This tag is for questions about cardinals and related topics such as cardinal arithmetics, clubs, stationary sets, cofinality, and principles such as $\lozenge$. Do not confuse with [large-cardinals] which is a technical concept about strong axioms of infinity.

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1answer
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The function space from $n$ to $m$ and the exponent $m^n$ are equinumerous (proof)

Can someone provide a tip with creating the bijection for the titular problem? Any tip is helpful! Update: $n=\{ 0,\ldots,n-1 \}$, $m=\{ 0,\ldots,m-1 \}$, and $ m^n=\{0,\ldots,m^n-1\} $. In other ...
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2answers
52 views

Prove that the sets $S$ and $D$ have the same cardinality

Prove that the sets $S$ and $D$ have the same cardinality, where $S = \{(x,y)\mid-1\leq x \leq 1\text{ and }-1\leq y\leq 1\}$ and $D = \{(x,y)\mid x^2 + y^2 \leq 1\}$.
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2answers
58 views

Prove $|A| = |B|$

Let $A= \{a_1,a_2,a_3,\ldots\}$. Define $B = A − \{a_{n^2} : n \in\mathbb N\}$. Prove that $|A|=|B|$. I would say that $B = \{a_2,a_3,a_5,a_6,\ldots\}$. Thus $B$ is a infinite subset of $A$ and ...
3
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2answers
535 views

The Continuum Hypothesis & The Axiom of Choice

Does anyone here know of a reference to an analysis on a proposed relationship between The Continuum Hypothesis and The Axiom of Choice?
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0answers
22 views

$a\le b$ iff there exist $\left|A\right|=a, \left|B\right|=b$ and $A\subseteq B$ [duplicate]

$a\le b$ iff there exist $\left|A\right|=a, \left|B\right|=b$ and $A\subseteq B$ My Proof: $(\Leftarrow)$ $A\subseteq B$ implies immediately that $\left|A\right|\le\left|B\right|$. Hence, $a\le ...
1
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1answer
41 views

Proving $X\sim Y$

Let $f:A\rightarrow B$, a bijection. Suppose $X\subseteq A$ and $Y\subseteq B$ are two sets such that $f(X)\subseteq Y$ and $f^{-1}(Y)\subseteq X$. Show that $X\sim Y$ and $f/X$ is the bijection ...
2
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3answers
221 views

Upper bound on cardinality of a field

Is there an upper bound on the cardinality of a field? The "biggest" fields I know are the field of real numbers, or the field of complex numbers. Is there a field with cardinality greater than ...
1
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1answer
78 views

difference of infinities [closed]

So I haven't studied this subject In a while, and when i get home, I'm going to consult my book on the Theory of computation to reinforce my understanding, but here is my question. So I understand ...
3
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1answer
74 views

$\aleph_\omega$ and large ordinals?

Assuming the GCH, each time you take the powerset of an aleph number, the subscript increases by one. So there is $\aleph_0, \aleph_1, \aleph_2\ldots \aleph_\omega\ldots \aleph_{\epsilon_0}\ldots$ and ...
0
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1answer
60 views

How to prove there is no surjection [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 : X 2^X is a surjection and consider the set $M=\{ ...
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2answers
455 views

Does taking the power set give you the “next biggest cardinal”

I know that if you take the power set of a set, it has a higher cardinality. Therefore there are an infinite number of them as $P^{n}(\mathbb{N})$(the nth power set of the naturals) Let's say $$C_{n} ...
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3answers
89 views

What is the cardinality of free product $\mathbb{Z} * \mathbb{Z}$? [closed]

I want to know cardinality of $\mathbb{Z} * \mathbb{Z}$. Is it countable? or uncountable?
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1answer
101 views

Constructing a Banach space of cardinality $\beth_{\omega+1}$

This is related to yesterday's question Constructing a vector space of dimension $\beth_\omega$; it's the next exercise (I.13.35 (a)) in Kunen's Set Theory. Let $B_0 = \ell^1$ and let $B_{n+1} = ...
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2answers
57 views

An injection from $\mathbb{N}$ to $\mathbb{N}^n$.

I'm currently attempting to prove $\mathbb{N}^n \sim \mathbb{N}$ via Cantor-Schroeder-Berstein (because I found no other way). In my work so far I've managed to find an injective function $f$ from ...
3
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2answers
98 views

Constructing a vector space of dimension $\beth_\omega$

I'm trying to solve Exercise I.13.34 of Kunen's Set Theory, which goes as follows (paraphrased): Let $F$ be a field with $|F| < \beth_\omega$, and $W_0$ a vector space over $F$ with $\aleph_0 ...
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1answer
428 views

How can one rigorously determine the cardinality of an infinite dimensional vector space?

Suppose $V$ is a vector space over a scalar field $F$. If $\dim(V)=n$, then $|V|=|F|^n$. How can I rigorously determine the cardinality of $V$ when $V$ is infinite dimensional? My thought was that if ...
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1answer
38 views

Functions with cardinals as domain

How do we actually define a function with cardinals as domain? For example, take domain of the function as $\aleph_1$. Do we define it for all cardinal numbers strictly less than $\aleph_1$?
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1answer
28 views

Cardinal Arithmetic, Regular Cardinals, and Exponentiation

I cannot solve a simple cardinal exponentiation/regularity exercise. Let $\kappa$ be a regular cardinal. Why is the cardinal $\kappa^{\lt \kappa}=\sum_{\alpha<\kappa}\kappa^\alpha$ regular as ...
0
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1answer
207 views

Finding a limit using arithmetic over cardinals

What is the value of: $$\lim_{n \to \infty} \frac{n}{2^n} (n \in \mathbb{N})$$ It seems to me that I can use L'Hopital's rule, but does that rule take into account types of infinity? More precisely, ...
2
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1answer
58 views

Proving there's a set with the cardinality $\mathfrak c$ on the $x$ axis of points that do not belong to the set of disks

Prove/disprove: On the $x$ axis there's a set with the cardinality $\mathfrak c$ of points that do not belong to any disk of a set $O$ of disjoint disks of positive radius $\{(x,y)\in \mathbb ...
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0answers
63 views

Can I assign a distinct homomorphism to every function?

I consider an algebraic structure and particular structure preserving morphisms (think groups and group homomorphism). I wonder if I can assign a distinct such morphisms to every function between any ...
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0answers
26 views

Does there exist a cardinal $\kappa$ such that $\aleph_{\kappa} = \kappa$? [duplicate]

Does there exist a cardinal $\kappa$ such that $\aleph_{\kappa} = \kappa$ ? Moreover is there one that is regular? Thanks in advance!
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1answer
35 views

A question about cardinal numbers in ZF set theory.

It is well known that cardinal numbers and the relations between them can be defined in ZF set theory (using the notion of "rank"), without the need of additional axioms. Can the following statement ...
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0answers
22 views

Cardinality of Reals vs that of the power set of the natural numbers. [duplicate]

How does the cardinality of the reals (c) compare to the cardinality of the Power Set of the natural numbers?
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9answers
4k views

Is the sum of all natural numbers countable?

I do not even know if the question makes sense. The point is rather simply. If I have the sum of all natural numbers, $$\sum_{n\in \mathbb{N}}n$$ this is clearly "equal to infinity". But since ...
0
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2answers
78 views

How can you compare the number of real numbers in the interval [0,1] and [0,10]?

There are infinite number of real numbers between 0 and 1,i.e in the interval [0,1]. So definitely there should be more numbers in the interval [0,10] because it includes the numbers in the first case ...
5
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1answer
182 views

Why continuum function isn't strictly increasing?

Is there any example that for cardinal numbers $\kappa < \lambda$, we have $2^\kappa = 2^\lambda$? My guess is that it only depends on whether GCH holds. Is it true?
6
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1answer
92 views

Does any uncountable set contain two disjoint uncountable sets?

Is it true that for any uncountable $S$, there exits two uncountable subsets $S_1,S_2 \subseteq S$ with $S_1 \cap S_2 = \emptyset$? I can find no counter example, but no proof either. I am aware of ...
2
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1answer
49 views

$l_{\infty}$ is the quotient of $l_{1}(\aleph_{1})$?

I wonder if $l_{\infty}$ is the quotient of $l_{1}(\aleph_{1})$. If so, how to prove it?
2
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1answer
30 views

Let $\alpha, \beta, \gamma$ be cardinals, $\beta \leq \gamma$, prove $\alpha ^{\beta}\le \alpha ^{\gamma}$

Let $|A|=\alpha, |B|=\beta, |C|= \gamma$ be cardinals and $\beta \leq \gamma$. Prove $\alpha ^{\beta}\le \alpha ^{\gamma}$. So from the given we know that there's an injection $f:B\to C$ and some ...
2
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2answers
32 views

$|A|=\mathcal c \ \ |B|=\aleph_0 \ \ A\cap B=\emptyset$ prove that $ |A\cup B|=\mathcal c$

Let $|A|=\mathcal c, \ |B|=\aleph_0, \ A\cap B=\emptyset,$ Prove that $ |A\cup B|=\mathcal c$ So $|A\cup B|=|A|+|B|$ but this just leads to cardinal arithmetic which I don't think is the right ...
2
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1answer
45 views

Cardinal Arithmetic 2: $\beth_{\omega_1}$

This is question is simple to write but (I think) hard to solve. Does the following equality hold? $$\bigcup_{{\beta}<\beth_{\omega_1}}\beth_{\omega+1}(|\beta+\omega|) =\beth_{\omega_1} $$ Where ...
18
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2answers
283 views

Covering $\mathbb R^2$ with function graphs

Suppose we have a countable family of function graphs (each function is $\mathbb R\to\mathbb R$, not necessary continuous). Obviously, they cannot cover the whole plane $\mathbb R^2$, because they ...
2
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0answers
32 views

What is exactly meant by a countable collection in the defintion of a sigma-algebra

This a just a small question about the definition of a sigma-algebra and what is eaxactly meant by countable? Would be grateful for any clarification on this. Most texts define a sigma-algebra ...
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1answer
71 views

Partitioning an infinite cardinal number

Can we partition an infinite cardinal greater than aleph null, into countable number of cofinal subsets? Can we have restriction on the cardinality of cofinal subsets? For example, suppose $\alpha$ ...
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1answer
58 views

Cardinal Arithmetic: $\beth_\omega$

Is the following equality independent of ZFC: $$\bigcup_{\aleph_{\beta}<\beth_{\omega}}2^{\aleph_{\beta}}=\beth_\omega $$ Consistent: Now that the equality is consistent with ZFC since it holds ...
3
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2answers
29 views

cardinality of infinite sets with cartesian product

claim: $A,B,C,D$ are infinite if $|A\times B|=|C\times D|$ then $|A|=|C|$, $|B|=|D|$ , prove or give a counter example. So imo, the claim is false, using $A=D=\mathbb{R}$ , $B=C=\mathbb{N}$ , is it ...
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1answer
57 views

Is it true , if $|A|=|B|$ and $|C|=|D|$, then $|A \times C| = |B \times D|$?

Check my proof, please. Divide into subsets $A \times C$ and $B \times D$ so that , all pairs with the same element belong to the same subset. Each such subset $|A \times C|$ bijective $C$, $|C|=|D|$ ...
3
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1answer
69 views

In $ \mathsf{ZFC} $, is it true that $ \text{cf}(\kappa) < \text{cf}(2^{\kappa}) $ for all cardinals $ \kappa $?

Question. In $ \mathsf{ZFC} $, is it true that $ \text{cf}(\kappa) < \text{cf}(2^{\kappa}) $ for all cardinals $ \kappa $? I am particularly interested in the case when $ \kappa = \mathfrak{c} ...
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1answer
44 views

What is the cardinality of the set of all higher order functions mapping real functions to real functions?

What is the cardinality of the set of all higher order functions mapping real functions to real functions? To be specific, this set includes all higher order functions with the type signature: ...
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1answer
1k 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)?
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1answer
40 views

Is the cardinality of the continuum weakly Mahlo?

Is $2^{\aleph_0}$ a weakly Mahlo cardinal? Can it be? That is, are there conditions (such as the negation of the continuum hypothesis or something) under which it is, and other conditions under which ...
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0answers
22 views

Collection of sets with a given cardinality $\kappa$ is not set [duplicate]

Show that collection of all sets with cardinality $\kappa\neq0$, is not set. I'll state my approach and I need to see whether this idea is precise/precisable or not : First let $K$ be the set ...
4
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1answer
37 views

Extension of ZFC models preserves cardinals

Let $M \subseteq N$ be countable transitive ZFC set models. Assume that this extension preserves cardinals, i.e. if $\alpha$ is an ordinal number (this notion is absolute) such that $(\alpha \text{ is ...
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1answer
65 views

Partial order on cardinalities without the axiom of choice

Cardinality can still be defined without choice, e.g. as equivalence class of equipotent sets, see Defining cardinality in the absence of choice. Injections define partial order on cardinalities by ...
0
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1answer
30 views

Find the number of vertices in the graph

Let $n\ge 1$ and $V_n = (\left\{ 1,2,...n \right\}\rightarrow\left\{ 0,1,2 \right\})$. Let us define $G_n = \left<V_n, E_n \right>$. $f,g$, are two vertices. They are connected iff: $$\left|\{ i ...
3
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2answers
36 views

Why this function defines a bijection?

Show that if $B\subset A$ and there is an injective function $f:A\to B$, then $\operatorname{card}(A)=\operatorname{card}(B)$. This exercise suggest a way to solve the problem: define $A_1=A, ...
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1answer
21 views

Cardinality of a set with a recurrence relation.

Let $A = \left\{ f\in \mathbb{N}\rightarrow \mathbb{C} \mid \forall n\in \mathbb{N}. f(n+3) + 3f(n+1) = f(n+2)+f(n) \right\}$ What is $\left|A\right|$? Well, I tried to treat $f$ as a recurrence ...
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1answer
31 views

How to show $A_1 \approx A_2 \iff \mathrm{card}(A_1)=\mathrm{card}(A_2)$

For any set $A_1$ and $A_2$, let us define the relation $\approx$ if there exists a bijection between $A_1$ and $A_2$. Then I want to show that $A_1 \approx A_2 \iff ...
2
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1answer
31 views

Show that $\left|X\right|=\left|Y\right|$

Let $A, B$, two sets. $X$ is the set of all relations from $A$ to $B$, and $Y$ is the set of all functions from $A$ to $P(B)$ (power-set of $B$). Prove that $\left|X\right|=\left|Y\right|$. My ...