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. (shrink)

We characterize the ordinals α of uncountable cofinality such that α is the standard part of a nonstandard model of ZFC (or equivalently KP).

The known results about Σ 1 -completeness, Σ 1 -compactness, ordinal omitting etc. are given a unified treatment, which yields many new examples. It is shown that the unifying theorem is best possible in several ways, assuming V = L.

Let $\mathscr{M}$ be a structure for a language $\mathscr{L}$ on a set $M$ of urelements. $\mathrm{HYP}(\mathscr{M})$ is the least admissible set above $\mathscr{M}$. In $\S 1$ we show that $pp(\mathrm{HYP}(\mathscr{M})) \lbrack = \text{the collection of pure sets in} \mathrm{HYP}(\mathscr{M}\rbrack$ is determined in a simple way by the ordinal $\alpha = \circ(\mathrm{HYP}(\mathscr{M}))$ and the $\mathscr{L}_{\propto\omega}$ theory of $\mathscr{M}$ up to quantifier rank $\alpha$. In $\S 2$ we consider the question of which pure countable admissible sets are of the form $pp(\mathrm{HYP}(\mathscr{M}))$ for (...) some $\mathscr{M}$ and show that all sets $L_\alpha (\alpha$ admissible) are of this form. Other positive and negative results on this question are obtained. (shrink)