Prove that \(\displaystyle H(\beth_{\omega})\) has cardinality \(\displaystyle \beth_{\omega}\).
Here, for a given cardinal \(\displaystyle \kappa\), \(\displaystyle H(\kappa)\) denotes the collection of sets whose transitive closure has cardinality less than \(\displaystyle \kappa\). Also, \(\displaystyle \beth_{\alpha}\) is defined by \(\displaystyle \beth_{0}= \aleph_0=|\mathbb{N}|=\omega\), \(\displaystyle \beth_{\beta +1}=2^{\beth_{\beta}}\), \(\displaystyle \beth_{\lambda} = \text{sup} \{ \beth_{\beta} | \beta < \lambda \}\), if \(\displaystyle \text{Limit}(\lambda)\).
Here, for a given cardinal \(\displaystyle \kappa\), \(\displaystyle H(\kappa)\) denotes the collection of sets whose transitive closure has cardinality less than \(\displaystyle \kappa\). Also, \(\displaystyle \beth_{\alpha}\) is defined by \(\displaystyle \beth_{0}= \aleph_0=|\mathbb{N}|=\omega\), \(\displaystyle \beth_{\beta +1}=2^{\beth_{\beta}}\), \(\displaystyle \beth_{\lambda} = \text{sup} \{ \beth_{\beta} | \beta < \lambda \}\), if \(\displaystyle \text{Limit}(\lambda)\).