Prove that every set has a transitive closure. [If you apply the Axiom of Replacement to a function-class $F$, you must explain clearly why $F$ is indeed a function-class.]

State the Axiom of Foundation and the Principle of $\epsilon$-Induction, and show that they are equivalent (in the presence of the other axioms of $\mathrm{ZFC}$ ).

State the $\epsilon$-Recursion Theorem.

Sets $C_{\alpha}$ are defined for each ordinal $\alpha$ by recursion, as follows: $C_{0}=\emptyset, C_{\alpha+1}$ is the set of all countable subsets of $C_{\alpha}$, and $C_{\lambda}=\cup_{\alpha<\lambda} C_{\alpha}$ for $\lambda$ a non-zero limit. Does there exist an $\alpha$ with $C_{\alpha+1}=C_{\alpha}$ ? Justify your answer.