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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1answer
76 views

Russell's definition of finite cardinals

whether the thought had been previously adumbrated, perhaps confusedly, i know not, but the name of Bertrand Russell has become associated with the assertion that: the number $2$ is the set of all ...
8
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2answers
91 views

Proof of non-existence of non-principal $\kappa$-complete ultrafilter

Let $\lambda$ be a cardinal. I would like to prove that for all cardinals $\lambda < \kappa \leq 2^\lambda$, there can't be a $\kappa$-complete non-principal ultrafilter on $\kappa$. Here is my ...
1
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1answer
62 views

Cardinality of the set of differentiable functions

Is the cardinality of $$X = \{f: \Bbb R \to \Bbb R \;|\; f \text{ is differentiable everywhere}\}$$ the same as $\Bbb R$? How to prove it?
2
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1answer
104 views

Godel's pairing function and proving c = c*c for aleph cardinals

I have a few questions about Godel's pairing function and proving that c = c * c for aleph cardinals. Mostly, though, I'm concerned that most of the proofs I've seen are erroneous, and this concerns ...
3
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2answers
58 views

Regularity of Limit Cardinals

My intuition is that co-finality is a non-decreasing function on the cardinals. If that's true, it seems to follow that all infinite cardinals are regular. In particular, $\aleph_0$ is clearly regular ...
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2answers
39 views

Prove cardinals equality

Let $x,c$ two cardinals, such that: $1\lt x \le c$ $c^2 = c$ Prove: $x^c = 2^c$ So, from the second statement, we know $c$ is infinite, because it cannot be true for a finite cardinal. I know the ...
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2answers
162 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?
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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
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2answers
65 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
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1answer
33 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
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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
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1answer
74 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
162 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
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2answers
93 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
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1answer
135 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
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0answers
29 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
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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
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4answers
184 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
132 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 ...
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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 ...
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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 ...
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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 ...
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0answers
43 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
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1answer
55 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
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1answer
65 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
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2answers
54 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
58 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
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1answer
226 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
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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
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1answer
57 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
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3answers
299 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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2answers
75 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
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2answers
134 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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3answers
284 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
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1answer
64 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
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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
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4answers
224 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
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3answers
103 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
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1answer
71 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 ...
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1answer
70 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
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1answer
111 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
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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
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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
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1answer
67 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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1answer
302 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 ...
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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
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 ...