§1. Let α be an admissible ordinal which is also a limit of admissible ordinals (e.g. take any α such that α = α*, its projectum [5]). For any admissible γ ≦ α, let [γ) denote the initial segment of ordinals less than γ. A very general question that one might ask is the following: What conditions should one put on γ so that a certain statement true in Lα is ‘reflected’ to be true in Lγ? We cite some examples: (a) If β < γ < α, then Lα ⊨ “β cardinal” is ‘reflected’ to Lγ ⊨ “ β cardinal” (⊨ is just the satisfaction relation). (b) If β < γ < α and γ is a cardinal in Lα (called α-cardinal for short), then Lα ⊨ “β is not a cardinal” is ‘reflected’ to Lγ ⊨ “β is not a cardinal” This fact is used in Gödel's proof that V = L implies the Generalized Continuum Hypothesis. Our objective in this paper is to study a ‘reflection’ property of the following sort: Let A ⊆ [α) be an α-recursively enumerable (α-r.e.), non-α-recursive set. Under what conditions will A restricted to a smaller admissible ordinal γ be γ-r.e. and not γ-recursive?

The notations used here are standard. Those that are not explained are adopted from the paper of Sacks and Simpson [5], to which we also refer the reader for background material.