Suppose that \(\displaystyle \kappa\) is a regular cardinal such that \(\displaystyle 2^{\lambda} < \kappa\) for all \(\displaystyle \lambda < \kappa\). Prove that \(\displaystyle H(\kappa)=\mathcal{V}_{\kappa}\).
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, a well orderable infinite cardinal \(\displaystyle \kappa\) is regular if \(\displaystyle \text{cf}(\kappa) =_c \kappa\), where \(\displaystyle \text{cf}\) denotes the cofinality. The set \(\displaystyle \mathcal{V}_{\alpha}\) is defined by \(\displaystyle \mathcal{V}_{0}= \emptyset\), \(\displaystyle \mathcal{V}_{\alpha+1}=\mathcal{P}(\mathcal{V}_{\alpha})\), and \(\displaystyle \mathcal{V}_{\lambda}= \bigcup_{\alpha< \lambda} \mathcal{V}_{\alpha}\) if \(\displaystyle \text{Limit}(\lambda)\).
I am not sure how to prove this. I would appreciate a few hints or suggestions. Thank you.
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, a well orderable infinite cardinal \(\displaystyle \kappa\) is regular if \(\displaystyle \text{cf}(\kappa) =_c \kappa\), where \(\displaystyle \text{cf}\) denotes the cofinality. The set \(\displaystyle \mathcal{V}_{\alpha}\) is defined by \(\displaystyle \mathcal{V}_{0}= \emptyset\), \(\displaystyle \mathcal{V}_{\alpha+1}=\mathcal{P}(\mathcal{V}_{\alpha})\), and \(\displaystyle \mathcal{V}_{\lambda}= \bigcup_{\alpha< \lambda} \mathcal{V}_{\alpha}\) if \(\displaystyle \text{Limit}(\lambda)\).
I am not sure how to prove this. I would appreciate a few hints or suggestions. Thank you.