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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4
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1answer
40 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 ...
1
vote
1answer
23 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 : ...
1
vote
2answers
28 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 ...
1
vote
1answer
33 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 ...
1
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1answer
40 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
84 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
votes
1answer
56 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 ...
1
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2answers
59 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 ...
1
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2answers
43 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
votes
2answers
130 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
votes
2answers
98 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
votes
1answer
73 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). ...
2
votes
3answers
320 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 ...
1
vote
1answer
42 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 ...
0
votes
3answers
53 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 ...
4
votes
0answers
76 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 ...
1
vote
3answers
76 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
votes
1answer
72 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 ...
3
votes
0answers
58 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 ...
1
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3answers
78 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 ...
2
votes
2answers
126 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 ...
1
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1answer
31 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 ...
0
votes
2answers
59 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
62 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 ...
0
votes
0answers
25 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
vote
1answer
47 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
votes
3answers
262 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
votes
1answer
99 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
votes
1answer
69 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=\{ ...
8
votes
2answers
485 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} ...
1
vote
2answers
67 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 ...
7
votes
1answer
124 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} = ...
4
votes
2answers
118 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 ...
-1
votes
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 $\aleph_1$?
0
votes
1answer
46 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
68 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
vote
1answer
67 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 ...
2
votes
1answer
65 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 ...
0
votes
2answers
94 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
votes
1answer
58 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
36 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
36 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
49 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 ...
5
votes
1answer
156 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
0answers
39 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
85 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
votes
1answer
75 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 ...
2
votes
2answers
133 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 ...
1
vote
1answer
58 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 $C$, $|C|=|D|$ ...
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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 ...