The Isomorphism Problem for Computable Abelian p-Groups of Bounded Length

Journal of Symbolic Logic 70 (1):331 - 345 (2005)
Abstract
Theories of classification distinguish classes with some good structure theorem from those for which none is possible. Some classes (dense linear orders, for instance) are non-classifiable in general, but are classifiable when we consider only countable members. This paper explores such a notion for classes of computable structures by working out a sequence of examples. We follow recent work by Goncharov and Knight in using the degree of the isomorphism problem for a class to distinguish classifiable classes from non-classifiable. In this paper, we calculate the degree of the isomorphism problem for Abelian p-groups of bounded Ulm length. The result is a sequence of classes whose isomorphism problems are cofinal in the hyperarithmetical hierarchy. In the process, new back-and-forth relations on such groups are calculated
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Ekaterina B. Fokina (2009). Index Sets for Some Classes of Structures. Annals of Pure and Applied Logic 157 (2):139-147.
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Paul C. Eklof (1972). Some Model Theory of Abelian Groups. Journal of Symbolic Logic 37 (2):335-342.
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