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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Infinite Series of infinite cardinals in ZFC
$\sum_{n=0}^\omega 2^{\aleph_n}=2^{\aleph_\omega}$
Is this true?
And is there a way in ZFC to let $\infty$ range over ALL infinite ordinals (not a concrete one as in the example above) ?
$\sum_{n=0}^\...
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Show the existence of family of sets
Show that there exists a family with cardinality of $c$, of subsets of $\mathbb{N}$, such that an intersection of any three elements of the family is an infinite (countable) set and the intersection ...
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How is transifnite recursion applied?
I've been struggling to understand how ordinal addition, multiplication, and exponentiation, along with the Aleph function $\aleph$, are defined using Transfinite Recursion in Jech's Set Theory or ...
- 5,166
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Correctness of Proof and Use of Axiom of Choice (Analysis I by Terence Tao)
I've skipped over some of the references in my proof for brevity. The following is an exercise from Terence Tao's Analysis I, specifically section 8.1. on countablity. ("countable" here ...
- 135
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What is cardinality of ordinal exponentiation?
Using von Neumann definition of ordinals, is it true that for all cardinal numbers $a$ and $b$ the following equation holds:
$$
a^b = |a^{(b)}|
$$
where on the left side is the cardinal exponentiation ...
- 59
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If $\kappa$ is weakly inaccessible, then it is the $\kappa$th element of $\{\alpha: \alpha =\aleph_\alpha\}$
This is an exercise from Kunen:
Exercise I.13.17 If $\kappa$ is weakly inaccessible, then it is the $\kappa$th element of $\{\alpha: \alpha =\aleph_\alpha\}$. If $\kappa$ is strongly inaccessible, ...
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The cardinal number of a set from Cylindrical Sigma-algebra.
Let $\mathbb{R}^{[0,1]}$ be the set of all function on $[0,1]$, and $\mathcal{B}(\mathbb{R}^{[0,1]})$ be the sigma-algebra generated by all cylinder sets:
$$\{ x=x(t):(x(t_1),\ldots,x(t_n))\in B \}$$ ...
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Basis for a $\sigma$-algebra of cardinality $\beth_1$. [duplicate]
Given a $\sigma$-algebra $\Sigma$ on a set $\Omega$, let's borrow language from topology, and call $B\subseteq\Sigma$ a basis for $\Sigma$ if $\Sigma$ is generated solely from countable unions of $B$, ...
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Shouldn't ℵ₀ be the cardinality of the reals?
If in ZFC any set can be well ordered, and that $\aleph_0$ is the cardinality of every infinite set that can be well ordered, shouldn't $\aleph_0$ be the cardinality of the real numbers?
I know this ...
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How does $2^{\aleph_2} \leqslant \aleph_{\omega}$ imply $\aleph_{\omega}^{\aleph_2} = \aleph_{\omega}^{\aleph_0}$? [duplicate]
I'm currently studying for my set theory exam and I got stuck trying to prove that if $2^{\aleph_2} \leqslant \aleph_{\omega}$ then $\aleph_{\omega}^{\aleph_2} = \aleph_{\omega}^{\aleph_0}$. Any help ...
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Does a cardinal with uncountable cofinality imply that the cardinal is regular?
In our book we use for our classes, we often require cardinals to be uncountable regular cardinals (when proving stuff with cofinality/stationarity...). We often use that by creating some sort of ...
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Does a separable, $US$, sequential space have cardinality at most the continuum?
Let $X$ be a separable sequential space with unique sequential limits ($US$). Can we prove that $X$ has cardinality at most $\mathfrak c=2^{\aleph_0}$?
Context.
If $X$ were Fréchet-Urysohn instead of ...
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What ways are there to define $\aleph$?
I've seen some posts on this website that consist in providing and comparing different proofs of a theorem (e.g. for Taylor's Theorem or trigonometric identities). Currently I'm reading Holz's ...
- 5,166
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Why does regularity of an ordinal $\gamma$ imply the existence of a sequence $(\delta_n)$ such that $\delta_n<\gamma$ for all $n$?
Source: Set Theory by Kenneth Kunen.
Lemma III.6.2: Let $\gamma$ be any limit ordinal, and assume that $\kappa:=cf(\gamma)>\omega$. Then the intersection of any family of fewer than $\kappa$ club ...
- 1,467
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Ordering and dividing orders of infinity.
I read there are an infinite number of orders of infinity.
Can they all be ordered, or are there different orders we can identify where we do not know which has the greater cardinality?
Is the ratio ...
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