116c- Lecture 9

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

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.

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2 Responses to 116c- Lecture 9

1. David Gray Carlson says:

What does it mean for the continuum function to be “eventually constant” below K. I am trying to read Thomas Jech. He uses the phrase “eventually constant” frequently but does not define it.

• Hi David,

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

(Although the first time one sees this, it is a bit puzzling, this situation is consistently possible.)