Recursive p-adic numbers and elementary theories of finitely generated pro-p-groups

The authors propose a method of effective presentation of finitely generated pro-p-groups, and use it to study the elementary theories of such groups. They prove that elementarily equivalent finitely generated pro-p-groups are isomorphic. The main result is the following criterion: The elementary theory of a finitely generated nilpotent pro-p-group G is decidable if and only if G is effectively presented.