We proved König’s theorem and results of Hausdorff and Tarski on cardinal exponentiation, indicated some of their consequences (for example, ), and showed how to compute under the function .

We stated Easton’s result essentially saying that without additional assumptions, in nothing can be said about the exponential function 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 is *strong limit*, then is regular and smaller than .

This result is beyond the scope of this course. Instead, we will prove a particular case of an earlier result of Silver, namely, that is *not* the first counterexample to .

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.