Exercises for Week 7

Due Wednesday, November 20

  1. 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 $x \in V_\alpha \iff \text{rank}(x) \lt \alpha$.
  2. Prove that if $\kappa$ is a regular strong limit cardinal, then $|V_\kappa| = \kappa$.
  3. Prove that $H_\kappa$ is a set, for every cardinal $\kappa$.
  4. Prove that $\textit{HC}$ does not have the form $V_\alpha$ for any ordinal $\alpha$.