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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6
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6answers
968 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 ...
0
votes
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 ...
0
votes
1answer
17 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 ...
0
votes
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 $X$ ...
2
votes
0answers
26 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 ...
1
vote
1answer
36 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, ...
3
votes
5answers
878 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, ...
6
votes
3answers
715 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 ...
-2
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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.
2
votes
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 ...
0
votes
0answers
85 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 $\alpha$?
1
vote
2answers
45 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 ...
1
vote
2answers
77 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 ...
1
vote
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 ...
2
votes
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| = ...
3
votes
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 ...
1
vote
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 : ...
1
vote
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 ...
1
vote
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 ...
2
votes
1answer
67 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 ...
1
vote
1answer
29 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
8
votes
0answers
2k views

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
votes
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 ...
1
vote
2answers
45 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
vote
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 ...
4
votes
2answers
111 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
87 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
69 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). ...
23
votes
6answers
3k 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?
2
votes
3answers
278 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
35 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 ...
14
votes
2answers
226 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 ...
1
vote
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 ...
0
votes
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 ...
4
votes
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 ...
1
vote
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 ...
4
votes
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 ...
1
vote
3answers
75 views

Show that $2^\mathbb{N} $ is equinumerous with $2^\mathbb{N\times N} $.

I need to show that $2^\mathbb{N} $ is equinumerous with $2^\mathbb{N\times N} $. I already found a function from $2^\mathbb{N} $ to $2^\mathbb{N \times N} $, wich just returns a pair containing ...
1
vote
1answer
49 views

Cardinality of set and its power set [duplicate]

Prove that for any set $X$ we have the $|X| < |\mathcal{P}(X)|$ (power set of $X$) How would you prove this using the definitions of bijection, surjection, and injection? Also, does this mean ...
2
votes
0answers
45 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 Silver´s 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
vote
3answers
71 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
114 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
vote
1answer
29 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
58 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 $D = \{(x,y)\mid x^2 + y^2 \leq 1\}$.
1
vote
2answers
60 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 ...
3
votes
2answers
593 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?
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
45 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
248 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
86 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 ...