Tall $\alpha$-recursive structures

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 $L$.