I am starting to read the Kunen's book set theory. I could not understand the why we need cofinality. Why is it important?
Also, I'm trying to solve some exercises (on page 146) about absoluteness for $R(\kappa)$, where $\kappa$ is strongly inaccessible.
The exercise in question is the following (Exercise 2):
(AC) Let $\kappa$ be strongly inaccessible. Check that the following are absolute for $R(\kappa)$:
(a) $\mathcal{P}(x)$.
(b) $\omega_\alpha$.
(c) $\gimel_\alpha$.
(d) $R(\alpha)$.
(e) $\mathrm{cf}(\alpha)$.
(f) $\alpha$ is strongly inaccessible.
I think that $\mathcal{P}(x)$, $\omega_\alpha$ and the sequence $(\omega_\alpha)$ (which is defined by transfinite recursion on $\alpha$) are absolute for $R(\kappa)$ because all of them are transitive sets. But in (e) and in (f) I could not prove anything. Could you give me a hint?