Π 1 1 Borel sets

• Published in: Journal of Symbolic Logic
• Main Authors: Kechris, Alexander S., Marker, David, Sami, Ramez L.
• Format: Article in Journal/Newspaper
• Language: English
• Published: Cambridge University Press (CUP) 1989
• Subjects: Logic; Philosophy; sami
• Online Access: http://dx.doi.org/10.2307/2274751; https://www.cambridge.org/core/services/aop-cambridge-core/content/view/S0022481200041608

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 ω ω 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 ⊆ ω ω 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.