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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2answers
150 views

Cardinality of algebraic extensions of an infinite field.

An exercise in Lang's algebra book is: let $k$ an infinite field, and $E$ an algebraic extension of $k$. Then $E$ has the same cardinality as $k$. How can one can prove this?
0
votes
4answers
56 views

Cardinal number of a group

I have the following group: $$ A = \{f \in \Bbb N \to \Bbb N \mid \exists S\subseteq \Bbb N, \left.f\right|_S = \operatorname{Id}_S \land f(\Bbb N \setminus S)\subseteq S\} $$ I need to find $|A|$?
2
votes
2answers
64 views

How to show $|S| = |S \times A|$, where $S$ is infinite and $A$ is finite.

Exactly what the title says. I have a proof where at one point, I use the fact that two copies of $S$ can be put in bijection with $S$, but I don't know how I'd prove that. For countable sets, I can ...
1
vote
1answer
30 views

Cardinality of a linear continuum

A linear continuum is a totally ordered set with more than one element, which is both dense and satisfies the least upper bound property. Is it true that every linear continuum has the same ...
2
votes
3answers
60 views

Countability of a set: only 2 options?

So I know sets can be countable (bijection between set and $\mathbb{N}$, finite) or uncountable. Is there another option or are all sets either or?
1
vote
1answer
71 views

$\aleph(X) < \aleph(\mathcal{P}(\mathcal{P}(\mathcal{P}(X))))$

Prove: $\aleph(X) < \aleph(\mathcal{P}(\mathcal{P}(\mathcal{P}(X))))$ With $W(X)=\{\langle A,R\rangle: A \in \mathcal{P} (X),R \in \mathcal{P}(X \times X)$ and $ R $ wellorders $ A \}$ And ...
3
votes
4answers
73 views

Is $n <\aleph_0, n \in \mathbb{N}$ well defined?

Is $n <\aleph_0, n \in \mathbb{N}$ well defined? That is, can I "compare" a natural number to a transfinite number in a strict sense? My intuition says yes since if $S$ is a finite set then we ...
5
votes
5answers
160 views

$\mathrm{card} ( \mathbb{Q})=\mathrm{card}( \mathbb{Q^c})$: Overcoming Wrong Intuition

This is a widespread intuitive argument, asserting that $\mathrm{card} ( \mathbb{Q})=\mathrm{card}( \mathbb{Q^c})$: Between any two rational numbers there's an irrational one and vice versa. So ...
1
vote
2answers
90 views

Does $a \le b$ imply $a+c\le b+c$ for cardinal numbers?

Let $a, b, c$ be cardinalities. Prove or disprove: If $a \le b$ then $a+c\le b+c$ I realize that $a \le b$ means that there's a bijection between A and B. But I don't really know what ...
2
votes
1answer
131 views

What is the cardinality of $\omega^\omega$?

I have started reading about Baire Space as a prerequisity for the subject of Borel hierarchies. And if I understand correctly, Baire space is $\omega^\omega$ (The space of all functions from $\omega$ ...
0
votes
0answers
28 views

cardinality of finite subset

I have the following question: Prove that a set A has the same cardinality of a subset of a Set B, if and only if exists an injective function A to B. I find it hard to prove it because I can easily ...
0
votes
2answers
65 views

Elementary set theory - are these sets empty? [duplicate]

we are asked to answer if the following statements are true or false, and why: 1) The set ${\{\emptyset\}^{\mathbb N}}$ has exactly $1$ element. 2) The set ${{\emptyset}^{\mathbb N}}$ is empty. 3) ...
3
votes
4answers
182 views

Does cardinality really have something to do with the number of elements in a infinite set?

I've seen some videos and read some texts (non-rigorous ones) that explained the concept of cardinality, and sometimes I see someone asking if there are more numbers between the reals in $[0,1]$ then ...
4
votes
1answer
119 views

Jech's proof of Silver's Theorem on SCH

Jech's textbook proves Silver's Theorem–that if the Singular Cardinals Hypothesis holds for all singular cardinals of countable cofinality, then it holds for all singular cardinals–by breaking up the ...
1
vote
3answers
91 views

Why $\omega+1$ and $\omega^2$ are not cardinal numbers?

I see the following definition of cardinal number in notes: An ordinal $\alpha$ is a cardinal number if $|\beta|<|\alpha|$ for all $\beta\in\alpha$. Why $\omega+1$ and $\omega^2$ are not cardinal ...
1
vote
2answers
72 views

For all infinite cardinals $\kappa, \ (\kappa \times \kappa, <_{cw}) \cong (\kappa, \in).$

I don't understand the proof to the a/m claim. How we know that $\eta < \kappa$ and $(\alpha,\beta)<_{cw} (0, \eta) $ and hence $h: \mu \to \eta \times \eta$ is injective? Appreciate if anyone ...
0
votes
0answers
42 views

Has $\mathbb{R}$ same cardinality with $\mathbb{R}\times\mathbb{R}$? [duplicate]

Has $\mathbb{R}$ same cardinality with $\mathbb{R}\times\mathbb{R}$? Is the above correct? If yes how we can prove it? I think we can prove that $\mathbb{R} \succeq \mathbb{R}\times ...
0
votes
1answer
59 views

A seeming absurdity [duplicate]

I'm having a hard time getting over the following question, which appears in Schimmerling's "A Course on Set Theory." (Problem) Given that $\kappa$ and $\lambda$ are infinite cardinals with ...
2
votes
1answer
63 views

How to show that $\mathcal P(\Bbb N)\sim\mathcal P(\Bbb N)^\Bbb N$?

I am trying to proof now that $\mathcal P(\Bbb N)$ is of the same cardinality as $\mathcal P(\Bbb N)^\Bbb N$ - the set of all functions $f:\Bbb N\to\mathcal P(\Bbb N)$. Currently I've tried to use the ...
0
votes
1answer
54 views

If $\gamma >0,$ then $ \alpha < \beta $ implies that $\gamma \ . \alpha<\gamma \ . \beta$ and that $\alpha \ . \ \gamma \leq \beta \ . \ \gamma.$

Show that if $\gamma >0,$ then $ \alpha < \beta $ implies that $\gamma \ . \alpha<\gamma \ . \beta$ and that $\alpha \ . \ \gamma \leq \beta \ . \ \gamma.$ (All operations are cardinal ...
1
vote
1answer
64 views

Cardinals, bijections, and a general inquiry

I'm trying to solve the following problem from Schimmerling's "A Course on Set Theory." (Problem) Prove that there exists a family $\mathcal G\subseteq\mathcal P(\omega)$ such that $|\mathcal ...
2
votes
2answers
53 views

cardinal of a quotient space

Suppose that we have an equivalence relation $R$ defined on an infinite set $X$ and that all equivalence classes are finite. Is it so that the cardinal of the quotient space of $R$ is that of $X$? If ...
1
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1answer
54 views

Addition of Alephs

Prove: $\aleph_{\alpha} + \aleph_{\alpha} = \aleph_{\alpha} $ The textbook I am using has a long proof done by transfinite induction. I am looking for a direct proof. Can I do this: ...
3
votes
1answer
220 views

Find the cardinality of these sets

Question from my homework im struggling with Find the cardinality of these sets: 1) the set of all sequences of natural numbers 2) the set of all arithmetic series (difference between 2 ...
0
votes
1answer
86 views

Cardinal of $X^2$

The axiom of choice is equivalent to the following statement: if $X$ is an infinite set then the cardinal of $X^2$ is the same as that of $X$. Is there an elementary proof of this statement?
2
votes
1answer
56 views

Elementary set theory, Cantor-Bernstein-Schröder usage, check my proof

I have a question, I was asked to show that $[0,1]$ and $\mathbb R$ are of equal cardinality using the Cantor-Bernstein-Schröder theorem. I would just like some feedback, if I solved it correctly: ...
3
votes
3answers
275 views

cardinal numbers and the power-set

Natural numbers have an operation of incrementation defined on them. For every natural number $n+1$ is a bigger number. Also we can obtain all natural numbers from 0 by way of incrementation. ...
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votes
2answers
74 views

What's the motivation for discerning infinite cardinality? [closed]

Why define cardinality to distinguish between $|\mathbb{Z}|$ and $|\mathbb{R}|$? They are completely different objects. One is countable, the other has the least upper bound property. In my mind it's ...
2
votes
2answers
120 views

Countably infinite set of real numbers with a complement that is infinite but not countably infinite

How can I show that if a set of real numbers is countably infinite, then its complement is infinite but not countably infinite? Thanks a lot in advance!
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votes
3answers
260 views

Using Schröder-Bernstein theorem to show same cardinality [closed]

Use the Schröder-Bernstein theorem to show that $(0,1)\subseteq \Bbb R$ and $[0,1]\subseteq\Bbb R$ have the same cardinality. Firstly I'm not even entirely sure about what the syntax even means. The ...
0
votes
1answer
63 views

Largest Useful Sets [closed]

I'm just asking this out of curiosity, what are the largest sets that are actually meaningful (infinite sets)? I know that there is no highest cardinal number, but there must come a point where we ...
0
votes
0answers
23 views

Proving Hausdorff's formula for $\alpha = 1$ [duplicate]

I have tasked myself with proving Hausdorff's formula for $\alpha = 1$ from the basics. So we want to show that $\aleph_2^{\aleph_0} = \aleph_2\aleph_1^{\aleph_0}$. The progress I have made is: ...
3
votes
4answers
222 views

Hilbert's Hotel and Infinities for Pre-university Students

Hilbert's paradox of the grand hotel is a fun and exciting ground to base a talk on the set theoretic concept of infinity for interested students - even in middle- and high school. However, it does ...
2
votes
3answers
102 views

Is every function from $\aleph_0 \to \aleph_2$ bounded?

If $f$ is a function $f:\aleph_0 \to \aleph_2$, does it mean that the range of f is bounded in $\aleph_2$? Does this hold for all regular cardinals?
3
votes
1answer
70 views

why $card(P(A))=2^a$ if $card(A)=a$ and $a$ is infinite?

On Page 100 of Naive set Theory by Halmos, He asserted that the cardinal number of the power set of a set $A$ is $2^a$ if $card(A)=a$, because "the proof is immediate from the fact that P(A) is ...
0
votes
1answer
68 views

How big is the set of hyper-naturals?

Consider the set $\mathbb N^*$, the set of hypernaturals. How big is this set? Is it the same size as $\mathbb R^*$?
5
votes
1answer
109 views

Aleph arithmetic question

We want to prove that: $$\aleph_2^{\aleph_0} = \aleph_2\aleph_1^{\aleph_0}$$ My idea was to approach this by doing a Schroder-Bernstein style argument and proving this by showing two inequalities, ...
0
votes
1answer
59 views

Constructing a bijection between intervals [closed]

So I am trying to solve questions below Let $A = \{(\alpha_1,\alpha_2,\alpha_3,\ldots): \alpha_i \in \{0,1\}, i \in N\}$, i.e., $A$ is the infinite cartesian product of the set $\{0,1\}$. Show ...
0
votes
1answer
28 views

determine the cardinalities of the set [closed]

So I am trying to figure out the cardinalities of the following sets (either finite, denumerable or uncountable ): the set of all open intervals with rational midpoints the set of all open intervals ...
2
votes
1answer
66 views

Proving $\kappa^{\lambda} = |\{X: X \subseteq \kappa, |X|=\lambda\}|$ [duplicate]

Let $\kappa , \lambda$ be cardinals with $\omega \leq \lambda \leq \kappa.$ Prove $\kappa^{\lambda} = |\{X: X \subseteq \kappa, |X|=\lambda\}|$. i.e could anyone advise me on how to construct a ...
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votes
1answer
298 views

Question about Cantors Diagonal Argument [closed]

Lets be honnest I don't understand cantors diagonal argument. http://en.wikipedia.org/wiki/Cantor%27s_diagonal_argument While I can understand that this proof proves that you cannot make the list of ...
0
votes
2answers
55 views

$[\![n]\!]\times[\![m]\!]\sim[\![nm]\!]$ where $[\![n]\!] = \{1,\ldots,n\}$

We have got: Let $n,m\in \Bbb N$ and denote $[\![n]\!]=\{1,\dots,n\}\subseteq \Bbb N$. Prove that: $$[\![n]\!]×[\![m]\!]\sim[\![nm]\!]$$ So conclude that, for the finite sets $A$ and $B$, ...
1
vote
1answer
40 views

cardinality of $S_{\mathbb{N}}$

When I am proving something, I got a doubt. what is the cardinality of $S_{\mathbb{N}}$, the set of all bijections from $\mathbb{N}$ to $\mathbb{N}$? I hope it is countable, because that will make my ...
1
vote
1answer
48 views

Show A is countable infinity

One more question about set theory: $A\subseteq R$ is an infinite set of positive numbers. Assume there is a value $k \in Z$ such that for any $B \subseteq A$: $\sum_{i=0}^\infty b(i) \le k$ where ...
3
votes
1answer
62 views

Homework - countable infinity

I'm trying to solve 2 problems, but I'm having some issues and would appreciate help. Here are the questions and what I thought could be done: 1) A is the set of all series of numbers, where in an ...
0
votes
2answers
51 views

Show that for a set $A$, $|^A 2|$ = $|^{|A| }2|$.

Question: Show that for a set $A$, $|^A 2|$ = $|^{|A| }2|$. Comments: This is a part of a larger problem that I'm attempting to solve and it seems like the equality is true, but I'm not quite sure ...
6
votes
5answers
310 views

Are there more real numbers than we can actually imagine?

I mean, if we could imagine all the real numbers then we could assign each number a finite sentence (or a finite book). Since the set of the finite books is countable then the set of real numbers ...
3
votes
0answers
65 views

Which Field Would You Use to Represent a Group Larger than $\aleph _1$?

I understand that in representation theory we try to understand a group $G$ by studying the homomorphisms $\rho\ \colon G \to $ GL$(V)$ where $V$ is a vector space over some field. I believe complex ...
1
vote
1answer
59 views

The set of all countably-infinite subsets of an infinite set

Let $A$ be an infinite set and $D(A)$ denote the set of all countably-infinite subsets of $A$ and let $P(A)$ denote ,as usual, the power set of $A$, then (i) does there exist a surjection of $D(A)$ ...
0
votes
1answer
58 views

Is the set of all sums-of-rationals-that-give-one countable?

Some (but not all) sums of rational numbers gives us 1 as a result. For instance: $$\frac12 + \frac12 = 1$$ $$\frac13 + \frac23 = 1$$ $$\frac37 + \frac{3}{14} + \frac{5}{14} = 1$$ Is the set of all ...