Let A be a standard transitive admissible set. Σ1-separation is the principle that whenever X and Y are disjoint Σ1A subsets of A then there is a ⊿1A subset S of A such that X ⊆ S and Y ∩ S = ∅.
Theorem. If satisfies Σ-separation, then
(1) If 〈Tn∣n < ω) ϵ A is a sequence of trees on ω each of which has at most finitely many infinite paths in A then the function n ↦ (set of infinite paths in A through Tn) is in A.
(2) If A is not closed under hyperjump and α = OnA then A has in it a nonstandard model of V = L whose ordinal standard part is α.
Theorem. Let α be any countable admissible ordinal greater than ω. Then there is a model of Σ1-separation whose height is α.