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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Comparing cardinalities

Why these two sets are equinumerous? $$[0,1]^\Bbb N\text{ and }\Bbb Q^\Bbb N$$ Here is my reason: The set of rational numbers $\Bbb Q$ is countably infinite. However, $[0, 1]$ is not countable and ...
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Cardinal Arithmetic Example Wikipedia

Hello I am studying cardinal arithmetic, and found out that: $$\mathfrak{c}^{\aleph_0} = (2^{\aleph_0})^{\aleph_0} = 2^{\aleph_0 \aleph_0} = 2^{\aleph_0} = \mathfrak{c} $$ However I found this ...
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Prove that the set is countable [duplicate]

Question: We call a real number $x$ $algebraic$ if $x$ is the root of a polynomial equation $c_{0}+c_{1}x+...+ c_{n}x^{n}$ where all $c_{i}$'s are integers. For example $\pm \sqrt{3}$ are algebraic ...
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Is it possible to define a map from $\aleph_0$?

In an assignment question I am asked to show that the cardinality of the set of all functions from $\mathbb{N}$ to $\mathbb{N}$ is equal to $2^{\aleph_0}$. To proceed with my proof I am trying to ...
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61 views

Show that $\aleph_{\alpha} +\aleph_{\alpha} = \aleph_{\alpha}$ and $n \cdot \aleph_{\alpha} = \aleph_{\alpha}$

a) Give a direct proof of $\aleph_{\alpha} +\aleph_{\alpha} = \aleph_{\alpha}$ by expressing $\omega_{\alpha}$ as a disjoint union of two sets of cardinality $\aleph_{\alpha}$. b) Give a direct proof ...
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Logarithms of Cardinals [duplicate]

Given any infinite cardinal $\lambda\neq\omega$, is it the case that there's a cardinal $\kappa$ such that $2^{\kappa}=\lambda$? Does this depend on whether the Continuum Hypothesis is true? Clearly, ...
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Does $\sf GCH$ imply that every uncountable cardinal is of the form $2^\kappa$?

I think that this is a popular fallacy that GCH implies that every uncountable cardinal is of the form $2^\kappa$ for some $\kappa$. I think it does imply that for successor cardinals only. It cannot ...
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If $2^{\aleph_0}$ is weakly inaccessible, can every cardinal $\kappa$ in the interval $[\aleph_0,2^{\aleph_0})$ satisfy $2^\kappa = 2^{\aleph_0}$?

Question. Is the following consistent with ZFC? $2^{\aleph_0}$ is weakly inaccessible Every cardinal $\kappa$ in the interval $[\aleph_0,2^{\aleph_0})$ satisfies $2^\kappa = ...
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71 views

$0^\sharp$ and the regularity of $\aleph_\omega$

I'm sure I'm missing something trivial, and the most likely of it is that I'm simply wrong on my understanding of the constructible universe $L$, or maybe one of the Wikipedia entries I'm about to ...
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3answers
152 views

An easy to understand definition of $\omega_1$?

I have two things I'm not sure in 100% about them. The first, is $\omega_1$. I have a little "feeling" of it, but if I'll be asked to define it - I don't know where to begin from. Perhaps it is ...
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Can every cardinal number between $\kappa^+$ and $2^\kappa$ be realized in this way?

(Assume ZFC for the entire question. By a tree, I mean a tree in the sense of set theory. I write $h(T)$ for the height of a tree $T$, and $h(f)$ for the height of an element $f \in T$.) Definition ...
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1answer
36 views

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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42 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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Is it true that $\dim(X) \leq \dim(X^{\ast})$ for every infinite dimentional banach space $X$?

So given an arbitrary infinite dimensional Banach space $X$ can we deduce that it's dimension $\dim(X)$ (the cardinality of one of it's Hamel bases) is less or equal of the dimension $\dim(X^{\ast})$ ...
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1answer
85 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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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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60 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
16 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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56 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
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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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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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52 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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47 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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24 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
22 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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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
67 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
58 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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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
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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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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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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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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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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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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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1answer
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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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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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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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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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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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87 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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62 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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68 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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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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133 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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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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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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327 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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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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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 ...