Uncountable admissibles II: Compactness

AssumeV=L. Let κ be a cardinal and forX⊆κ, n<ω let α _n (X) denote the least ordinal α such thatL _αX] is Σ _n admissible. In our earlier paperUncountable admissibles I: forcing, we characterized those ordinals of the form σ _n (X) when κ is regular. This paper treats the singular case using Barwise compactness, an effective version of Jensen's covering lemma and β-recursion theory.