Exercises for Week 9

1. Let $\kappa$ be an infinite cardinal. Show that $H_\kappa$ is closed under the function $x \mapsto \bigcup x$ if and only if $\kappa$ is regular. Use the definition of $H_\kappa$ that says $x \in H_\kappa \iff \forall y \in \text{tc}(\{x\})\,(|y|\lt \kappa)$.
2. Let $\mu$ be an infinite cardinal. Show that the model $H_{\mu^+}$ does not satisfy the Power Set axiom. Hint: it's not quite as easy as it looks.
3. Let $\kappa$ be a regular strong limit cardinal. Show that $H_\kappa = V_\kappa$. Use the definition of $H_\kappa$ that says $x \in H_\kappa \iff \forall y \in \text{tc}(\{x\})\,(|y|\lt \kappa)$.