Let A be a standard transitive admissible set. \mathbf{\Sigma}_1-separation is the principle that whenever X and Y are disjoint \mathbf{\Sigma}^A_1 subsets of A then there is a \mathbf\Delta^A_1 subset S of A such that X \subseteq S and Y \cap S = \varnothing. Theorem. If A satisfies \mathbf\Sigma_1-separation, then (1) If \langle T_n\mid n < \omega \rangle \in A is a sequence of trees on \omega each of which has at most finitely many infinite paths in A then the function n\mapsto (set of infinite paths in A through T_n) is in A. (2) If A is not closed under hyperjump and \alpha = On^A then A has in it a nonstandard model of V = L whose ordinal standard part is \alpha. Theorem. Let \alpha be any countable admissible ordinal greater than \omega. Then there is a model of \mathbf\Sigma_1-separation whose height is \alpha.