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 $\alpha$.