Abstract
AssumeV=L. Let κ be a cardinal and forX⊆κ, n<ω let αn(X) denote the least ordinal α such thatLα[X] is Σn admissible. In our earlier paperUncountable admissibles I: forcing, we characterized those ordinals of the form σn(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.
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