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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.

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cardinality of all cardinal numbers less than a cardinal number

For a cardinal number $\alpha$ what is the cardinality of the set $X=\{\beta, \beta$ is a cardinal number with $\beta<\alpha\}?$ Can we say for example that $card(X)=\alpha$ or $card(X)\leq \...
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Cardinality of all cardinal numbers less than a given cardinal

For a given cardinal number $\aleph_{\alpha}$ we define $$X_{\alpha}= \{\aleph_{\beta}; \aleph_{\beta}<\aleph_{\alpha}\}.$$ We can easily prove that 1) $card(X_{\alpha}) \leq \aleph_{\alpha}^{+}=...
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1answer
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Value of the cardinal product $\aleph_1 \cdot \mathfrak{c}$

Suppose we want to know how many well-orderings of the naturals there are. That is, not up to isomorphism, but how many individual ways to well order the naturals there are. It's easy to see that for ...
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1answer
31 views

Set of various order types of a set

Starting from the cardinal $|\Bbb N| = \aleph_0 = \beth_0$, we can generate a larger cardinal in two ways: Take the set of all subsets, generating the cardinal $\beth_1$ Take the set of all well-...
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0answers
31 views

The “number” of complete vector fields in a manifold

I was wondering about the cardinality of complete vector fields in relation to the cardinality of general vector fields. Let $$\mathcal{A}=\{X:M\rightarrow TM| X \text{ is a vector field}\}$$ and $$\...
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1answer
45 views

Question about the existence of sets

If there is a sequence of set $a_{i}$ and $i \in I \wedge |I|>|\mathbb{R}|$, is the set $A$ which $\forall i (i \in I \rightarrow a_{i}\in A)$ and the set $\bigcup \limits_{i \in \mathbb{R}} a_{i}$ ...
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1answer
57 views

Are infinite indexes multiplicative?

For a group $G$ and its subgroup $H$, the index of $H$ relative to $G$, denoted by $[G:H]$, is the cardinality of the set $\{ gH \mid g \in G \}$. It is known that $|G| = [G:H]|H|$ even if all ...
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1answer
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Explicit example of $C \subset \mathbb{R}$ with a single certain type of condensation point… if such is even doable.

In Abbott's Understanding Analysis 2e, part of exercise 1.5.10 asks Let $C \subset [0,1]$ be uncountable, let $A := \{ a \in (0,1): C \cap [a,1] \text{ is uncountable} \}$, and let $\alpha := \sup ...
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1answer
86 views

Find the cardinality of $A= \left\{ f : \mathbb N \rightarrow \mathbb N \mid \forall x:f(x)\le x\right\}$

How can I solve tasks like this one? Example task Find the cardinality of $$A= \left\{ f : \mathbb N \rightarrow \mathbb N \mid \forall x:f(x)\le x\right\}.$$ I know that $|A| \le \mathfrak{c}$...
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2answers
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Show that the cardinal of $ A := \left\{ k \in \mathbf{Z} | 0 \leq k \leq n \text{ and } \binom{n}{k} \text{ is odd} \right\} $ is a power of 2 [duplicate]

Let $ A := \left\{ k \in \mathbf{Z} | 0 \leq k \leq n \text{ and } \binom{n}{k} \text{ is odd} \right\} $. I must show that the cardinal of $A$ is a power of 2. I have tried to show that there ...
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1answer
45 views

How to prove $\mathbb{R}^n < \mathbb{N}^\mathbb{R}$

I know that the cardinality of $\mathbb{R}^n$ is equal to $\mathbb{R}$. I also know that the cardinality of $\mathbb{N}^n$ is equal to $\mathbb{N}$, but how do I prove that the cardinality of $\...
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2answers
59 views

Does it make sense to say that “there are more non-Abelian groups than Abelian groups”?

Firstly, does the family of all non-isomorphic non-Abelian groups have a well defined cardinality? How about the family of all non-isomorphic Abelian groups? If they are both defined, how do they ...
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0answers
26 views

cardinality of sets intersections

Given that: $$| A \cap \mathbb{R}| = |\mathbb{R}| $$ $$| B \cap \mathbb{R}| = |\mathbb{N}| $$ Can we say that: $$|A| = |\mathbb{R}|$$ $$|B| = |\mathbb{N}|$$ ? If not, can we say something for ...
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0answers
21 views

Cardinality of set of words [duplicate]

How can we prove that for the countable alphabet $A$ the cardinality of set of all finite words over this alphabet $A^*$ is equal to $\aleph_0$
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0answers
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Is the least inaccessible cardinal equivalent to the first aleph fixed point? [duplicate]

Let $I$ be the least / first inaccessible cardinal. As inaccessible cardinas are all aleph fixed points, and they are regular, so each inaccessible cardinal is an aleph fixed point after the previous ...
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2answers
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Cardinality of sets of functions $\mathbb{R}\to \mathbb{N}$ and $\mathbb{N}\to \mathbb{R}$. [closed]

Let $B^A$ denote the set of all functions $A \to B$. Prove that $\left|\mathbb{R}^\mathbb{N}\right|<\left|\mathbb{N}^\mathbb{R}\right|$.
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1answer
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$\operatorname{card}(\bigcup\limits_{n \in \mathbb{N}} \underbrace{A\times…\times A}_{n})=k$ if $\operatorname{card}(A)=k$ infinite.

I was reading a proof of a theorem that goes like this: Let $A$ be an infinite set of cardinality $k$ and $A^{<\omega}$ the set of finite sequences of elements of A. Then $\operatorname{card}(A^{&...
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1answer
22 views

Proving sets of functions have same cardinality

Prove that $$card(A^{B \times C})=card(A^{B^C})$$ where $A^B$ is a set of all functions from $B$ to $A$ and $A \times B$ is cartesian product of sets. Is the bijection supposed to be sending $f$ to $...
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1answer
88 views

Cardinality of infinite dimensional vector space

Assume that V is an infinite dimensional vector space. I know that if V is a vector space over a field F, then |V|=max{dimV,|F|}. So if we take V=$\mathbb{R}$ and F=$\mathbb{Q}$ then |V|>|F| and |V|=...
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0answers
19 views

Cardinality of $[\lambda]^\kappa$

Let $\kappa \leq \lambda$ cardinals with $\lambda$ infinite, and $[\lambda]^\kappa=\{Y\subseteq\lambda : ot(Y,\in)=\kappa\}$. I want to show that $[\lambda]^\kappa \asymp\ ^\kappa\lambda$. I've ...
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2answers
36 views

How to show $\operatorname{card}(\omega+1)=\omega$

Apparently $\operatorname{card}(\omega+1)=\omega$. This means that there is an order $<$ on $\omega+1$ such that there is an isomorphism of ordered set $f$, $(\omega+1,<) \cong (\omega,\in)$, ...
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1answer
63 views

finding bijection such that $|\{ x\in A : x \neq f(x)\}| =\mathfrak{c} $

Let $|A| = 2^{\mathfrak{c}}$. I am finding function $f$ is bijection from $A$ to $A$ such that $|\{ x\in A : x \neq f(x)\}| =\mathfrak{c} $. Any ideas? I will try to prove it later.
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1answer
26 views

Proving that exists equivalence relation $r$ in set $A$ such that $ |A \setminus r| = n$

I am trying to show that if $|A| = m$ and $0\neq n \le m $ then exists equivalence relation $r$ in set $A$ such that $ |A \setminus r| = n$. Could someone help me deal with it?
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1answer
44 views

Understanding power of $\left\{ f | f: \mathbb R \rightarrow \mathbb R \right\}$

I have problem with understanding power of $\left\{ f | f: \mathbb R \rightarrow \mathbb R \right\}$ Generally, from lecture I know that power of set $$\left\{ f | f: A \rightarrow B \right\}$$ is $ ...
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4answers
26 views

Cardinality of set of sequences from $\mathbb{Q}$ that converge to $0$

I'm looking for cardinality of a set of sequences from $\mathbb{Q}$ which are convergent to $0$. I think the answer is continuum, but I don't know how to prove it.
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0answers
27 views

Hereditary product of non-hereditary cardinal functions

Does there exists two cardinal functions in topology, each of them not hereditary but their product is hereditary? For cardinal function i mean a rule to associate a cardinal number to every ...
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1answer
59 views

Partially ordered sets cardinality

What is the cardinality of the set of all partially ordered sets of natural numbers which have one least element and infinity number of maximal elements? I only noticed that upperbound for this set ...
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3answers
78 views

Find cardinality of $X = \left\{ A : A \subset \mathbb R \wedge \text{c}(A) \right\} $

I have some doubts with this task: Find cardinality of a) $X = \left\{ A : A \subset \mathbb R \wedge \text{c}(A) \right\} $ b) $X = \left\{ A : A \subset \mathbb Q \wedge \text{c}(A) \right\} ...
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0answers
68 views

weakly normal filters

Kanamori (Ultrafilters over Uncountable Cardinals) in his Phd Thesis defines a filter $\mathcal F$ as weakly normal whenever every function $f$ such that $\{\xi<\kappa\mid f(\xi)<\xi\}\in\...
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1answer
62 views

Cardinal exponentiation without generalized continuum hypothesis

First I have to confess that I don't know about set theory language. Let $A$ and $B$ be infinite cardinals with $A>B$. My question is: $A^B=A$? (without assuming generalized continuum hypothesis) ...
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2answers
37 views

How does my bijection between the natural numbers and the powerset of natural numbers break down? [duplicate]

Lets consider some natural number x in binary. Let the least significant digit represent the inclusion or exclusion of 0, the next least significant represent 1, and so on upwards. For some examples: ...
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1answer
81 views

Measurable cardinals: non-trivial two-valued measures

while doing some exercises about measurable cardinals, I got stuck on this one: If $κ$ is the minimal cardinal that carries a non-trivial two-valued measure, then how can one prove that $κ$ is ...
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2answers
71 views

Let $A_{n}=\left\{ f \in \left\{ 0,1\right\}^{\mathbb N}: f(n)=0 \right\} $

Let $$A_{n}=\left\{ f \in \left\{ 0,1\right\}^{\mathbb N}: f(n)=0 \right\} $$ Find (a) $$| \bigcap_{m \in \mathbb N}^{} \bigcup_{n \ge m}^{} A_{n}|$$ (b) $$|\left\{ 0,1\right\}^{\mathbb N} \...
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2answers
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If $A\sim B$(both dedekind infinite), is it then that $A\sim B\cup \{x\}$

If the symbol $A\sim B $signifies that there is a bijection between A and B, and We take our sets to be dedekind infinite, then is the following correct? If not, what is the counter example?:$$A \sim ...
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0answers
82 views

Proof of the Solovay Theorem in Jech

The Solovay Theorem says: Let $\kappa$ be a regular uncountable cardinal. Then every stationary subset is the disjoint union of $\kappa$-many stationary subsets. [Jech, Theorem 8.8, p. 95] One ...
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1answer
77 views

Finite chain condition - Variation of Martin's Axiom statement

In the following $k$ and $w$ will be cardinal numbers. Consider the classical statement $MA(k)$: For any partial order $P$ satisfying the countable chain condition (hereafter $ccc$) and any family ...
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3answers
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Questions about Aleph-Aleph-Null

Note: I apologize in advance for not using proper notation on some of these values, but this is literally my first post on this site and I do not know how to display these values correctly. I ...
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1answer
68 views

If $\bigcup_{i=1}^\infty A_i$ has cardinality $\kappa$, then some $A_i$ has cardinality $\kappa$?

Is the following claim true? If $\bigcup_{i=1}^\infty A_i$ has cardinality $\kappa$, then some $A_i$ has cardinality $\kappa$. Here $\kappa$ is uncountable. I'm interested in the particular case $\...
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1answer
42 views

A question about the proof about strongly inaccessible cardinal

My textbook Introduction to Set Theory by Hrbacek and Jech presents Theorem 3.13 and its corresponding proof as follows: Since the authors refer to Theorem 2.2, I post it here for reference: ...
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1answer
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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 ...
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1answer
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A question about a proof of Hausdorff's Formula

My textbook Introduction to Set Theory by Hrbacek and Jech presents Hausdorff's Formula: and its corresponding proof: I am unable to deduce 3. from 1. and 2. as stated in the proof. Each ...
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0answers
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Counting equivalence classes. [duplicate]

Let $X$ denote the set of real transcendental numbers. Define the relation $\sim$ on $X$ by $x\sim y $ iff $x-y \in \mathbb{Q}$. Let $Y$ denote the set of equivalence classes generated by $\sim$ ...
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0answers
50 views

Is my understanding of this proof about cardinality correct?

In my textbook Introduction to Set Theory by Hrbacek and Jech, there is a theorem: and its corresponding proof: I would like to ask if my understanding of the proof in case $\color{blue}{\...
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1answer
50 views

Is my understanding of a proof from textbook Introduction to Set Theory by Hrbacek and Jech correct?

3.8 Therorem Let us assume the Generalized Continuum Hypothesis. If $\aleph_\alpha$ is a regular cardinal, then $$\aleph_\alpha^{\aleph_\beta}=\begin{cases} \aleph_\alpha&\text{if }\beta<\alpha\...
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1answer
45 views

Is every (possibly infinite) sum of cardinal numbers defined?

Hrbacek and Jech gives the following definition of cardinal addition: My question is: given an indexed system of cardinals $\left \langle \kappa_{i} |i\in I \right \rangle$ does there exist a system $...
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1answer
80 views

Let $\aleph_\alpha$ be a singular cardinal and $2^{\aleph_\xi}=\aleph_\beta$ for all $\xi<\alpha$. Then $2^{\aleph_\alpha}=\aleph_\beta$

Let $\aleph_\alpha$ be a singular cardinal. Let us assume that the value of $2^{\aleph_\xi}$ is the same for all $\xi<\alpha$, say $2^{\aleph_\xi}=\aleph_\beta$. Then $2^{\aleph_\alpha}=\aleph_\...
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1answer
75 views

A question regarding Brian M. Scott's proof that $\text{cf}(\aleph_{\omega_1})=\omega_1$

$\text{cf}(\aleph_{\omega_1})=\omega_1$ From here, I quote Brian M. Scott's proof: Suppose that $\langle\alpha_n:n\in\omega\rangle$ is an increasing sequence cofinal in $\omega_{\omega_1}$. For ...
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1answer
65 views

Let $(\alpha_\xi\mid\xi<\kappa)$ be a sequence such that $\{\alpha_\xi\mid\xi<\kappa\}=\alpha$. Find an increasing subsequence that has limit $\alpha$

Let $\alpha$ be a limit ordinal which is not a cardinal, and $\kappa=|\alpha|$. Then there exists a bijection from $\kappa$ to $\alpha$, or equivalently, a one-to-one sequence $\langle \alpha_\xi \...
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1answer
81 views

How can we determine $2^\kappa$ for singular $\kappa$ assuming that $2^\lambda=\lambda^+$ whenever $2^{\operatorname{cof}(\lambda)}<\lambda$?

I got an exercise in set theory and can't seem to solve it: If we assume that $2^\lambda = \lambda^+ $ holds for every singular cardinal with $2^{\operatorname{cof}(\lambda)}<\lambda$, then how can ...
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1answer
80 views

If $\alpha$ is a limit ordinal, then $\operatorname{cf}(\alpha)$ is a limit ordinal [duplicate]

In the textbook Introduction to Set Theory by Hrbacek and Jech, Section 9.2, the authors first introduce the definition of increasing sequence of ordinals: Then they introduce cofinality: My ...