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.


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<\kappa 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.)

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