Exercises for Week 9

Due Tuesday, March 12
  1. Prove that $|V_{\omega+\alpha}| = \beth_\alpha$ for every ordinal $\alpha$.
  2. Prove that if $\lambda$ is a limit ordinal, then $\beth_\lambda$ is a strong limit cardinal. (A cardinal $\kappa$ is a strong limit cardinal if $|2^\alpha| < \kappa$ for all $\alpha < \kappa$.)
  3. Prove that if $\lambda$ and $\lambda'$ are limit ordinals and there is a cofinal increasing function $f: \lambda \to \lambda'$, then $\text{cf}(\lambda) = \text{cf}(\lambda')$.
  4. Prove that if $\kappa$ is a measurable cardinal, then $\kappa$ is regular.