|<|Y|$, $|Y|<|Z| \Longrightarrow |X|<|Z|$ without CSB

how to prove that if $|X|<|Y|$, $|Y|<|Z|$ then $|X|<|Z|$ without CSB theorem? it is immediate that $|X|\leq |Z|$ so I tried to assume that $|X|=|Z|$ and reach a contradiction but so far I ...
1
vote
2answers
35 views

Characterizing uncountable connected topological spaces

We know that if $X$ is a connected metric space with more than one point , then $X$ is uncountable ; can we characterize those connected topological spaces for which more than one point implies ...
3
votes
1answer
30 views

How to prove that if $A$ is infinite and $B$ is finite, then $|A\cup B|=|A|$?

I'm studying logic and unfortunately, I'm a newbie at this, so I don't see the stuff everyone sees at the moment. I want to solve following exercise, but get nowhere: Let $A$ be an infinite set ...
0
votes
1answer
94 views

How to prove there is no surjection $f\colon X \rightarrow 2^X$ [duplicate]

This is the following problem: Let $X$ be a set. Prove that there is not a surjection from $X \rightarrow 2^X$ (Hint: Assume to the contrary that $f\colon X \rightarrow 2^X$ is a surjection and ...
3
votes
0answers
37 views

basis for $\mathbb{R}^{\mathbb{N}}:=\left\{f:\mathbb{N}\to\mathbb{R}\right\}$, and its cardinality.

I know that all vector space has a basis. My question is "concrete" example for basis for $\mathbb{R}$-vector space $\mathbb{R}^{\mathbb{N}}:=\left\{f:\mathbb{N}\to\mathbb{R}\right\}$. If I refer ...
1
vote
1answer
43 views

How to calculate the cardinality of the complement of two countable sets of reals?

Let $A,B\subseteq\Bbb R$ be countable sets. Denote by $A'$ and $B'$ the complements (in $\Bbb R$) of $A$ and $B$ respectively. What is the cardinality of $C=A'\cap B'$? I cant figure this ...
3
votes
1answer
18 views

Let $\Gamma$ be a $\kappa$-based monotone operator where $\kappa$ is regular. Then the closure ordinal of $\Gamma$ is $\kappa$.

A monotone operator $\Gamma: \mathcal{P}(A) \to \mathcal{P}(A)$ is an operator such that, if $X \subseteq Y \subseteq A$, then $\Gamma(X) \subseteq \Gamma(Y)$. A monotone operator is $\kappa$-based ...
3
votes
1answer
57 views

Anatomy of $\mathcal P(\mathbb{N})$

How many proper subsets of $\mathcal P(\mathbb{N})$ there is that have cardinality of $2^{|\mathbb{N}|}$ ?
3
votes
1answer
46 views

Counting quantification and the cardinality of a set

A counting quantifier is a quantifier that denotes how many elements satisfy a predicate. I will use the notation $C_n x P[x]$ to denote that there are $n$ elements that satisfy $P$. I was thinking ...
3
votes
1answer
83 views

What else does ZFC prove about the “spectrum” of a cardinal number?

An auxiliary definition: Definition 0. Given an infinite set $X$ and a filter $\mathcal{F}$ on $X$, let $\sim_\mathcal{F}$ denote the equivalence relation on $\mathcal{P}(X)$ defined as follows: ...
1
vote
0answers
38 views

Prove that $|\mathbb R^n | = |\mathbb R|$. [duplicate]

Prove that $|\mathbb{R}^n| = |\mathbb{R}|$. It will be enough to prove $|\mathbb{R}^{2}|=|\mathbb{R}|$. We can further simplify by proving $|(0,1)\times(0,1)| = |(0,1)|$ (because ...
3
votes
1answer
117 views

How do I show the existence of a weakly inaccessible cardinal is not provable in ZFC?

So I know that if we assume the existence of inaccessible cardinals than we can show ZFC is consistent. How do I show that the existence of such cardinals is not provable in ZFC?
-2
votes
2answers
124 views

Why the number of all reals is $2^{\aleph_0}$ and not ${\aleph_0}^{\aleph_0}$? [closed]

Here is a small intuition why it should be the later. Let $\omega$ be the number of all natural numbers. Then what is the smallest real number? We can write reals in binary form. Usual logic would ...
1
vote
2answers
31 views

Is this proof about the countability of $\Bbb Q \times \Bbb Q \times \cdots \times \Bbb Q$ sound?

If $\Bbb{Q}$ is countable, prove that the set $\Bbb{Q}^n$ for $n = 2,3,...$ is countable. Base case: $n = 2 \rightarrow \Bbb{Q}^2 = \Bbb{Q}\times\Bbb{Q}$ which, by Proposition 4.5 (see bottom of ...
2
votes
2answers
98 views

Example of a $\kappa$-long sequence of disjoint club subsets of regular cardinal $\kappa$

I'm self-studying set theory and got stuck on this exercise: Let $\kappa$ be a regular cardinal. Give an example of a sequence $\langle C_\alpha\mid\alpha<\kappa\rangle$ such that $C_\alpha$ is ...
15
votes
1answer
181 views

There's no cardinal $\kappa$ such that $2^\kappa = \aleph_0$

I am trying to prove that there is no cardinal $\kappa$ such that $2^\kappa = \aleph_0$ . My attempt: We suppose it exists. Since $\kappa<2^\kappa$, in particular, $\kappa<\aleph_0$. But ...
0
votes
1answer
50 views

Cardinal number for a subset of $\mathbb{N}$

Following simple statement came to my mind when I was thinking about infinite sets. Statement: There is no set $X\subset\mathbb{N}$ that has cardinality strictly between any finite set ...
11
votes
1answer
176 views

Sets of Cardinals without choice

I have a theory that the axiom of choice is equivalent to the statement that sets of distinct cardinals are well ordered by cardinality. I can prove that the axiom of choice implies this. However I am ...
13
votes
2answers
427 views

What's the difference between saying that there is no cardinal between $\aleph_0$ and $\aleph_1$ as opposed to saying that…

... $\aleph_1$ is the immediate successor of $\aleph_0$? I was reading the wiki article on $\aleph_1$ where a distinction is made between the two. If there's isn't a cardinal between $\aleph_1$ and ...
1
vote
1answer
32 views

Sets,transversals,PT property,cardinals

A transversal of a family $S$ of sets is an injective choice function. $PT(\lambda,\chi)$ means, if $S$ is a family of $\lambda$ sets,each of cardinality $<\chi$,and every subfamily with ...
2
votes
2answers
58 views

Why is $\aleph_0$ the smallest cardinal number? [duplicate]

It is a well-known fact that $\aleph_0 = \vert \mathbf{N} \vert$ is the smallest infinite cardinal number. But I'm wondering why; does anyone know a proof? Thanks!
1
vote
1answer
70 views

Will $\kappa_1, \kappa_2, m$ cardinals. Given $\kappa_1 \leq \kappa_2$. prove: $\kappa_1 \cdot m \leq \kappa_2 \cdot m$

Will $\kappa_1, \kappa_2, m$ cardinals. Given $\kappa_1 \leq \kappa_2$. prove: $\kappa_1 \cdot m \leq \kappa_2 \cdot m$. Hi, I would be happy if someone could help me with this. What I did until ...
0
votes
2answers
61 views

Why is $\mathfrak{c}$ the cardinality of the lower limit topology on $\mathbb{R}$?

Why is $\mathfrak{c} = |\mathbb R|$ the cardinality of the lower limit topology on $\mathbb{R}$? An open set in the lower limit topology is of the form $[a,b)$. I can clearly see why the cardinality ...
20
votes
1answer
3k views

Overview of basic results on cardinal arithmetic

Are there some good overviews of basic formulas about addition, multiplication and exponentiation of cardinals (preferably available online)?
5
votes
1answer
87 views

Iterated Forcing, to force $2^{\omega}=\kappa$ and $2^{\omega _1}=\lambda$

Hellow i'm stuck on some details in this iterated forcing exercise. Let $M$ be a countable transitive model of $ZFC+GCH$ and assume that $\kappa<\lambda$ are cardinals with $\aleph _0 ...
4
votes
3answers
45 views

some basic cardinal arithmetic on $\text{cf}(\aleph_{\omega_1})$

I'm reading The Joy of Sets by K. Devlin, by self-study. I've just seen a statement $\text{cf}(\aleph_{\omega_1})=\omega_1$ without proof, but I think this is slightly harder to prove than more ...
2
votes
2answers
363 views

Proof of cardinality inequality: $m_1\le m_2$, $k_1\le k_2$ implies $k_1m_1\le k_2m_2$

I have this homework question I am struggling with: Let k1,k2,m1,m2 be cardinalities. prove that if $${{m}_{1}}\le {{m}_{2}},{{k}_{1}}\le {{k}_{2}}$$ then $${{k}_{1}}{{m}_{1}}\le {{k}_{2}}{{m}_{2}}$$ ...
0
votes
0answers
17 views

Will κ1,κ2,m cardinals. Given κ1≤κ2. prove: κ1⋅m≤κ2⋅m. [duplicate]

Will κ1,κ2,m cardinals. Given κ1≤κ2. prove: κ1⋅m≤κ2⋅m. Hi, I would be happy if someone could help me with this.. What I did until now:I replaced the cardinals with sets: |K1|=k1, |K2|=k2, |M|=m. From ...
1
vote
1answer
28 views

Few questions about the basics of Cardinality

I am looking for some help to either conform that my reasoning is sound, or to please elaborate to me more on the subject so I can gain a better understanding. I am studying some from my class notes, ...
3
votes
1answer
27 views

Find the cardinality of $S=\{(x,y,z) \in \Bbb R^3: x^2+y^2=4\}$

Find the cardinality of $S=\{(x,y,z) \in \Bbb R^3: x^2+y^2=4\}$. I know that as $S\subseteq \Bbb R^3 \implies |S|\leq \mathfrak{c}$. My conjecture is that $|S|= \mathfrak c$, I think this is true ...
2
votes
1answer
28 views

Cardinality of $X^n$

I asked the following question before: Is this proof by induction that $|\Bbb Q^n|=\aleph_0$ correct? I want to know if the proccess I did that can be generalized to the case $|X|=\kappa; \kappa$ ...
6
votes
2answers
722 views

Why do we distinguish between infinite cardinalities but not between infinite values?

More specifically, why are we "allowed" to denote $|\mathbb{N}|<|\mathbb{R}|$ but not $\sum\limits_{n\in\mathbb{N}}1<\sum\limits_{r\in\mathbb{R}}1$? Can we distinguish between "countable ...
3
votes
1answer
48 views

Existence of infinite set and axiom schema of replacement imply axiom of infinity

I'm self-teaching an intro to set theory course, and came across this exercise: Show that the existence of an infinite set is equivalent to the existence of an inductive set. For the notion of ...
0
votes
1answer
24 views

Confused by how to proof some statements about cardinals

I have a set of statements such as: Proof $\aleph_0+\aleph_0=\aleph_0$ I know that $|\Bbb Z|=\aleph_0$ and that for countable $A,B$ $A\cap B=\emptyset$: $|A\cup B|=|A|+|B|$. To this I add that ...
0
votes
1answer
33 views

Find the cardinality of $S=\{(x,y) \in \Bbb R^2 : 2x+3y<5\}$.

Find the cardinality of $S=\{(x,y) \in \Bbb R^2 : 2x+3y<5\}$. Attempt: I graphed this set, and I noticed that the simpler set $(0,1)^2=B\subset S$, and I thought these two sets had the same ...
1
vote
1answer
32 views

Is this proof by induction that $|\Bbb Q^n|=\aleph_0$ correct?

Is this proof by induction that $|\Bbb Q^n|=\aleph_0$ correct? Suppose I know that $|\Bbb Q|=|\Bbb N|=|\Bbb N^2|=\aleph_0\cdot\aleph_0=\aleph_0$. Proof: Suppose $|\Bbb Q^n|=\aleph_0$, then ...
1
vote
1answer
29 views

What is the cardinality of $L^p(\mathbb R)$, $1 \le p < \infty$?

$L^2(\mathbb R)$ is isomorphic to $\ell^2(\mathbb R)$ (which has the cardinality of $\mathbb R$ since there is an injection to the space of continuous functions which has the cardinality of $\mathbb ...
2
votes
1answer
30 views

Can it be proved without the axiom of choice that every cardinal is comparable with every finite cardinal?

Can it be proven in ZF, without using the axiom of choice, that every finite set is a universal size comparator, meaning, is comparable with every set in terms of size? And what is the proof?
4
votes
2answers
62 views

Proving that $(\omega_n)^\omega=\omega_n$ providing CH but not GCH

This is an exercise from a book from Kunen - SET THEORY, An Introduction to Independence Proofs Assume CH but don't assume GCH. Show that $(\omega_n)^\omega=\omega_n$ for $1 \le n < \omega$. I ...
1
vote
2answers
67 views

Is this a valid equivalent expression of the twin prime conjecture?

The twin prime conjecture states basically that it is possible to find two primes $p$, $p+2$ at a distance $2$ that are as big as wanted (Wikipedia). I am learning about the basic properties ...
5
votes
4answers
146 views

Can a number have an uncountably infinite amount of digits?

I'm an extreme mathematical layman, so please excuse the probable ignorance and awkward phrasing of this question. Is there such thing as a kind of number which has an uncountably infinite amount of ...
4
votes
1answer
49 views

Cardinality of a set of natural sequences

Let $a=(a_n)_{n\ge 1}$ a sequence such that for every $n\ge 1$ we have: a) $a_n \in\mathbb{N}$ b) $a_n\lt a_{n+1}$ c) Exists $\displaystyle\lim_{n\to \infty} \frac{\#\{j\mid a_j\le n\}}{n}$ Let ...
3
votes
4answers
107 views

$S$ and $T$ are two sets. Prove that if $|S-T|=|T-S|$, then $|S|=|T|$.

Here is the problem that I am currently working on: $S$ and $T$ are two sets. Prove that if $|S-T|=|T-S|$, then $|S|=|T|$. I have access to the answer for this proof, and wanted help with the first ...
3
votes
1answer
31 views

Show that there exists a sequence of functions $\{f_n:[0,1]\to\mathbb C\}$ satisfying the given condition.

Show that there exists a sequence of functions $\{f_n:[0,1]\to\mathbb C\}$ satisfying: 1) $f_n\to0$ pointwise; 2) $\gamma_nf_n\not\to0$, for all $\gamma_n\in\mathbb C$ such that ...
3
votes
0answers
74 views

Height of an ordered field

I'm studying ordered fields, and a specific notion regarding ordered fields that I will denote here by their "height". If $k$ is an ordered field, and $\alpha$ is a non-empty ordinal, a ruler of ...
1
vote
2answers
62 views

Cardinal Exponentiation Inequality

Let $\lambda, \kappa$ be infinite cardinals with $\lambda<\kappa$, what is known about $\kappa^\lambda$? specially in the case either $\kappa$ is regular. Or is there very little that can be ...
2
votes
4answers
98 views

What is the cardinality of the set of roots of unity?

Consider the geometric interpretation of "roots of unity": My intuition says that you can place arbitrarily many equidistant points on the unit circle and catch every point that lies on it. ...
2
votes
2answers
58 views

a question about analysis, how to find the largest cardinality in the following examples

This is a GRE math question: My thoughts: I guess as for the cardinality, (A)=(B) and (D)=(E),but I couldn't prove whether it is true or not. Also, how to find the cardinality of (C), can someone ...
0
votes
0answers
21 views

The size of the set of continuous function of periode T

I have a naive question. The Fourier series give an injection between continuous function of periode $T$ and the set of real valued sequences. But, don't we expect the set of continuous function of ...