Stack Exchange Network

Stack Exchange network consists of 175 Q&A communities including Stack Overflow, the largest, most trusted online community for developers to learn, share their knowledge, and build their careers.

Visit Stack Exchange

Questions tagged [cardinals]

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.

0
votes
1answer
52 views

Can the cardinality of a set $\mathbb{S}^n$ be written in terms of $\lvert S\rvert$?

The question is pretty simple: Can the cardinality of a set $\mathbb{S}^n$ be written in terms of $\lvert \mathbb{S}\rvert$? This includes transfinite cardinals. Here, $\mathbb{S}^n=\{(x_1,\cdots,...
2
votes
1answer
56 views

Weakly inaccessible cardinal such that $\kappa^{<\kappa}=\kappa$

Was looking at the Cantor's attic page for weakly compact cardinals and thought it was kind of odd that they only stipulate $\kappa^{<\kappa}=\kappa$ when many of the equivalent conditions ...
-4
votes
0answers
28 views

Equinumerousity of sets [on hold]

Are the following pairs of sets equinumerous? Question 1) $\mathbb{N}$ and $\mathbb{R}$ Question 2) $\mathbb{N}$ and $\mathbb{N} \times \mathbb{N}$ I need to prove the answers.
-2
votes
4answers
137 views

What is wrong with this paper on arXiv which claims that the set of real numbers is countable?

This is a follow up to: Cantor's diagonal argument and alternate representations of numbers What is wrong in this paper on arXiv: The Continuum is Countable: Infinity is Unique, Laurent Germain? ...
0
votes
0answers
41 views

Lemma I.13.19 on Kunen's set theory [duplicate]

I'm not even sure how to approach this problem. Kunen defined for arbitary sets $B$ and cardinals $\theta$, $$ [B]^\theta = \{ x \subseteq B : |x| = \theta \} $$ $$ [B]^{< \theta} = \{ x \subseteq ...
1
vote
1answer
48 views

Does $ | X | = | \mathbb{R} |$ hold for all Banach spaces $X$?

I remember reading somewhere that the cardinality of a Banach space $X$, denoted $|X|$ is equal to that of the continuum, but couldn't find it. If this is true, the set of all bounded functions from $...
2
votes
1answer
34 views

$KG(\mathfrak a_1,\mathfrak b_1)\cong KG(\mathfrak a_2,\mathfrak b_2)$ with $(\mathfrak a_1,\mathfrak b_1)\neq(\mathfrak a_2,\mathfrak b_2)$

Let $\mathfrak a,\mathfrak b$ be cardinals (a cardinal being identified with the smallest ordinal of that cardinality), $\mathfrak b\subseteq\mathfrak a$. The Kneser graph $KG(\mathfrak a,\mathfrak b)$...
10
votes
3answers
473 views

Are there non-equivalent cardinal arithmetics?

‎Generalizing a concept in mathematics is always a problematic situation. In most cases there are several ways to generalize a notion and it is not easy to decide if a particular generalization is ...
1
vote
1answer
76 views

Undecidable cardinality

Let $S$ be an infinite subset of $\mathbb{R}$ with the property that the existence of $S$ can be proved within ZFC (and in particular the definition of $S$ does not invoke the negation of the ...
0
votes
1answer
42 views

Does every graph have a maximum stable set?

(Don't confuse this question with that one about maximal stable sets.) The answer is positive for finite graphs of course. In infinite graphs the matter becomes interesting. Let's revisit the ...
0
votes
1answer
64 views

An exercise from Herrlich

Construct a set $X$ with the following properties: X is well orderable. $|X| \not\lt |\mathbb{R}|.$ $|X|\not\gt |\mathbb{R}|.$ $|X| = |\mathbb{R}|$ iff $|\mathbb{R}|$ is well orderable. This is an ...
4
votes
2answers
71 views

What is the cardinality of the set of subgroups of $(\mathbb R, +)$?

Let $X$ be the set of subgroups of $(\mathbb R, +)$. What is $|X|$? An attempt at a proof that $|X| = 2^{2^{\aleph_0}}$: Clearly $|X| \le 2^{2^{\aleph_0}}$, because $X \subset P(\mathbb R)$. For a ...
0
votes
0answers
18 views

Generalized Ramsey numbers, possibly infinite?

The Ramsey number $R(m,n)$ is easy to describe. It's the smallest positive integer such that any graph with at least $R(m,n)$ vertices has at least a clique of size $m$ or an independent set of size $...
4
votes
1answer
1k views

finding a bijective function from the real plane to the real line

As part of a HW assignment in the course elementary set theory, I was given the following question: Prove explicitly (don't use any theorems or known facts, but find a bijective function) that $\...
-2
votes
1answer
30 views

κ=sup{λ:λ<κ ∧ “λ is a cardinal number”} [closed]

How can I prove that If κ is an infinite cardinal number then we have κ=sup{λ:λ<κ ∧ “λ is a cardinal number”}
-3
votes
2answers
48 views

Showing that $cf(\aleph_\alpha) \leq cf(\alpha)$

How can I prove this direction? Let $\alpha$ be a limit ordinal. Then $cf(\aleph_\alpha) \leq cf(\alpha)$.
0
votes
1answer
46 views

Arithmetic on infinite cardinal numbers

I am stuck on the following problem that says: Assuming the Generalized Continuum Hypothesis (GCH), that is, the statement $2^{\aleph_{\alpha}}$ = $\aleph_{\alpha+1}$ for every ordinal $\alpha$,...
2
votes
1answer
34 views

Hamel dimension of a vector space, and dimension of the dual

I have the following (possibly trivial) observation: Let $K$ be an $\mathbb{F}$-vector space (I believe the argument also works for free modules), and let $X\subseteq K$ be it's basis with ...
2
votes
1answer
811 views

The class of cardinal numbers is well ordered

I'm looking for a proof that the class of cardinal numbers is well ordered under the order relation $|A|\leq |B| \Leftrightarrow$ exists an injection $f:A \to B$. In fact, I've found a very beautiful ...
0
votes
1answer
30 views

Prove or disprove: $\lvert\mathcal{R}\rvert=\lvert\mathcal{R}^{-1}\rvert$

Prove or disprove: If $\mathcal{R}$ is a relation then $\lvert\mathcal{R}\rvert=\lvert\mathcal{R}^{-1}\rvert$. I think it is true but I do not know how to prove it. Facts: $\mathcal{R}^{-1}=\{(...
1
vote
1answer
51 views

Formally representing “arbitrarily large” as a distinct entity from “countably infinite”

There is often an important difference to be made between "countably infinite" things and "arbitrarily large finite" things. There are so many examples of this that I need not list them all here. The ...
-2
votes
6answers
859 views

About the Continuum Hypothesis [duplicate]

Possible Duplicate: Are there more rational numbers than integers? The Continuum Hypothesis states: There is no set whose cardinality is strictly between that of the integers and that of the ...
0
votes
0answers
31 views

Is the standard model of the reals the only model up to isomorphism of cardinality $\beth_1$? [duplicate]

TL;DR: I'm trying to figure out what you need to do to pin down the standard model of the reals without explicitly constructing such a model and then defining the standard model $W$ of the reals up to ...
5
votes
3answers
2k views

Every infinite subset of a countable set is countable.

Here is the proof I tried to weave while trying to prove this theorem: Theorem. Every infinite subset of a countable set is countable. Proof. Let $A$ be a countable set and $E\subset A$ be ...
1
vote
0answers
22 views

Finite sets of a infinite set A is in bijection with A [duplicate]

Suppose $A$ is a infinite set. Let $P_f(A)$ be the collection of all finite subsets of $A$. Now we want to show that $P_f(A)$ is in bejection with $A$. I can do this for the case that $A$ is countable,...
3
votes
4answers
589 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 ...
3
votes
2answers
44 views

Cardinality of the set of all total orders on $\Bbb{N}$

I need to compute the cardinality of the set of all total orders on $\Bbb{N}$. Now, by definition there is an inclusion of this set into $\mathcal{P}(\Bbb{N}\times\Bbb{N})$, and so has cardinality $\...
3
votes
2answers
41 views

Countability of a subset of sequences

Let $\mathcal{A} = \{a \in \{1,2,3,4,5\}^\Bbb N : |a_i- a_{i+1}| = 1 \; \forall i\}.$ Is the set $\mathcal{A}$ countable? I tried an argument like Cantor's diagonalization process but without success....
2
votes
1answer
41 views

Cardinality of all infinite subsets with an infinite complement.

Working in $\text{ZF}$... Let $X$ be an infinite set with a given well-ordering relation $\le$. Define $\tag 1 \mathcal B(X) = \{ S \in \mathcal P(X) \, | \, S \text{ is infinite } \text{ and } X \...
2
votes
2answers
69 views

$\aleph_0\times 2^{\aleph_0} = 2^{\aleph_0}$ requires the Axiom of Choice?

In a model solution it is stated that the cardinality of a set which is the countable union of sets of cardinality $2^{\aleph_0}$ is $\aleph_0\times 2^{\aleph_0}$, and, using the Axiom of Choice, $\...
0
votes
1answer
36 views

find cardinality of A [duplicate]

$A$ = $\{X\subseteq \mathbb N : |X| = \aleph_0\land |\bar X| \lt\aleph_0 \} $ I need to find the cardinality of group A. As I understood, I need to find a function that helps me determine the ...
0
votes
1answer
55 views

Which stage in the Neumann hierarchy do powers of the reals fit in?

To be more specific than the short title, I try to gauge the size of some "normal" function spaces as e.g. found in functional analysis against set universe sizes at certain stages. For the sake of ...
55
votes
7answers
11k views

Why is the Continuum Hypothesis (not) true?

I'm making my way through Thomas W Hungerfords's seminal text "Abstract Algebra 2nd Edition w/ Sets, Logics and Categories" where he makes the statement that the Continuum Hypothesis (There does not ...
4
votes
2answers
7k views

Union of Uncountably Many Uncountable Sets

I know that the union of countably many countable sets is countable. Is there an equivalent statement for uncountable sets, such as the union of uncountably many uncountable sets is uncountable? ...
0
votes
0answers
23 views

what is an Injective function from A to B when A=$P(\mathbb N)$ B=$\{T⊆N|∀S⊆T.|T|<א_0→\sum_{x∈S} x≠\sum_{x∈T/S} x\}$?

what is an Injective function from A to B when A=$P(\mathbb N)$ B=$\{T⊆N|∀S⊆T.|T|<א_0→\sum_{x∈S} x≠\sum_{x∈T/S} x\}$ ?
0
votes
1answer
34 views

Cardinal of all well-orders of $\mathbb{N}$ [duplicate]

Lets consider the set $$ A = \{R\subset\mathbb{N}^2:R \text{ is a well-order of } \mathbb{N}\} $$ Now, it's clear that $\aleph_1\leq|A|\leq2^{\aleph_0}$(Since all the well-orders of $\mathbb{N}$ are ...
2
votes
1answer
20 views

Does a (complete) $\mathfrak{a}$-partite graph always have a maximum in AND out degree?

A graph $G$ is called $\mathfrak{a}$-partite (with $\mathfrak{a}$ being any non empty cardinal), if there exists a partition $\mathcal P$ of $V(G)$ such that any two vertices in the same class $A\in\...
2
votes
0answers
85 views

Show that continuous function from omega_1 to R is eventually constant [duplicate]

Let $f: \omega_1 \to \mathbb{R}$ be a contionuous function. Prove that $f$ is eventually constant. I was trying to prove it by contradiction but I did not have any idea.
1
vote
0answers
31 views

Why is cofinite topology on a countable set a second countable space? [duplicate]

Consider a countable set $X$ with cofinite topology. Here the claim is that $X$ is second countable. The simple argument is as follows : $\color{red}{\text{Since $X$ is countable, the number of ...
3
votes
1answer
40 views

Are there uncountable cardinals $\kappa$ such that $|\kappa\cap\mathsf{Card}| = \kappa$?

All the cardinals $\kappa\leq\aleph_0$ have the property that there are precicely $\kappa$ cardinals less than $\kappa$. Of course, $\aleph_1$ lacks this property since there are only $\aleph_0 +1= \...
1
vote
2answers
95 views

What is the analogue to this cardinal arithmetic theorem for infinite products?

$\begin{array}{l}{\text { 1.3 Theorem Let } \lambda \text { be an infinite cardinal, let } \kappa_{\alpha}(\alpha<\lambda) \text { be nonzero cardinal}} \\ {\text {numbers, and let } \kappa=\sup \...
1
vote
0answers
36 views

cardinality of multiplication of groups A,B when $A\leq B$

if $A$ and $B$ are infinite groups and $$ |A|\leq|B|, $$ where $|{}\cdot{}|$ denotes the group cardinality, is it right to say that $|A|\cdot|B| = |B|$?
2
votes
1answer
43 views

How $c^{\aleph_0}=c$

I am reading about coutabaility, uncountability and cardinal numbers. I have attached herewith a screenshot of the wikipedia page. I am not able to understand how in the third equation we show that $...
3
votes
2answers
1k views

Cardinality of the set of at most countable subsets of the real line?

I'm exploring an unrelated question about power series with complex coefficients. While exploring this question, I wondered: What is the cardinality of the set of all such power series? Or with ...
8
votes
2answers
701 views

Cardinality of the Set of all finite subset of $\mathbb{R}$

Find the Cardinality of the set of all finite subsets of $\mathbb{R}$. I have proved that the set of all finite subsets of $\mathbb{N}$ is countable . But I cannot find the cardinality of the set ...
1
vote
1answer
39 views

Commutative Noetherian ring with distinct ideals having distinct index

Let $R$ be a Noetherian commutative, infinite ring with unity such that distinct ideals have distinct index i.e. if $I,J$ are ideals of $R$ and $I \ne J$ , then $R/I$ and $R/J$ are not bijective as ...
2
votes
1answer
43 views

If $X$ is an infinite set and $N$ is a countable set, then what is the cardinality of $X \times N$? [duplicate]

If $X$ is countable, then the cartesian product is countable. However, what about general cases? The googling suggests that the answer is the cardinality of $X$. But why? Could anyone please provide ...
2
votes
0answers
27 views

Binomial coefficients for infinite cardinalities [duplicate]

Let’s define $C_\alpha^\beta$ as the cardinality of the set of all subsets with cardinality $\beta$ of a set with cardinality $\alpha$: $$C_\alpha^\beta = |\{T \subset S| |T| = \beta \}|$$ where $|S|...
0
votes
1answer
20 views

what is the cardinality of equivalence classes of relation $ R=\{<A,B>\in P(\mathbb{N} )|A\cap T=B\cap T\} $?

given :$$T\subseteq \mathbb{N} $$ $$ R=\{<A,B>\in P(\mathbb{N} )|A\cap T=B\cap T\} $$ what is a equivalence relation what is the cardinality of equivalence classes of relation R ? how can I ...
0
votes
2answers
31 views

List all the possible sizes of N. Which of these answers divides 60?

Suppose you have disjoint sets $A, B, C, D $, and $F$ with sizes 1, 12, 12, 15, and 20, respectively. Suppose N is a set formed by taking the union of A with one or more of the other sets. List all ...