Effectively Dense Boolean Algebras and Their Applications

A computably enumerable Boolean algebra B is effectively dense if for each x ∈ B we can effectively determine an F (x) ≤ x such that x 6= 0 implies 0 < F (x) < x. 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 E (the lattice of computably enumerable sets under inclusion) which are not Boolean algebras. We derive a similar result for theories of certain initial intervals [0,a] of subrecursive degree structures, where a is the degree of a set of relatively small complexity, for instance a set in exponential