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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3
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1answer
33 views

Question about cardinals.

I have heard that $2^{\omega_1} = \omega_1^{\omega_1}$. Is that true? Why is that? I have tried to find a bijection between the set of all subsets of $\omega_1$ and the set of all functions $\alpha: ...
4
votes
2answers
174 views

Ordinals that satisfy $\alpha = \aleph_\alpha$ with cofinality $\kappa$

I am trying to solve the following question: Prove that for every regular cardinal, $\kappa \gt \aleph_0$, there is a exists an $\alpha$ with cofinality $\kappa$ such that $\alpha = ...
2
votes
1answer
44 views

What is the defenition of $\mathcal{c}$ and $\aleph_1$ if we assume ZFC without CH.

I am reading an intro to a chapter of Andreas Blass called "Combinatorial Cardinal Characteristics of the Continuum" and I am getting a bit confused. When I studied "Discrete Mathematics", it was ...
3
votes
1answer
63 views

Prove that for every $ \quad n\in\mathbb{N},\quad \mathbb{R}^{n} = \mathfrak{c}$?

I was thinking about induction like: Base: $$\#\mathbb{R}^{1} = \#\mathbb{R} = \mathfrak{c}$$ And for $n+1$ $$\#\mathbb{R}^{n+1} = \#\mathbb{R}^{n}\mathbb{R} ...
2
votes
1answer
66 views

chain A s.t. $|X|<|A|\leq |P(X)|$ [duplicate]

Can we prove that there exists at least one chain $A$ in P(X), where X is a non-empty set (finite or infinite), s.t. $ |X|<|A|\leq |P(X)|$? If you can't solve it, ideas/possible directions are ...
7
votes
1answer
136 views

What would a world where $\mathsf{CH}$ is false look like?

My question is a little more specific than the title may lead to believe. In the article The set-theoretic multiverse (J.D. Hamkins), the author writes the following: [...] the continuum is ...
2
votes
1answer
139 views

Every club of $\kappa$ in $M[G]$ contains a club in $M$.

I'm trying to solve exercise (H1) of chapter VII on Kunen's Introduction to Independence Proofs and I would like some hint. I would prefer a hint instead of the full solution :) Assume in M that ...
0
votes
2answers
157 views

Cantor's infinities, and the cardinality of reals vs. complex

Cantor devised 1:1 mappings to prove that the set of integers was the same cardinality as positive integers, odd, etc. And he proved that reals are infinitely more dense. As I recall he called the ...
3
votes
3answers
133 views

Cardinality of all sequences of non-negative integers with finite number of non-zero terms. (NBHM 2012)

Consider the set $S$ of all sequences of non-negative integers with finite number of non-zero terms. Is the set $S$ countable or not? What is the cardinality of the set $S$ if it is not countable? ...
1
vote
1answer
67 views

Proving that $|\{B\subseteq S: |B|<\infty \}|=|S|$ [duplicate]

I've some elementary set theory problem that I came across with: Let $S\subseteq\mathbb{R}$ be infinite set, and let $A=\{B\subseteq S: |B|<\infty \}$. I'm interested in showing that cardinality ...
1
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3answers
98 views

countable or not countable

Good evening everyone; Can you tell me whether are they countable or not ? $$ 2^\mathbb{R}\\ 2^\mathbb{Z}\\ 2^\mathbb{N} $$ where $\mathbb{R}$ is the set of real numbers. $\mathbb{Z}$ is set of ...
3
votes
2answers
93 views

What is the meaning of $n\in \aleph$

Using mathematical induction, prove that, for each $n\in \aleph$ $$n<3^n$$ Dear all, what is the meaning of "$n\in \aleph$" . How to substitute it and prove that? please give me one step ...
3
votes
1answer
71 views

$(k^{<\lambda})^{<\lambda}=k^{<\lambda}$ if $\lambda$ is regular

I'm in need of some help... Why does $(k^{<\lambda})^{<\lambda}=k^{<\lambda}$ if $\lambda$ is regular? I can't see why... Any hints?
1
vote
1answer
27 views

Trying to understand an equality between sets

Consider the following equivalence class: $${[\mathbb{N}]_s} = \{ A \in P(\mathbb{Z}) : |A| = |A \cup \mathbb{N}| \wedge |\mathbb{N}| = |A \cup \mathbb{N}|\} $$ So, $A$ must be infinite set with the ...
2
votes
1answer
140 views

Largest infinite cardinal used in a proof

I've heard before that Knuth holds the record for the largest constant used in a mathematical proof. I was wondering what is the largest cardinal ever explicitly considered in set theory. I presume ...
4
votes
1answer
115 views

Bijection between countable ordinals and reals

The set of all countable ordinals is $\omega_1$, which has a cardinality of $\aleph_1$. When accepting the continuum hypothesis, $2^{\aleph_0} = \aleph_1$, so a bijection between countable ordinals ...
1
vote
1answer
63 views

Cardinality of cantor set $K$

Is the Cantor function bijective from $[0,1]$ to Cantor set $K$? As $K$ is uncountable I think cardinality of $K$ must be $\mathfrak c$ as $K$ is a subset of $[0,1]$. But I am surprised whether there ...
2
votes
1answer
82 views

prove that $\Bbb{Z \times ((0,1]\cap Q)}$ and $\Bbb Q$ have the same cardinality

I have to prove that $\Bbb{Z \times ((0,1]\cap Q)}$ and $\Bbb Q$ have the same cardinality. I think I have bijective function($f(\langle a,b\rangle)=a-b$) between thse sets, but I don't know how to ...
2
votes
0answers
24 views

(S is a proper class wrt model M) implies ((|T|=|S| under model N) implies (T is a proper class wrt M)). Why?

If I understand correctly (which is far from guaranteed, so a reply telling me that it is rubbish will be better than no reply at all): Suppose we have two models N and M such that N is a ...
1
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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
votes
2answers
92 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
vote
1answer
63 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?
3
votes
1answer
116 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
votes
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 ...
1
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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 ...
-1
votes
2answers
174 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
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
35 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
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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
vote
2answers
94 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
142 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
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
votes
2answers
69 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
185 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 ...
5
votes
1answer
136 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 ...
2
votes
2answers
81 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 ...
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 ...
0
votes
0answers
45 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
60 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
57 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
66 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
55 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
vote
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
votes
1answer
243 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 ...