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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66 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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2answers
78 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.
2
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1answer
25 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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2answers
92 views

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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1answer
43 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
91 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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2answers
34 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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0answers
96 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 $\alpha$?
4
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1answer
44 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
25 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
30 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
38 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
41 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
2
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1answer
92 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 ...
2
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1answer
67 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
80 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
45 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 ...
4
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2answers
146 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 ...
3
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2answers
109 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 ...
2
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1answer
80 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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3answers
369 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
47 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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3answers
54 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
81 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
84 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 ...
4
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1answer
74 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 ...
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0answers
65 views

A problem with an assumption in a previous lemma for the proof of Silver's Theorem on SCH in Jech's “Set Theory”

In the Jech's textbook proof of Silvers Theorem, specifically in the Lemma 8.15, there are an assumption in the beginning of the proof. First, the Lemma 8.15 says that, under the assumption that ...
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3answers
89 views

Cantor diagonalization and fundamental theorem

Can the Cantor diagonal argument be use to check countability of natural numbers? I know how it sounds, but anyway. According to the fundamental theorem of arithmetic, any natural number can be ...
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2answers
160 views

Is the Axiom of Choice necessary to prove $\mathbb R \approx \mathcal P(\omega)$?

I am self-studying Horst Herrlich, Axiom of Choice (Lecture Notes in Mathematics, Vol. 1876), and I'm getting quite confused about cardinal arithmetic without AC. Here (Which sets are well-orderable ...
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1answer
37 views

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
61 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
64 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 ...
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0answers
26 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 ...
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1answer
48 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
284 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 ...
3
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1answer
122 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 ...
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1answer
76 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
495 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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2answers
75 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 ...
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1answer
142 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
133 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
40 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
50 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 ...
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0answers
69 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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1answer
75 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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1answer
71 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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2answers
100 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 ...
3
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1answer
59 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?
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1answer
42 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 ...
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2answers
40 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 ...