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
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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)....
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1
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1
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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 $\mathbb{Q}$ ...
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3
answers
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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
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1
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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
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1
answer
50
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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 \...
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votes
1
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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 ...
- 3,630
1
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1
answer
56
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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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1
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1
answer
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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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0
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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 ...
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1
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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
4
votes
2
answers
167
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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<\...
- 310
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1
answer
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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 ...
- 1,296
0
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2
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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 ...
1
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0
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29
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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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1
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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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2
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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 ...
- 3,325
1
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0
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54
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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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0
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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 $G=\bigcup_{i<\omega} H_i$?
2
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4
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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 ...
- 3,325
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0
answers
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How does measure theory deal with higher cardinalities?
The second part of the definition of a sigma-algebra is that countable unions of measurable sets are measurable. The second property of a measure is that the measure of countable unions of measurable ...
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0
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56
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On the Singular Cardinal Hypothesis
I'm trying to find the proof of this result.
If for each $\lambda\geq2^\omega$, $\lambda^\omega\le\lambda^+$, then the SCH holds.
I'm not sure where to look. So if you have any info about this, please ...
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1
answer
66
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Why is $\{0,1\}^{\Bbb N}$ uncountable? [duplicate]
In the book Measure and Integral : An introduction to real analysis, in chapter 8 Lp spaces, theorem 8.18, the authors give a counterexample to show that $l^\infty$ is not separable:
Question: why is ...
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0
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29
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Cardinal sum of powers
I'm trying to solve this exercise. Can anybody please help me:
If $\kappa$ ist an infinite cardinal number with $cf(\kappa) = \kappa$ and for all $\mu < \kappa$ the inequality $2^{\mu} \leq \kappa$ ...
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0
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34
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For all cardinals $\kappa, \lambda$ with $\lambda \geq cf(\kappa)$ the inequality $\kappa^{\lambda} > \kappa$ holds [duplicate]
I genuinely have no idea why the proposition in the title holds or how to show it. I am kind of new to cardinals and ordinals and very confused. If someone could explain, I would really appreciate ...
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0
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Cardinal power towers
I am not an expert on large cardinals. I could not find any reference (and terminology) for the following question:
We start with
$$\lambda:=\aleph_0 \text{ [tet] } \omega = \aleph_0 ^ {\aleph_0 ^ {\...
- 11
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0
answers
25
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Would well-founded Scott cardinals work in ZCA + Ranks?
Does original Zermelo's set theory + Regularity + Ranks, prove that every set is of equal size to some element of a Scott cardinal? The original Zermelo does include an axiom of Choice, and it admits ...
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1
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119
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The cardinality of specific set $A\subset \mathbb{N}^{\mathbb{N}}$
Let $A$ be a set of total functions from the naturals to the naturals
such that for every $f\in A$ there is a finite set $B_f\subset \mathbb{N}$ , such that for every $x\notin B_f$ , $f(x+1)=f(x)+1$.
...
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1
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Induction does not preserve ordering between cardinality of sets?
Consider building a binary tree and consider it as a collection of points and edges. Here is one with five levels, numbered level $1$ at the top with $1$ node to level $5$ at the bottom with $16$ ...
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1
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A family of $\kappa^{<\omega}$ such that for every member in $\kappa$ is contained by all but finitely many elements of the family
Suppose that $\kappa$ is an uncountable cardinal. Let $\kappa^{<\omega}$ denote the family of all finite subsets of $\kappa$. Does there exist a family $S\subset\kappa^{<\omega}$ such that for ...
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0
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How should I should prove $\mathbb{R}\sim\{0,1\}^{ \mathbb{N}}$ [duplicate]
I've seen some argument about the binary representation, but I think it is not accurate because under some extreme cases, the rounding or bit constraint would results distinct reals also have the same ...
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