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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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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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 ...
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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{...
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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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Proving Power Set of $\mathbb N$ is Uncountable [duplicate]
I'm getting hung up on a proof that I remember being fairly easy... Showing that the power set of $\mathbb N$ is uncountable. Supposing it's countable, say $A=\{A_1,...\}$, we choose a set $B$ ...
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Clarification about Infinite Sums in Jech's Set Theory 3rd Edition
On page 57 in the 3rd Edition of Jech's Set Theory, he begins with the elaboration on cardinal exponentiation. To that end the Hausdorff formula is introduced by noticing that for a regular cardinal $\...
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Can the unit interval map bijectively to a region?
The Hilbert Curve shows that there exists a surjection from the unit interval to the unit square.
I was wondering, does there exist a bijection from the unit interval to the unit square?
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A weak version of König's Theorem - Does this depend on Choice?
Let $X_1,X_2,Y_1,Y_2$ be nonempty disjoint sets, where $|X_i|<|Y_i|$. Here, we take this to mean that there is an injection $X_i\to Y_i$, but there is no bijection $X_i\to Y_i$. My question is, is ...
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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 ...
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what is the cardinality of powerset of a union set?
Is there exist something like P(X+Y) (P STANDS FOR POWERSET)? I am confuse because power set is the set of all subset of Cartesian product, and X+Y wont give Cartesian product but (x,0) U (y,1), and ...
CommunityBot
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(CH) implies $\omega_2 = \omega_2^\omega$: reference?
Just by chance I found out that (CH) implies $\omega_2 = \omega_2^\omega$, which I found quite surprising, since this an almost trivial conclusion of
$2^{\omega_1} = \omega_2$ rather than of $2^{\...
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Assuming $(GCH)$ but not the axiom of choice, strongly inaccessible and weakly inaccessible coincide
My book says
"... If $GCH$ holds, then the notions of strongly inaccessible and weakly inaccessible cardinals coincide, ..."
In $ZFC$ I can prove this. But the paragraph from which I have excerpted ...
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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 ...
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