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

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

Link:

Zermelo?Fraenkel set theory - Wikipedia, the free encyclopedia