Let $(V, \in)$ be a model of ZF. Give the definition of a class and a function class in $V$. Use the concept of function class to give a short, informal statement of the Axiom of Replacement.

Let $z_{0}=\omega$ and, for each $n \in \omega$, let $z_{n+1}=\mathcal{P} z_{n}$. Show that $y=\left\{z_{n} \mid n \in \omega\right\}$ is a set.

We say that a set $x$ is small if there is an injection from $x$ to $z_{n}$ for some $n \in \omega$. Let HS be the class of sets $x$ such that every member of $\mathrm{TC}(\{x\})$ is small, where $\mathrm{TC}(\{x\})$ is the transitive closure of $\{x\}$. Show that $n \in \mathbf{H S}$ for all $n \in \omega$ and deduce that $\omega \in \mathbf{H S}$. Show further that $z_{n} \in \mathbf{H S}$ for all $n \in \omega$. Deduce that $y \in \mathbf{H S}$.

Is $(\mathbf{H S}, \in)$ a model of ZF? Justify your answer.

$[$ Recall that $0=\emptyset$ and that $n+1=n \cup\{n\}$ for all $n \in \omega .]$