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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2
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1answer
11 views

Does every set have choice sequences as long as the original set?

Given a set $X$, we say that $X$ has choice sequences of length $|I|$, denoted $CS(|X|,|I|)$, if for any $f:I\to{\cal P}(X)\setminus\{\emptyset\}$ there is a function $g:I\to X$ such that $g(x)\in ...
2
votes
1answer
20 views

Give an example of a structure of cardinality $\omega_2$ that has a substructure of $\omega$ but no substructure of $\omega_1$

Give an example of a structure of cardinality $\omega_2$ that has a substructure of $\omega$ but no substructure of $\omega_1$ This is from Hodges' A Shorter Model Theory. My idea is to take ...
1
vote
1answer
38 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
31 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 ...
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 ...
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 ...
1
vote
1answer
44 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
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 ...
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 ...
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 ...
-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
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!
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 ...
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 ...
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 ...
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 ...
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 ...
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, ...
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 $|X|=\kappa; \kappa$ ...
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 ...
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 ...
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 ...
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 ...
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 ...
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 ...
-5
votes
1answer
65 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?
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 ...
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. ...
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 ...
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 ...
1
vote
1answer
56 views

For Infinite Cardinals does $A > B \Rightarrow A^C > B^C$?

It seems clear that for $A, B, C$ infinite cardinals with $A > B $ one could define an injection from $B^C \to A^C$ and so $A > B \Rightarrow A^C \ge B^C$, but is the inequality strict and ...
2
votes
1answer
37 views

Cardinality of $A^B$ when $A > B \ge \aleph_0$

For an infinite cardinal A, then if $B$ is finite $A^B = A$ If $B$ is infinite and $B \ge A$ then $A^B \ge A^A \ge 2^A > A$ What if $B$ is infinite, but $B < A$, i.e. $A > B \ge \aleph_0$. ...
2
votes
2answers
28 views

Confused about one to one functions and cardinality

There is something that I'm not getting about functions and cardinality of sets. I've read the following: If $F$ is a one to one function, then $F^{-1}$ (the inverse) is also a one-to-one ...
4
votes
1answer
35 views

Cardinality of a set of non-continuous functions [duplicate]

I have to find the cardinality of the set of the non-continuous functions $f:\mathbb{R}\rightarrow\mathbb{R}$. I think we should look for function that at least have a point of discontinuity, but i ...
2
votes
0answers
38 views

Is the set of all of possible uncountable cardinalities uncountably or countably infinite? [duplicate]

I assume that the number of unique possible cardinalities that are uncountably infinite is either uncountable or countable because it is possible to take the powerset of each set, resulting in an ...
2
votes
1answer
41 views

Cardinality of $P_\mathfrak{c}(\mathbb R)$

Calculate the cardinality of $\mathcal P_\mathfrak{c}(\mathbb R)=\{\mathcal R \in \mathcal P(\mathbb R) \, /\, \#(\mathcal R)=\mathfrak{c}\}$. I thought that instead of $\mathcal ...
3
votes
0answers
62 views

Proof check:$ \left | \mathbb{R} \right |= 2^{\left|\mathbb{N} \right |}$

This is my first time to post here. Sorry if this post is too simple or naive. Here I would like to prove that $\left | \mathbb{R} \right |= 2^{\left |\mathbb{N} \right |}$ I would first ...
2
votes
2answers
44 views

If $n < \aleph^*(m)$, then $n < 2^m$.

Without $AC$ Let $\aleph^*(m)$ be the least aleph that $\not\leq^* m$. I need a help or hint that if $n < \aleph^*(m)$, then $n < 2^m$. $a \leq^* b$ means we can define a surjective map from ...