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.

Link:
Zermelo?Fraenkel set theory - Wikipedia, the free encyclopedia