State the Axiom of Replacement.

Show that for any set $x$ there is a transitive set $y$ that contains $x$, indicating where in your argument you have used the Axiom of Replacement. No form of recursion theorem may be assumed without proof.

Which of the following are true and which are false? Give proofs or counterexamples as appropriate. You may assume standard properties of ordinals.

(a) If $x$ is a transitive set then $x$ is an ordinal.

(b) If each member of a set $x$ is an ordinal then $x$ is an ordinal.

(c) If $x$ is a transitive set and each member of $x$ is an ordinal then $x$ is an ordinal.