116c- Lecture 9

April 30, 2008

We proved König’s theorem and results of Hausdorff and Tarski on cardinal exponentiation, indicated some of their consequences (for example, {\mathfrak c}\ne\aleph_\omega), and showed how to compute under {\sf GCH} the function (\kappa,\lambda)\mapsto\kappa^\lambda.

We stated Easton’s result essentially saying that without additional assumptions, in {\sf ZFC} nothing can be said about the exponential function 2^\lambda beyond monotonicity and König’s theorem.

For singular cardinals the situation is much more delicate. We stated as a sample result Shelah’s theorem that if \aleph_\omega is strong limit, then 2^{\aleph_\omega} is regular and smaller than \aleph_{\min(\omega_4,{\mathfrak c}^+)}.

This result is beyond the scope of this course. Instead, we will prove a particular case of an earlier result of Silver, namely, that \aleph_{\omega_1} is not the first counterexample to {\sf GCH}.

In order to prove Silver’s result, we need to develop the theory of club and stationary sets. We defined these notions and proved some of their basic properties.