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Proceedings of the American Mathematical Society

ISSN 1088-6826(online) ISSN 0002-9939(print)



Tall $ \alpha $-recursive structures

Authors: Sy D. Friedman and Saharon Shelah
Journal: Proc. Amer. Math. Soc. 88 (1983), 672-678
MSC: Primary 03C70; Secondary 03D60
MathSciNet review: 702297
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Abstract: The Scott rank of a structure $ M$, $ \operatorname{sr}(M)$, is a useful measure of its model-theoretic complexity. Another useful invariant is $ {\text{o}}(M)$, the ordinal height of the least admissible set above $ M$, defined by Barwise. Nadel showed that $ {\operatorname{sr}}(M) \leqslant {\text{o}}(M)$ and defined $ M$ to be tall if equality holds. For any admissible ordinal $ \alpha $ there exists a tall structure $ M$ such that $ {\text{o}}(M) = \alpha $. We show that if $ \alpha = {\beta ^ + }$, the least admissible ordinal greater than $ \beta $, then $ M$ can be chosen to have a $ \beta $-recursive presentation. A natural example of such a structure is given when $ \beta = \omega _1^L$ and then using similar ideas we compute the supremum of the levels at which $ {\Pi _1}({L_{\omega _1^L}})$ singletons appear in $ L$.

References [Enhancements On Off] (What's this?)

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  • [78] Sy D. Friedman, An introduction to $ \beta $-recursion theory, Generalized Recursive Theory. II (Fenstad, Gandy, Sacks, eds.), North-Holland, Amsterdam, 1978.
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Keywords: Scott rank, admissible ordinals, Barwise compactness
Article copyright: © Copyright 1983 American Mathematical Society