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
-5
votes
1
answer
40
views
Cantor Diagonalization argument disproved? [duplicate]
It seems I found a loophole in the Cantor's diagonalization proof - that states that there are more real numbers between 0 and 1 than there are natural numbers (i.e., the set of all positive integers)....
- 91
1
vote
1
answer
96
views
Irrational numbers Cardinality.
The real numbers, $\mathbb{R}$, are uncountable and the rational numbers, $\mathbb{Q}$, are countable. We can write $\mathbb{R} = \mathbb{Q} \cup (\mathbb{R} \setminus \mathbb{Q})$. Since $\mathbb{Q}$ ...
-1
votes
3
answers
87
views
Find the cardinality of $A \cup B$ [closed]
Let the sets $A=\{\frac{1}{1\times 2} , \frac{1}{2\times 3}, ... , \frac{1}{2021\times 2022}\}, B=\{ \frac{1}{2\times 4}, \frac{1}{3\times 5}, ..., \frac{1}{2020\times 2022}\}$. Find the cardinality ...
0
votes
1
answer
47
views
For $k$-algebras $B_1, \dots, B_n$, $\# \operatorname{Hom}_k( \prod_{i=1}^nB_i, \Omega) = \Sigma_{i=1}^n \# \operatorname{Hom}_k(B_i, \Omega)$?
Let $k$ be a field with $ k \subseteq \Omega$ a algebraically closed field. Let $B_1 , \dots, B_n$ be ( possibly finite local ) $k$-algebras. Then next equality of cardinals holds
$$ \# \operatorname{...
- 2,656
0
votes
1
answer
50
views
Proving the Equality of Infinite Cardinal Products and Powers
Theorem: Let $\Xi$ be an infinite set, $\{\kappa_i\}_{i \in \Xi}$ be a family of cardinal numbers, and $\lambda$ be a cardinal number. Then:
$\prod_{i \in \Xi} \kappa_i^{\lambda} = \left(\prod_{i \in \...
- 85
5
votes
1
answer
113
views
Absoluteness of inaccessible cardinals
I'm studying large cardinals and I'm hoping to fully understand the proof that says ZFC is not able to prove the existence of inaccessibles (given ZFC is consistent, of course).
I've already fully ...
- 3,630
1
vote
1
answer
56
views
Brun's theorem and the twin prime conjecture
According to the following extract taken from Wikipedia, almost all prime numbers are isolated given Brun's theorem. Doesn't that mean that there is only a finite number of twin prime numbers (they ...
- 37
1
vote
1
answer
56
views
Cardinal of a set of integers with symmetry relations
Context
In computational chemistry, there are two-electron integrals noted $(ij|kl)$ for integers (i,j,k,l) between 1 and K. The explicit expression of $(ij|kl)=\int dx_1dx_2 \chi_i(x_1)\chi_j(x_1)\...
- 13
2
votes
0
answers
43
views
Lemma 0 in Hajnal's Paper "Embedding Finite Graphs into Graphs Colored with Infinitely Many Colors"
I am looking for a proof of the following lemma.
Let $E_0$ be the set of edges of an undirected graph with no loops with vertex set a cardinal $\kappa$. Let $E_1$ be the family of two-element subsets ...
- 417
2
votes
1
answer
70
views
Assuming GCH holds, calculate $\aleph_{\omega_1}^{\aleph_0}$
I'm working through the book Discovering Modern Set Theory by Just and Weese, and this question comes right after this theorem:
Here's what I've worked out so far:
I believe the cofinality of $\...
- 849
4
votes
2
answers
167
views
Does the Cardinal Supremum Commute with the Cardinal Power?
Let $\kappa,\tau$ be two cardinals and $\{\varkappa_\alpha\}_{\alpha<\kappa}$ an indexed set of cardinals. Is it true that
$$\sup_{\alpha<\kappa}(\varkappa_\alpha^\tau)=\left(\sup_{\alpha<\...
- 310
0
votes
1
answer
84
views
For infinite cardinals $\kappa$, we have $\kappa \otimes \kappa = \kappa$.
I am aware that other questions are quite similar to this; however, it seems like the other questions regarding the same statement are looking at proofs that seem somewhat different from the one I am ...
- 1,296
0
votes
2
answers
58
views
Cardinality of a set of disjoint open sub intervals of $( 0 ,1)$
Let $A$ be any collection of disjoint open subintervals of $(0 ,1)$ . Then what is maximum cardinality of $A$ ?
I know one easy way to prove its countable is that every open interval has rational ...
1
vote
0
answers
29
views
A sequence of continuum hypotheses
The continuum hypothesis asserts that $\aleph_{1}=\beth_{1}$. Both it and its negation can be consistent with ZFC, if ZFC is consistent itself.
The generalised continuum hypothesis asserts that $\...
- 290
1
vote
1
answer
83
views
Can cardinality $\kappa$ exist where $\forall n\in\mathbb{N} \beth_n<\kappa$,$\kappa<|\bigcup_{n\in\mathbb{N}}\mathbb{S}_n|$,$|\mathbb{S}_n|=\beth_n$
The Wikipedia article on Beth numbers defines $\beth_\alpha$ such that $\beth_{\alpha} =\begin{cases}
|\mathbb{N}| & \text{if } \alpha=0 \\
2^{\beth_{\alpha-1}} & \text{if } \alpha \text{ is a ...
- 171