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.
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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 ...
- 3,632
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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 ...
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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)\...
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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
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"Finitely distinguishable" family of functions $X \to Y$ [closed]
It's a well-known mathematical puzzle to find an uncountable subset of $\mathcal P(\mathbb N)$ such that any two sets have finite intersection. There's various ways to approach this, such as ...
- 4,762
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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
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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<\...
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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 ...
- 1,222
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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 ...
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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 $\...
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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 ...
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proving the set of natural numbers is infinite (Tao Ex 2.6.3)
Tao's Analysis I 4th ed has the following exercise 3.6.3:
Let $n$ be a natural number, and let $f:\{i \in \mathbb{N}:i \leq i \leq n\} \to \mathbb{N}$ be a function. Show that there exists a natural ...
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Injective monotonic mapping from rationals $\mathbb Q^2$ to $\mathbb R$
Exercise: $f: \mathbb Q^2\to\mathbb R$. Where $\mathbb Q$ is the set of rational
numbers.
$f$ is strictly increasing in both
arguments.
Can $f$ be one-to-one?
This question is related to many ...
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Can a countable union of subgroups of uncountable index in G be equal to G? [closed]
Let G be a group and $\{H_i\}_{i<\omega}$ be a countable family of subgroups of $G$, each of them of uncountable index. Can $G=\bigcup_{i<\omega} H_i$?
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Is the cardinality of $\varnothing$ undefined?
It is intuitive that the cardinality of the empty set is $0$.
However we are asked to demonstrate this using given definitions/axioms in Tao Analysis I 4th ed ex 3.6.2.
My question arises as I think ...
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