Abstract

Let α be any countable admissible ordinal greater than ω. There is a transitive set A such that A is admissible, locally countable, On A = α, and A satisfies Σ 1 -separation. In fact, if B is any nonstandard model of $KP + \forall x \subseteq \omega$ (the hyperjump of x exists), the ordinal standard part of B is greater than ω, and every standard ordinal in B is countable in B, then HC B ∩ (standard part of B) satisfies Σ 1 -separation