Tall $\alpha$-recursive structures
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- by Sy D. Friedman and Saharon Shelah PDF
- Proc. Amer. Math. Soc. 88 (1983), 672-678 Request permission
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
- Jon Barwise, Admissible sets and structures, Perspectives in Mathematical Logic, Springer-Verlag, Berlin-New York, 1975. An approach to definability theory. MR 0424560, DOI 10.1007/978-3-662-11035-5 Sy D. Friedman, An introduction to $\beta$-recursion theory, Generalized Recursive Theory. II (Fenstad, Gandy, Sacks, eds.), North-Holland, Amsterdam, 1978. —, Model theory for ${L_{\infty {\omega _1}}}$, (Proc. 1980-81 Logic Year, Jerusalem) (to appear).
- Mark Nadel, Scott sentences and admissible sets, Ann. Math. Logic 7 (1974), 267–294. MR 384471, DOI 10.1016/0003-4843(74)90017-5
Additional Information
- © Copyright 1983 American Mathematical Society
- Journal: Proc. Amer. Math. Soc. 88 (1983), 672-678
- MSC: Primary 03C70; Secondary 03D60
- DOI: https://doi.org/10.1090/S0002-9939-1983-0702297-7
- MathSciNet review: 702297