Representing sets of ordinals as countable unions of sets in the core model

Abstract:

We prove the following theorems. Theorem 1 $(\neg {0^\# })$. Every set of ordinals which is closed under primitive recursive set functions is a countable union of sets in $L$. Theorem 2. (No inner model with an Erdàs cardinal, i.e. $\kappa \to {({\omega _1})^{ < \omega }}$.) For every ordinal $\beta$, there is in $K$ an algebra on $\beta$ with countably many operations such that every subset of $\beta$ closed under the operations of the algebra is a countable union of sets in $K$.