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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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
49 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
89 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 ...
2
votes
1answer
426 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 ...
-1
votes
1answer
35 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 $\aleph_1$?
0
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1answer
21 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
votes
1answer
175 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
56 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
61 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 ...
-1
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0answers
24 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
27 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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votes
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
votes
2answers
76 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
votes
1answer
180 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
votes
1answer
91 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
votes
1answer
48 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
votes
1answer
29 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
votes
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
votes
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
votes
2answers
279 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
votes
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 ...
1
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1answer
69 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 $\alpha$ ...
5
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1answer
56 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
518 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?
3
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2answers
28 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
54 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
votes
1answer
68 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} ...
0
votes
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: ...
15
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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)?
2
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1answer
39 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
votes
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
62 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
votes
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 ...
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2answers
35 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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vote
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 ...
0
votes
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
votes
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 ...
2
votes
1answer
60 views

How far is it possible to develop cardinals without ordinals?

I'm wondering which of the usual facts about cardinals in ZFC can be established without using ordinal arithmetic at all. After all the definitions of a cardinal (as a class of equivalence), and also ...
1
vote
1answer
44 views

show that for an infinite cardinal $k$, $k + k = k$

Show that for an infinite cardinal $k$, $k + k = k$ So far I have that $k + k = 2k$ Is it possible to somehow show that $2k = k$? I've been trying to understand some cardinal arithmetic, and I ...
2
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1answer
62 views

Behaviour of sum of $2^\kappa$ for all $\kappa<\lambda$ when $\lambda$ is singular [closed]

What can we say about the conditions under which $\sum\limits_{\kappa<\lambda}2^\kappa \leq \lambda$ holds when $\lambda$ is singular?
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2answers
52 views
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3answers
200 views

Is there a largest large cardinal?

In ZFC, a cardinal is an isomorphism class of sets. However ZFC doesn't explicitly have classes; NBG, which is a conservative extension of ZFC does. There is no largest cardinal by Cantors Theorem ...
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3answers
41 views

For all finite sets $A$ and $B$, $\{ f:B\to A \}$ is finite and its (finite) cardinality is $|A|^{|B|}$.

In Halmos' Naive Set Theory (towards the end of the "Arithmetic" chapter) he mentions the titular claim: For all finite sets $A$ and $B$, $\{ f:B\to A \}$ is finite and its (finite) cardinal is ...
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0answers
87 views

Why are large cardinal axioms actually axioms?

A cardinal is an isomorphism class in ZFC, or a representative of one. I'm not asking what the significant uses of large cardinals are, or why we would want to find them or construct them; I'm ...
3
votes
2answers
39 views

Question about partitions in intervals of the real numbers.

I have to prove the following: Let $ \mathcal{D} $ be a partition of $ \mathbb{R} $ in intervals of any kind, except intervals containing a single element. Prove $ \mathcal{D} $ is countable. ...
16
votes
1answer
688 views

Medial Limit of Mokobodzki (case of Banach Limit)

A classical Banach limit is very usefull concept for me, but there is a problem with the integration and even with the measurability, this means for a sequence $(f_n)_{n\in \mathbb{N}}$ of measurable ...
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0answers
10 views

A question about countability. [duplicate]

I would like to know if $\mathbb{N}$ has the same cardinality as the set $\mathcal{P}_{0}(\mathbb{N}) = \{ A \subset \mathbb{N}: A \text{ is a finite set } \}.$ My strategy was to prove that $\left| ...
3
votes
1answer
84 views

Proving König's lemma (technical problems)

the aim of my exercise is to give a proof of the König's lemma. So, let $\kappa, \lambda$, be cardinals such that $cf(\kappa)\leq \lambda$. My professor's suggested us to prove that there exists a ...