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

Abstract

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