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
votes
4answers
75 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
47 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
96 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
44 views

What is ordinal expression of $\infty$? [on hold]

$\infty$ - is cardinal expression. What is ordinal expression of $\infty$? If so, then examples.
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
71 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
votes
0answers
18 views

Is a class of sets indexed by all the abelian groups a proper class [duplicate]

Let $\{K^i \mid i \in \mathbb{N}\}$ be a set indexed by $G \in \textbf{Ab}$. Is the class of all $\{K^i \mid i \in \mathbb{N}\}_{G \in \textbf{Ab}}$ a proper class? My understanding is that since ...
1
vote
2answers
56 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
93 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
55 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
20 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 ...
3
votes
4answers
2k views

Proving $\mathbb{N}^k$ is countable

Prove that $\mathbb{N}^k$ is countable for every $k \in \mathbb{N}$. I am told that we can go about this inductively. Let $P(n)$ be the statement: “$\mathbb{N}^n$ is countable” $\forall n \in ...
1
vote
1answer
53 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 ...
7
votes
2answers
2k views

Proving the Cantor Pairing Function Bijective

How would you prove the Cantor Pairing Function bijective? I only know how to prove a bijection by showing (1) If $f(x) = f(y)$, then $x=y$ and (2) There exists an $x$ such that $f(x) = y$ How would ...
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
34 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
37 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 ...
14
votes
5answers
2k views

Cardinality of all cardinalities

Let $C = \{0, 1, 2, \ldots, \aleph_0, \aleph_1, \aleph_2, \ldots\}$. What is $\left|C\right|$? Or is it even well-defined?
3
votes
0answers
60 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
43 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 ...
1
vote
1answer
29 views

Hilbert's hotel prime powers method

To fit an infinite number of coaches each with an infinite number of passengers, we can assign the people in the hotel with the prime number 2, and coach $c$ is assigned with the $c$th odd prime ...
0
votes
1answer
19 views

For every $n \ge 2 $ , existence of uncountably many mutually disjoint closed balls , complement of whose union is path connected

For every $n \ge 2 $ , does there exist an uncountable family $\{D_\lambda: \lambda \in \Gamma\}$ of mutually disjoint closed balls in $\mathbb R^n$ , such that $\mathbb R^n \setminus \cup_{\lambda ...
-3
votes
1answer
57 views

Are $\mathbb{N}$ is isomorphic to $\mathbb{Q}$? [duplicate]

Are $\mathbb{N}$ isomorphic to $\mathbb{Q}$? There are any difference between isomorphism and cardinal equality? If $X$ and $Y$ are two sets and $\text{Cardinal}(X)=\text{Cardinal}(Y)$, is $X$ ...
6
votes
1answer
111 views

Proving that $\sf Add$$(\aleph_\omega , 1)$ collapses cardinals $\leq \aleph_\omega$

First, let me fix some notation. $\sf Fn$$(I, J, \kappa) = $ the poset of all partial functions $p$ such that $|p| < \kappa$, dom$(p) \subseteq I$ and rng$(p) \subseteq J$. $\sf Add$$(\kappa, ...
3
votes
1answer
66 views

Cardinals definable using ordinal arithmetics

Let $\kappa$ be an infinite cardinal. $\kappa$ is stable under ordinal addition $+$, ordinal multiplication $.$ and ordinal exponentiation $e: (a,b) \mapsto a^b$, so $\mathcal{K} = ...
1
vote
0answers
37 views

Cardinality of the set of all field automorphisms of $\mathbb C$ [duplicate]

Does $\mathbb C$ have infinitely many field automorphisms? Does it have uncountably many field automorphisms?
0
votes
3answers
301 views

How big is the size of all infinities?

"Not only infinite - it's "so big" that there is no infinite set so large as the collection of all types of infinity..." What does exactly mean? How many infinities are there? I've heard there are ...
5
votes
2answers
125 views

Is the cardinality of $\mathbb{Z^R}$=$\mathbb{R^Z}$?

Previously in this question, we have found that $\mathbb{R^Z}$ is uncountable and its multiset of components, denoted by $$K = \{ (..., 0, 0, w, 0, 0, ... ) : w \in \mathbb{R} \}$$ where for each ...
5
votes
2answers
209 views

Dimension of $\operatorname{Hom}(V, W)$

What is the dimension of $\operatorname{Hom}(V, W)$ if at least one of the two vector spaces $V, W$ is infinite dimensional? In the sense of cardinal numbers. Thanks
0
votes
1answer
25 views

Hilbert space and uncountable cardinal

Given an uncountable cardinal does there exist Hilbert space with orthonormal basis of that cardinality?
1
vote
1answer
19 views

Linear continuum poset with succ and pred

Is there a poset of continuum cardinality that satisfies conditions: it is linear there is a minimum element there is a maximum element each element (apart from maximum) has a successor each element ...
6
votes
2answers
49 views

Let $X \subseteq \mathbb R$ and $X$ has same cardinality as $\mathbb R$ , does there always exist a continuous surjection from $\mathbb R$ onto $X$ ?

Let $X \subseteq \mathbb R$ and $X$ has same cardinality as $\mathbb R$ , does there always exist a continuous surjection from $\mathbb R$ onto $X$ ? ( I know that there need not always be a ...
1
vote
1answer
47 views

Orthonormal Basis and Hamel Basis Cardinality

Will cardinality of orthonormal basis will always be strictly less than cardinality of Hamel Basis. It is true in case of seperable spaces. (Because Hilbert space is always uncountable but ...
6
votes
5answers
2k views

Are there fewer reals on $(0, 1)$ than on $(1,\infty)$?

I know that the cardinality of the sets of real numbers $(0, 1)$ and $(1, \infty)$ are equal. So what is the fallacy in this argument? For every real on $(0, 1)$, we can add any integer $n$ to it ...
0
votes
1answer
54 views

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

The twin prime conjecture states 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 associated to the ...
3
votes
0answers
51 views

Why is this not a proof of Schroeder-Bernstein?

We can show that if $f: A \rightarrow B$ is injective then $|A| \leq |B|$ and if $g: B \rightarrow A$ is injective then $|B| \leq |A|$ so $|A| = |B|$. By the definition of having equal cardinality, ...
26
votes
8answers
2k views

Is symmetric group on natural numbers countable?

I guess it is too difficult a question to ask about the cardinality of $S_{\mathbb{N}}$ so I would like to ask whether it is countable or not. I tried to prove it is uncountable somewhat mimicking ...
2
votes
6answers
102 views

More numbers between $[0,1]$ or $[1,\infty)$?

There are infinitely many real numbers between any two real numbers, therefore there are infinitely many real numbers in the range $[0,1]$ as there are in $[1, \infty)$. In a mathematical sense, are ...
4
votes
1answer
72 views

First Uncountable Ordinal Cofinality: Needs AC?

Say $\omega_1$ is the first uncountable ordinal. The reason I care about $\omega_1$ is Any countable subset of $\omega_1$ is bounded (or if you prefer, there is no countable cofinal subset). This ...
3
votes
4answers
93 views

Does there exist a path connected metric space , in which at least one open ball is countable ?

Does there exist a path connected metric space with more than one point , in which at least one open ball is countable ?
2
votes
1answer
26 views

Does there exist a connected metric space, with more than one point and without any isolated point, in which at least one open ball is countable?

Does there exist a connected metric space, with more than one point and without any isolated point, in which at least one open ball is countable?
2
votes
0answers
34 views

Why is the weight of a topological space a minimum?

If $(X, \tau_X)$ is a topological space, then the weight is usually defined as follows: $$w(X) = \min \{ \vert B \vert : B \subset \wp(X), B \mathrm{\; is \; a \; basis \; of \;} \tau_X \}$$ I was ...
0
votes
2answers
39 views

What is the cardinality of the power set $P(A \cup B)$

Let $A = \{1, 3, 5\}$ and $B = \{3, 4, 5\}$ be sets. What is the cardinality of the power set $P(A \cup B)$? If i'm not mistaking isn't it all the possible combination of these two: $\{\}, ...
4
votes
1answer
66 views

Singularity of small cardinals under AD

It appears to be quite a folklore result that under AD we know which small well-orderable cardinals (for my purposes I mean below $\omega_\omega$) are regular, namely only $\omega,\omega_1,\omega_2$. ...
1
vote
0answers
25 views

Why Is This Step Needed in Proving Bernstein-Schroeder?

Link to Original Text My question is in the lemma: If $f: A \rightarrow B$ is injective, where $B \subset A$, then there is a bijection between $A$ and $B$. The author commented that, with $Y = ...
0
votes
2answers
50 views

What is the cofinality of $2^{\aleph_\omega}$

There is a similar question in this site but I am not satisfied with the answer, which is basically the same as the proof in the mentioned textbook. The book(Karel Hrbacek&Thomas Jech, ...
1
vote
4answers
68 views

When does the cardinality disappear?

Pardon me, if this question sounds stupid. I am learning real analysis on my own and stumbled on this contradiction while reading this -- http://math.kennesaw.edu/~plaval/math4381/setseq.pdf. I ...
2
votes
1answer
54 views

Vaught's two cardinal theorem using Vaught pairs

I've been reading David Marker's Introduction to Model Theory, and found Vaught's two cardinal theorem (4.3.34): if a theory $T$ has a $(\kappa,\lambda)$-model, where $\kappa > \lambda \geq ...