Π 1 1 Borel sets
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....
| Published in: | Journal of Symbolic Logic |
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| Main Authors: | , , |
| Format: | Article in Journal/Newspaper |
| Language: | English |
| Published: |
Cambridge University Press (CUP)
1989
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| Subjects: | |
| Online Access: | http://dx.doi.org/10.2307/2274751 https://www.cambridge.org/core/services/aop-cambridge-core/content/view/S0022481200041608 |
| Summary: | 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. |
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