## 116c- Lecture 9

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.

I imagine K is some limit cardinal $\kappa$? The phrase means that there is some cardinal $\rho<\kappa$ such that for some fixed cardinal $\lambda$ and for all cardinals $\tau$ in the interval $(\rho,\kappa),$ we have that $2^\tau=\lambda.$