Tall α-recursive structures

The Scott rank of a structure M, sr(M), is a useful measure of its model-theoretic complexity. Another useful invariant is o(M), the ordinal height of the least admissible set above M, defined by Barwise. Nadel showed that sr(M) ≤ o(M) and defined M to be tall if equality holds. For any admissible ordinal a there exists a tall structure M such that o(M) = α. We show that if α = β⁺, the least admissible ordinal greater than β, then M can be chosen to have a β-recursive presentation. A natural example of such a structure is given when β = ω^L₁ and then using similar ideas we compute the supremum of the levels at which Π₁ (L_ωL1) singletons appear in L.