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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dimension of direct products

Suppose $\{V_i\}_{i\in I}$ is a family of $k$ vector spaces. Is it possible to calculate $\dim\oplus_{i\in I} V_i$ and $\dim\prod_{i\in I}V_i$?
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1answer
45 views

Why coefficients of Fourier series are countable, though the initial periodic function is described with an uncountable set of points

Coefficients in the Fourier series for any periodic square-integrable function $f(x)$ form a countable (though infinite) set, i.e., they have cardinality $\aleph_0$. As far as Fourier exponents form a ...
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1answer
33 views

Cardinality: is it true that $|X^\mathcal{inj}| \leq 2^{|X|}$?

Definition. Whenever $X$ is a set, write $X^\mathcal{inj}$ for the collection of all injections $f$ such that: $f$ has codomain $X$ There exists an ordinal $\alpha$ such that $f$ has domain ...
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3answers
295 views

How to prove that the set of all countable ordinals, $\omega_1$, is uncountable / has the cardinality $\aleph_1$?

The only reasoning I've seen given for this is that it's uncountable because it can't include itself an element. I'm a little unconvinced and was looking for a more proper formal proof demonstrating ...
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1answer
31 views

Need some help with this Cardinality/sets question.

I've got this problem about sets, and cardinality. I don't really understand it other than cardinality is the number of elements within each set, I don't understand a lot of the signs used within the ...
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1answer
40 views

How to force p<b?

Two cardinal characteristics (cardinals between $\aleph_1$ and $\mathfrak{c}$ are: $\mathfrak{b}$, the least size of an unbounded family in $\omega^{\omega}$ ordered under eventual domination ...
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2answers
48 views

When is an infinite set larger than another infinite set?

Somewhat of a basic question that I've been pondering about, suppose we have 2 finite sets $A,B$, arbitrary sets with arbitrary elements that we know nothing about, except that they are both finite. ...
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1answer
15 views

Building a function with codomain equal to a given set of reals.

I was discussing with friends the astounding fact that $\mathbb R$ and the set of real continuous functions were equipotent. I asked for a proof that $\mathbb R$ and $\mathbb R ^{\mathbb R}$ are not ...
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3answers
48 views

Does there exist this type of sequence of subsets of $\mathbb{R}$?

Let $A_n$ be a sequence of subsets of $\mathbb{R}$ with the following properties. $A_n$ is unbounded for all $n$ The union of all $A_n$ is $\mathbb{R}$ No two $A_n$ share elements For all $n$, given ...
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1answer
14 views

Would 3 to the n power where n is an element of Z be countably infinite?

I'm just learning about finite, countably infinite, and uncountable sets. My question is, which of the three categories would this fall into: {3^n|n ϵ Z} I first thought that it would uncountable, ...
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2answers
16 views

One-to-one correspondence of a set within a set

I need to find a one-to-one correspondence between each of the following pairs of sets: (a) {x, y, {a, b, c}} and {14, -3, t} (b) 2Z and 17Z For problem a, I have no idea if the inner set counts as ...
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1answer
25 views

Prove cardinal arithmetic (exponentiation)

Suppose $|K|=\kappa, |L|=\lambda, |M|=\mu$ and $L \cap M=\emptyset$. Prove that $$(\kappa^{\lambda})^{\mu}=\kappa^{\lambda \cdot \mu}$$ My attempt: Suppose $F : K^{ L \times M} \rightarrow (K^L)^M$. ...
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1answer
42 views

Dedekind finite set and a special well ordered set

In ZFC, Dedekind finite set and finite set are same things. So I have a set say A(which is equal to N in ZFC) all Dedekind finite set are equivalent to proper subsets of A and A is well ordered set. ...
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6answers
976 views

Proof that aleph null is the smallest transfinite number?

The wikipedia page on the cardinal numbers says that $\aleph_0$, the cardinality of the set of natural numbers, is the smallest transfinite number. It doesn't provide a proof. Similarly, this page ...
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0answers
23 views

Cardinality of a set of injections [duplicate]

Let $A$ be the set of all injections $f: \mathbb{Z}_+ \rightarrow \mathbb{Z}_+$ What can we say about the cardinality of $A$ with respect to the cardinalities of $\mathbb{Z}_+$ and ...
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1answer
18 views

For every ordinal $\alpha$, there is a cardinal number greater then $\alpha$.

I am trying to prove that for every ordinal $\alpha$, there is a cardinal number greater then $\alpha$. Proof: $\alpha$ is a (well ordered) set. Take the set $P(\alpha)$. Assume that we know that for ...
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0answers
22 views

Decomposition of an infinite set into pairwise disjoint subsets which exhaust set does not affect cardinality

Show that if $X$ is an infinite set and $A$ is subset of the power set of $X$ containing only finite pairwise disjoint sets such that the union of all elements of $A$ is $X$, then cardinalities of $X$ ...
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1answer
42 views

Cardinal exponentiation formula

Assume GCH and let $k,m$ be infinite cardinals. I would like to show that $k^m = \max \{ k,2^m \}$. We of course have $k=\beth_a$ and $m=\beth_b$ for ordinals $a,b$. If $a$ is a successor ordinal, ...
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5answers
898 views

cardinality of all real sequences

I was wondering what the cardinality of the set of all real sequences is. A random search through this site says that it is equal to the cardinality of the real numbers. This is very surprising to me, ...
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3answers
723 views

Number of countable subsets of $\mathbb{R}$

More generally, if a set $S$ has cardinality $\mathfrak{m}$, how many of its subsets have cardinality $\mathfrak{n}$? Clearly there are at least $2^\mathfrak{n}$ such subsets. I don't see how many ...
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2answers
54 views

What is aleph null times aleph one?

Could you shed some light on this? I am guessing it is aleph one, since one cannot pair every element of naturals with its subsets.
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1answer
22 views

Prove there's $\left|A-B\right| = \aleph$.

Let A a set such that $\left|A\right| \ge \aleph$. Prove there's a $B\subseteq A$ such that $\left|B\right|\ge \aleph$ and $\left|A-B\right| = \aleph$. Lets assume there's a $B\subseteq A$ such ...
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0answers
86 views

Existence of a Banach space with algebraic dimension $\alpha \neq \aleph_0$

For any given cardinal number $\alpha \neq \aleph_0$, is there a Banach space $X$ with algebraic dimension $\alpha$?
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2answers
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Cardinality of all injective functions from $\mathbb{N}$ to $\mathbb{R}$.

What is the cardianlity of: $$ A = \left\{ f:\mathbb{N}\to\mathbb{R} : \text{f is injective} \right\} $$ Trying to prove it using Cantor–Bernstein–Schroeder theorem, I have the obvious side: $$A ...
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2answers
79 views

Sizes of infinity

I was just thinking about infinity (as you do) and thought the following. "There are infinitely many reals in the interval $x\in[0,1]$ and an 'equal number of reals' $x\in[1,2]$, so there are 'double ...
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1answer
41 views

Show $\left|B^A\right| \cdot \left|B^A\right| = \left|B^A\right|$

Let $A$, an infinite set such that $\left|A\right|\cdot \left|A\right| = \left|A\right|$ and Let $B$, an arbitrary set. Show $\left|B^A\right| \cdot \left|B^A\right| = \left|B^A\right|$ I'd be ...
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2answers
31 views

Show $\left|B-A\right| = \left|B'-A'\right|$

Let $A,A',B,B'$ such that: $A\subseteq B$, $A'\subseteq B'$, $\left|B\right|=\left|B'\right| \gt \aleph_0$, $\left|A\right|=\left|A'\right| = \aleph_0$. Show that $\left|B-A\right| = ...
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1answer
32 views

Is there a cardinal $\kappa$ with uncountable cofinality such that $\kappa<\kappa^{\aleph_0}$?

Is there a cardinal $\kappa$ with uncountable cofinality such that $\kappa<\kappa^{\aleph_0}$? The question is motivated by the observation that $\kappa< \kappa^{{\rm cf}\kappa}$ for any ...
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1answer
22 views

Why is the equality right? (Set-Theory)

Let $A, B$ finite sets, and let $f,g\in A\to B$. Also, Let the equivalence class: $$f \sim g \iff \exists h\in Eq(A,A). f=g\circ h $$ Claim: $$f\sim g \iff \forall b\in B. \left| \left\{ a\in A : ...
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2answers
24 views

What is the cardinality of $\left|C_s\right|$?

Let $$C_s = \left\{ f\in \mathbb{N}/S \to \mathbb{N} : \forall M\in \mathbb{N} / S. f(M)\in M \right\}$$ Where $S$ is an equivalence class. I need to prove $$\left|C_s\right| > \aleph_0 \implies ...
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1answer
30 views

Division of cardinals

Given two distinct infinite cardinals, $\mu<\pi$, Wikipedia states that $\kappa=\pi$ is the only possible solution of the equation $\mu\cdot\kappa=\pi$, so that one could say that $\pi/\mu=\pi$. It ...
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1answer
69 views

Infinite Set has greater or equal cardinality that of N

For any infinite set, we can find a 1-1 function (not necessary onto) from N (set of natural no.) to that set. The proof of this theorem I know using axiom of choice. Can we prove it without using ...
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1answer
30 views

Prove that for any infinite set $A$, $|\mathbb{N}|\le |A|$ [duplicate]

How can you show that for any infinite set $A$, $|\mathbb{N}|\le |A|$? thanks
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0answers
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cardinality of set of all real continuous functions [duplicate]

Could somebody explain to me how to prove that the cardinality of all real continuous functions is $c$ ? The first problem is that I don't know how to show that each real continuous function $f: X ...
2
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1answer
49 views

A question about cardinals with countable cofinality

This question bothers me as I am not sure if we could drop the assumption of uncountable cofinality: Let $\kappa$ be an uncountable cardinal and let $x_\alpha, y_\alpha$ be collections of real ...
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2answers
46 views

Cardinal numbers with countable cofinality

What does this assumption mean: Let $k$ be any cardinal number with uncountable cofinality Which cardinals have countable cofinality? I know the definition of cofinality, but I'd like to see some ...
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2answers
40 views

Find a bijection to show $\left|B\right| = \mathfrak{c}$.

Let $B = \left\{ A \cup \mathbb{N}_\text{even} : A\subseteq \mathbb{N}_\text{odd} \right\}$ I need to show $\left|B\right| = \mathfrak{c}$ by using an equivalence function (bijection) to another set ...
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2answers
113 views

Proving that the cardinality of a set is even

Let $E$ be a set and $f:E\to E$ be a function such that $f\circ f=Id$. Let $A=\{x\in E, f(x)\neq x\}$. Suppose that $A$ is finite. Prove that the cardinality of $A$ is even. My ...
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2answers
89 views

Is there any infinite quantity small enough to be affected by finite changes?

Hilbert's paradox of the Grand Hotel shows us, among other useful things, that the cardinality of any infinite set is a quantity equal to n more than itself for any finite n. I am interested in ...
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1answer
71 views

What cardinals do we actually need?

I recently heard of a great variant of set theory called "Pocket Set Theory" in which the only two cardinalities are that of $\mathbb N$ and that of $\mathbb R$ (and the latter is a proper class). ...
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6answers
4k views

Cardinality of set of real continuous functions

The set of all $\mathbb{R\to R}$ continuous functions is $\mathfrak c$. How to show that? Is there any bijection between $\mathbb R^n$ and the set of continuous functions?
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3answers
286 views

Cardinality of the set of all two-element subsets of $\mathbb{N}$

Consider the set $\mathbb{N}$ of all natural numbers; we can assign each natural number a point on a single axis. Let $A$ be the set of all of these points; $A$ is a countable set (we can assign each ...
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1answer
36 views

Cardinal Arithmetic and a permutation function.

I am working on the following problem and am having difficulties getting started: We define a permutation of $K$ to be any one-to-one function from $K$ onto $K$. We can then define the factorial ...
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2answers
227 views

The preorder of countable order types

Consider the set $\mathcal{O}$ of order types corresponding to all posets of cardinality at most $\aleph_0$. The set $\mathcal{O}$ is a preorder under embeddability of its elements (note that some ...
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1answer
64 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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3answers
52 views

Understanding the proof of: If $|A| = \kappa$, then $|\mathcal{P}(A)|=2^{\kappa}$.

I don't quite understand the part where the author writes that $f$ is a one-to-one correspondence between the power set of $A$ and the set of all mappings from $A$ into $\{0, 1\}$. Isn't the ...
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0answers
73 views

$\aleph_\alpha^{\aleph_1} = \aleph_\alpha^{\aleph_0}\cdot 2^{\aleph_1}$ for all $\omega \le \alpha < \omega_1$

I'd like someone to check my proof that $\aleph_\alpha^{\aleph_1} = \aleph_\alpha^{\aleph_0} \cdot 2^{\aleph_1}$ for all $\omega \le \alpha < \omega_1$. First, let's prove that ...
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3answers
64 views

prove that set of reals numbers and complex numbers are equipotent.

I have to prove that set of reals R and set of complex C are equipotent. " i know that set A and B are equipotent iff there is one to one mapping of A onto B. " please anyone give me answer of ...
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1answer
69 views

Question about cardinals in ZF

In Lévy's basic set theory refers a theorem of Bernstein as exercise problem: Theorem. (Bernstein) Let $\mathfrak{a,b}$ be cardinals. If $\mathfrak{a+a=a+b}$, then $\mathfrak{b\le a}$. I try to ...