| We examine the relation between the uniformity of a collection of operators witnessing Turing computable embeddings, and the existence of an operator witnessing the universality of a class. The primary equivalence relation studied here is computable infinitary Sigmaalpha equivalence. This project of exploiting uniformity of Turing computable embeddings to construct a limit embedding is carried out entirely in the context of countable reduced abelian p-groups. One may look at this program as either a project in the computable structure theory of abelian p-groups, or as a project in the construction of limits of sequences of uniform Turing computable operators.;In an attempt to explore the boundary between computable infinitary Sigma alphaequivalence and isomorphism, we show that for any computable alpha, certain classes of countable reduced abelian p-groups are universal for ∼ca under Turing computable embedding. Further, the operators witnessing these embeddings are extremely uniform.;Exploiting the uniformity of the embeddings, we produce operators which are, in some sense, limits of the embeddings witnessing the universality of the classes of countable reduced abelian p-groups. This is approached in three different ways: transfinite recursion on ordinal notation, Barwise-Kreisel Compactness, and hyperarithemetical saturation. Finally, we work in admissible set theory, and use Barwise Compactness and Sigma A- saturation to generalize selected results. |
