Abstract: The class of Abelian p-groups are an example of some very interesting phenomena in computable structure theory. We will give an elementary first-order theory Tp whose models are each bi-interpretable with the disjoint union of an Abelian p-group and a pure set (and so that every Abelian p-group is bi-interpretable with a model of Tp) using computable infinitary formulas. This answers a question of Knight by giving an example of an elementary first-order theory of “Ulm type”: Any two models, low for ω1CK, and with the same computable infinitary theory, are isomorphic. It also gives a new example of an elementary first-order theory whose isomorphism problem is Σ11-complete but not Borel complete.
Keywords: p-groups, Borel reducibility, computable structure theory