The pure part of HYP(M)

Journal of Symbolic Logic 42 (1):33-46 (1977)
Let M be a structure for a language L on a set M of urelements. HYP(M) is the least admissible set above M. In § 1 we show that pp(HYP(M)) [ = the collection of pure sets in HYP(M] is determined in a simple way by the ordinal α = ⚬(HYP(M)) and the $\mathscr{L}_{\propto\omega}$ theory of M up to quantifier rank α. In § 2 we consider the question of which pure countable admissible sets are of the form pp(HYP(M)) for some M and show that all sets L α (α admissible) are of this form. Other positive and negative results on this question are obtained
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    Sy D. Friedman (1982). Steel Forcing and Barwise Compactness. Annals of Mathematical Logic 22 (1):31-46.
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