## transitive closure, cardinality

For a given cardinal $\kappa$, $H(\kappa)$ denotes the collection of sets whose transitive closure has cardinality less than $\kappa$. Prove that $H(\aleph_1)$ has cardinality $2^{\aleph_0}$. Which axioms of ZFC are satisfied by $H(\aleph_1)$?

I am confused on how to prove this. I would appreciate some help on this problem. Thanks.