My Math Forum cardinality, ZFC

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 April 24th, 2010, 04:39 PM #1 Newbie   Joined: Nov 2008 Posts: 25 Thanks: 0 cardinality, ZFC Prove that $H(\beth_{\omega})$ has cardinality $\beth_{\omega}$. Which axioms of ZFC are satisfied by $H(\beth_{\omega})$? Here, for a given cardinal $\kappa$, $H(\kappa)$ denotes the collection of sets whose transitive closure has cardinality less than $\kappa$. Also, $\beth_{\alpha}$ is defined by $\beth_{0}= \aleph_0=|\mathbb{N}|=\omega$, $\beth_{\beta +1}=2^{\beth_{\beta}}$, $\beth_{\lambda}= \text{sup} \{ \beth_{\beta} | \beta < \lambda \}=$, if $\text{Limit}(\lambda)$. I do not have any good ideas on how to prove this. I would appreciate a few hints. Thanks. Link: http://en.wikipedia.org/wiki/Zermelo%E2 ... The_axioms Problem statement: http://i719.photobucket.com/albums/ww19 ... 1272155858

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