Exercises for Week 7
Due Wednesday, November 20
- Define ranks of sets by $\in$-recursion: $\text{rank}(x) = \sup \{
\text{rank}(y) + 1 : y \in x\}$. Define $V_\alpha$ for $\alpha \in
\text{Ord}$ by transfinite recursion: $V_0 = \emptyset$, $V_{\alpha+1} =
\mathcal{P}(V_\alpha)$, and $V_\lambda = \bigcup_{\alpha \lt \lambda}
V_\alpha$ if $\lambda$ is a limit ordinal. Prove that for every set
$x$ and ordinal $\alpha$ we have $x \in V_\alpha \iff \text{rank}(x)
\lt \alpha$.
- Prove that if $\kappa$ is a regular strong limit cardinal, then $|V_\kappa| = \kappa$.
- Prove that $H_\kappa$ is a set, for every cardinal $\kappa$.
- Prove that $\textit{HC}$ does not have the form $V_\alpha$ for any ordinal $\alpha$.