Questions tagged [cardinals]
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.
3,622
questions
2
votes
1
answer
308
views
Dependence of the equation $a+1=a$ for infinite cardinals $a$ on the axiom of choice
let $A$ be a set such that for all $n \in $ N $ A ≉ N_n$
where $N_n = \{ 0 ,1 ,2 ...... n-1\} $
and $a$ be the Cardinality of $A$ meaning ($|A| = a$)
is it possible to prove that $a+1=a$ without ...
- 81
2
votes
1
answer
75
views
The cardinality of a convergent series [duplicate]
$A\subseteq \mathbb{R}^{+}$ is a set of positive real numbers ($0\notin \mathbb{R}^{+}$), for which there exists a positive real number $x$, such that for every finite subset $S\subseteq A$, the sum ...
- 83
2
votes
1
answer
44
views
Singular cardinals and $\kappa$-Lindelöf spaces
Say a space is $\kappa$-Lindelöf provided that for every open cover of the space, there exists a subcover of cardinality $<\kappa$. So $\aleph_0$-Lindelöf is compact, and $\aleph_1$-Lindelöf is ...
- 6,384
0
votes
2
answers
57
views
Are there any theorems that use the uncountability of the reals in their proof?
Can we use the uncountability of the reals as a tool to prove any theorem? Can we use this to calculate anything? Suppose I was trying to convince a pragmatist that uncountability is useful.
- 470
0
votes
0
answers
31
views
Prove $C ∼ P(P(\mathbb{N}))$ when $C$ is defined as the set of all $S$ s.t $(z − m, z + m) ∩ S = ∅ $ for every $z∈\mathbb{Z}$, $m∈\mathbb{R}$
First, I know that there is a very similar question - The cardinality of all sets $A$ such that $\forall \ z\in \mathbb{Z} \ , (z-k,z+k)\ \cap A=\emptyset : 0<k<0.5$ , but here I want to the ...
1
vote
1
answer
117
views
The cardinality of all sets $A$ such that $\forall \ z\in \mathbb{Z} \ , (z-k,z+k)\ \cap A=\emptyset : 0<k<0.5$
The question asks the following:
Given $k\in \mathbb{R}$ such that $0<k<0.5$ a set $A$ is "k-integer-avoidant" if $ \ \forall z\in\mathbb{Z}$ , $\left ( z-k,z+k \right ) \cap A=\...
- 83
1
vote
3
answers
92
views
$B\subseteq P(\mathbb{N}) : |B| = \aleph \ ?$
The question asked if there exist a set $B\subseteq P(\mathbb{N})$ with cardinality of $\aleph$ such that for all $A_1 ,A_2 \in B$ , if $A_1\neq A_2$ then $A_1 \cap A_2 = \phi$ .
I have looked at the ...
- 83
2
votes
1
answer
47
views
Saturation of infinite complete Boolean algebra is a regular cardinal
Similar question existed here. However there are still many gaps for stupid persons like me.
A Boolean algebra $B$ is called $\kappa$-saturated if there is no antichain with supremum $1$ (also called ...
- 147
0
votes
0
answers
59
views
Finite cardinals raised to the power of an infinite cardinal
I am trying to prove the fact that if $a$ and $b$ are finite cardinals, and $c$ is an infinite cardinal, then $a^c = b^c$. I am able to prove this fact by using $d \cdot d = d$ for all infinite ...
0
votes
0
answers
46
views
Prove that an infinite well ordered set X has equal cardinality to the set X∪{a}, where 'a' does not belong to X.
Found this question in a book of analysis as a corollary. Before the question is introduced (as an exercise), the book introduced Theorem of Recursion on Wosets and Comparability Theorem. For ...
- 1
1
vote
0
answers
42
views
What is the cardinality of the Diffeomorphism group $\text{Diff}(\mathbb{R})$ over the reals?
As a small set-up, we have $\aleph_0$ as the cardinality of $\mathbb{N}$, and then the cardinality of the reals $\mathbb{R}$ is $\beth_1 = 2^{\aleph_0}$. In particular, it turns out that this is also ...
- 11
0
votes
2
answers
63
views
Prove cardinality of R2 without creating a map to R1
I think we can prove $|\mathbb{R}^2|=\aleph_1$, by creating a bijection between $\mathbb{R}$ and $\mathbb{R}^2$. But this map is difficult to construct. Is there any easier way to show $|\mathbb{R}^2|=...
- 470
5
votes
2
answers
123
views
Why does countability misbehave in intuitionistic logic
On page 3 of this paper https://arxiv.org/pdf/2404.01256.pdf
I spotted the claim:
Definitions of countability in terms of injection into ℕ misbehave intuitionistically, because a subset of a ...
- 4,223
0
votes
1
answer
87
views
Is there a way to construct larger cardinals without choice axiom?
From Cantor's Theorem, we know that $|\mathcal{P}(X)| > |X|$. So, we can define inductively a set with cardinality $\aleph_n, \forall n \in \mathbb{N}$. Let $\lbrace A_i\rbrace_{i \in \mathbb{N}}$ ...
- 31
0
votes
0
answers
53
views
Is this Proof Involving Cardinal Arithmetic Correct?
Question
Show that for $n>0$,
$n.2^{2^{\aleph _0}}=\aleph _0.2^{2^{\aleph _0}}= 2^{\aleph _0}.2^{2^{\aleph _0}}= 2^{2^{\aleph _0}}.2^{2^{\aleph _0}}=(2^{2^{\aleph _0}})^n=(2^{2^{\aleph _0}})^{\...
- 451