Effectively dense Boolean algebras and their applications

A computably enumerable Boolean algebra is effectively dense if for each we can effectively determine an such that implies . We give an interpretation of true arithmetic in the theory of the lattice of computably enumerable ideals of such a Boolean algebra. As an application, we also obtain an interpretation of true arithmetic in all theories of intervals of (the lattice of computably enumerable sets under inclusion) which are not Boolean algebras. We derive a similar result for theories of certain initial intervals of subrecursive degree structures, where is the degree of a set of relatively small complexity, for instance a set in exponential time.