116c- Lecture 10

For $\kappa$ an uncountable regular cardinal, we studied basic properties of club sets, defined diagonal intersection and showed that the intersection of fewer than $\kappa$ many club subsets of an uncountable regular cardinal $\kappa$ is club and also the diagonal intersection of $\kappa$ many club sets is club.

We also showed that if ${\mathcal A}=(\kappa,\dots)$ is a structure in a countable language, then for club many $\alpha<\kappa$, $\alpha$ is the universe of an elementary substructure of ${\mathcal A}$. The proof in fact works as long as the language has size smaller than $\kappa$.

We closed with some basic examples of stationary sets and stated Fodor’s theorem.

Remark. The proof given in lecture of the fact that the diagonal intersection of club sets is club was purely combinatorial. In a forthcoming lecture, I will prove the reflection theorem, that any finite collection of sentences true in the universe is true in arbitrarily large stages $V_\alpha$. With this and a few absoluteness facts, one can give a different proof, in effect showing that many ordinals below a cardinal $\kappa$ “behave like” $\kappa$ itself. This idea, which we will formalize once reflection is presented, is a very useful heuristic that allows us to use (basic) model-theoretic techniques to prove combinatorial arguments.