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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questions with no upvoted or accepted answers
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Formulations of Singular Cardinals Hypothesis
I have stumbled on a few different formulations of the Singular Cardinals Hypothesis. The most common are:
SCH1: $\quad 2^{cf(\kappa)}<\kappa \ \Longrightarrow \ \kappa^{cf(\kappa)}=\kappa^+$ for ...
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Conditions needed to prove $|A|^{|P(A)|}=|P(P(A))|$
I'm trying to figure out if ${|A|}^{|P(A)|}=|P(P(A))|$ (where $A$ is infinite) is provable without the Axiom of Choice.
I know unconditionally we have the lower bound: $|A|^{|P(A)|}\geq 2^{|P(A)|}=|P(...
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How to solve probability when sample space is infinite?
I came up with a random problem yesterday:
Suppose that in a random trial, each point $(x,y)$ where $x,y \in \mathbb{R}$ and $0 \leq x,y \leq 1$ is assigned a value of $0$ with 50% chance and a ...
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Raising a partial function to the power of an ordinal
Consider a set $X$, and let $f : X \rightarrow X$ denote a partial function. Then for natural $n$, we can define $f^n$ as iterated composition, e.g. $f^2 = f \circ f$. Now suppose that $X$ is also ...
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Cardinality of Galois groups
We know that no Galois group of a Galois extension is countable. The question is: which cardinalities are possible for a Galois group? (or also: for profinite groups?)
I suspect that the theory of ...
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About singular $\beth_{\alpha}$ for limit ordinals $\alpha$
Why are there cofinitely many regular cardinals of type $\beth_{\beta}$ below any given singular $\beth_{\alpha}$, if $\alpha$ is a limit ordinal?
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On cardinality of varieties (Hartshorne I.4.8, following Hartshorne's hint)
In Hartshorne's Algebraic Geometry Exercise I.4.8,
Exercise: Show that any variety of positive dimension over $k $ has the same cardinality as $ k $.
Hints: Do $\mathbb{A}^{n} $ and $\mathbb{P}^{n}$ ...
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Two well orderings of an infinite cardinal agree on a large set
I've seen this question but I'm having trouble following the proof given.
This is an exercise from Kunen: If $\kappa$ is an infinite cardinal and $\triangleleft$ is a well ordering on $\kappa$, then $...
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$\aleph_\alpha^{\aleph_1} = \aleph_\alpha^{\aleph_0}\cdot 2^{\aleph_1}$ for all $\omega \le \alpha < \omega_1$
I'd like someone to check my proof that
$\aleph_\alpha^{\aleph_1} = \aleph_\alpha^{\aleph_0} \cdot 2^{\aleph_1}$ for all $\omega \le \alpha < \omega_1$.
First, let's prove that $\aleph_n^{\...
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Is there a simple example of a ring that satifies the DCC on two-sided ideals, but doesn't satisfy the ACC on two-sided ideals?
It follows from the Hopkins–Levitzki theorem that if a ring satisfies the DCC on left ideals, then it also satisfies the ACC on left ideals. I've been trying to find a counterexample to the following ...
user23211
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Is there an upper bound to the number of rings that can be obtained from a semigroup with zero by defining an additive operation?
Let $\mathscr S$ be the class of all semigroups with zero. For $(S,\times,0)\in\mathscr S,$ I want to count additive operations $+$ on $S$ such that $(S,+,\times,0)$ is a ring (possibly without unity)....
user23211
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What is the intuition behind this proof of Silver's Theorem?
Context
There is a well-known result on the Generalized Continuum Hypothesis for singular cardinals:
Silver's Theorem: Let $\aleph _{\lambda }$ be a singular cardinal such that its cofinality is ...
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Is there a structure on $\mathbb{R}$ with a given cardinality of isomorphisms
We have, for example
$|\mathrm{Aut} (\mathbb{R})|= |\mathbb{R}^\mathbb{R}|$
$|\mathrm{Aut} (\mathbb{R,+,0})|= |\mathbb{R}^\mathbb{R}|$
$|\mathrm{Aut} (\mathbb{R,<})|= |\mathbb{R}|$
$|\mathrm{Aut} (\...
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A question on algebraically closed field
Let $k$ be an algebraically closed field . Consider the commutative ring , with unity , $A=k^\mathbb N=\prod_{i\in \mathbb N}k$ . Consider the proper ideal $I=\oplus_{i\in \mathbb N}k(=k^{(\mathbb N)})...
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Property of Galvin-Hajnal rank
In what follows, $\|\Phi\|_I$ means the Galvin-Hajnal rank of a function $\Phi:\kappa \to Ord$ with respect to a $\omega_1$-complete ideal $I$ of $\kappa$.
Lemma 2.2.5 of Introduction to Cardinal ...
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