# Tagged Questions

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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 ...
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### Question about cardinals in ZF

In Lévy's basic set theory refers a theorem of Bernstein as exercise problem: Theorem. (Bernstein) Let $\mathfrak{a,b}$ be cardinals. If $\mathfrak{a+a=a+b}$, then $\mathfrak{b\le a}$. I try to ...
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### A problem with an assumption in a previous lemma for the proof of Silver´s Theorem on SCH in Jech´s “Set Theory”

In the Jech´s textbook proof of Silver´s Theorem, specifically in the Lemma 8.15, there are an assumption in the beginning of the proof. First, the Lemma 8.15 says that, under the assumption that ...
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### $\aleph_\omega$ and large ordinals?

Assuming the GCH, each time you take the powerset of an aleph number, the subscript increases by one. So there is $\aleph_0, \aleph_1, \aleph_2\ldots \aleph_\omega\ldots \aleph_{\epsilon_0}\ldots$ and ...
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### Functions with cardinals as domain

How do we actually define a function with cardinals as domain? For example, take domain of the function as $\aleph_1$. Do we define it for all cardinal numbers strictly less than $\aleph_1$?
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### Cardinal Arithmetic, Regular Cardinals, and Exponentiation

I cannot solve a simple cardinal exponentiation/regularity exercise. Let $\kappa$ be a regular cardinal. Why is the cardinal $\kappa^{\lt \kappa}=\sum_{\alpha<\kappa}\kappa^\alpha$ regular as ...
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### Cardinal Arithmetic 2: $\beth_{\omega_1}$

This is question is simple to write but (I think) hard to solve. Does the following equality hold? $$\bigcup_{{\beta}<\beth_{\omega_1}}\beth_{\omega+1}(|\beta+\omega|) =\beth_{\omega_1}$$ Where ...
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### Partitioning an infinite cardinal number

Can we partition an infinite cardinal greater than aleph null, into countable number of cofinal subsets? Can we have restriction on the cardinality of cofinal subsets? For example, suppose $\alpha$ ...
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### Cardinal Arithmetic: $\beth_\omega$

Is the following equality independent of ZFC: $$\bigcup_{\aleph_{\beta}<\beth_{\omega}}2^{\aleph_{\beta}}=\beth_\omega$$ Consistent: Now that the equality is consistent with ZFC since it holds ...
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### Is the cardinality of the continuum weakly Mahlo?

Is $2^{\aleph_0}$ a weakly Mahlo cardinal? Can it be? That is, are there conditions (such as the negation of the continuum hypothesis or something) under which it is, and other conditions under which ...
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### Collection of sets with a given cardinality $\kappa$ is not set [duplicate]

Show that collection of all sets with cardinality $\kappa\neq0$, is not set. I'll state my approach and I need to see whether this idea is precise/precisable or not : First let $K$ be the set ...
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Suppose $\lambda$ is a strong limit cardinal, i.e. $\forall \alpha<\lambda \ 2^\alpha<\lambda$, and the cofinality of $\lambda$: $cf(\lambda)=\omega$. How do we show that $2^\lambda \leq ... 2answers 126 views ### about the smallest$k$that$V_k$is a model of ZFC Let$k$to be the smallest ordinal that$V_k$is a model of ZFC. I know that$k$need not to be inaccessible cardinal,and$k$has confinality$\omega$. Then how big is$k$? How to write down$k$in ... 4answers 147 views ### Cardinality of Irrational Numbers I know and I have proved more than once that the set of irrational numbers ($\mathbb{I}$) is uncountable, but now I'm given to solve this problem: Show that$|\mathbb{I}|=|\mathbb{R}|$, How can I ... 2answers 123 views ### The regularity of successor cardinal I was looking at two different proofs of the fact that successor cardinals are regular. It struck me as odd that both proofs used AC. Looking at the concepts involved in defining cofinality I feel as ... 3answers 228 views ### Uncountable Cardinals without AC I am doing an exercise, proving that without AC or Replacement that there are uncountable cardinals. As a point of reference I looked at the proof in Kunen's "The Foundations of Mathematics" that ... 1answer 70 views ### About alephs and beths If$2^{\aleph_{0}} \ge \aleph_{\omega_1}$, show that$\beth_{\aleph_{\omega}} = 2^{\aleph_{0}}$, and that$\beth_{\aleph_{\omega_1}} = 2^{\aleph_{1}}$I don´t know how to start, can you give me a ... 1answer 313 views ### A question about splitting sets I've been looking into combinatorics and small cardinals, in particular, the splitting number$\mathfrak{s}$. By definition, a set$X \subseteq \omega$splits an infinite set$Y \subseteq \omega$if ... 1answer 47 views ### Are there ordinals other than the set of natural numbers which satisfy this property? Let$\alpha$be an ordinal. We say that$\alpha$is good iff for every$\beta\in \alpha$, there exists$\gamma\in \alpha$such that$|\scr{P}(\beta)|\leq |\gamma|$. Question: Is the set of natural ... 1answer 46 views ### Cofinality assuming GCH There is this statement that GCH holds iff any pair of regular cardinals$\kappa,\lambda$such that$\kappa<\lambda$satisfy that$\lambda^\kappa = \lambda$. Assume we do have two such cardinals. ... 1answer 29 views ### Identity on singular strong limit cardinals Let$\lambda$be a singular strong limit cardinal. Prove that$2^\lambda = \lambda^{\mbox{cf}\lambda}$. It has been a while since I had to prove anything relating to cardinals, and I am not sure ... 1answer 98 views ### Continuum Hypothesis$\iff ?$? I have read that CH cannot be proved nor disproved within ZFC, and I was wondering: Which (If any) branches/fields of Mathematics are built upon CH being true? Are there any subjects built upon ... 1answer 96 views ###$\aleph$function fixed points below a weakly inaccessible cardinal are a club set I am throwing yet another one of my solutions out here for the internets to debug and for future set-theory students. Let$\aleph_\delta$a weakly inaccessible cardinal. Prove that$A =\{\alpha ...
I am solving old exam questions and I came across this question: Let $\langle A_n \mid n < \omega\rangle$ disjoint sets such that $\bigcup_{n < \omega}A_n = \mathbb{R}$. Prove that there ...