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

We prove the following theorems.Theorem 1.(¬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 Erdös cardinal, i.e.k→(ω₁)^<ω.)For every ordinal β, there is in K an algebra on β with countably many operationssuch that every subset of β closed under the operations of the algebra is acountable union of sets in K.