Role of True Finiteness in the Admissible Recursively Enumerable Degrees

Discusses both codings of models of arithmetic into the recursively enumerable degrees, and non-distributive lattice embeddings into these degrees. This title shows that if an admissible ordinal $\alpha$ is effectively close to $\omega$ then such constructions may be performed in the $\alpha$-r.e. degrees, but otherwise they fail.