Abstract
The results in this paper were motivated by the following question of Sacks. Suppose T is a recursive theory with countably many countable models. What can you say about the least ordinal α such that all models of T have Scott rank below α? If Martin's conjecture is true for T then α ≤ ω · 2.Our goal was to look at this problem in a more abstract setting. Let E be a equivalence relation on ωω with countably many classes each of which is Borel. What can you say about the least α such that each equivalence class is ? This problem is closely related to the following question. Suppose X ⊆ ωω is and Borel. What can you say about the least α such that X is ?In §1 we answer these questions in ZFC. In §2 we give more informative answers under the added assumptions V = L or -determinacy. The final section contains related results on the separation of sets by Borel sets.Our notation is standard. The reader may consult Moschovakis [5] for undefined terms.Some of these results were proved first by Sami and rediscovered by Kechris and Marker.
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