What goes wrong shooting a club on $\kappa >\aleph_1$?

150 Views Asked by At

I'm reading through Jech's book, and in the section Stationary Sets in Generic Extensions (pages 444 and 445), he remarks that the poset used in shooting a club through a stationary $S\subseteq\omega_1$ can be generalized for regular $\kappa>\aleph_1$, but with additional restrictions on the stationary subset we use in order to preserve $\kappa$.

Why do we need these restrictions? What goes wrong in the usual argument if we consider, say, $\aleph_2$?

Is it that all club subsets of ordinals $\alpha<\omega_1$ have countable cofinality, while those $\alpha<\omega_2$ might have confinality $\omega_1$ or $\omega$?