Effectively dense Boolean algebras and their applications

A computably enumerable Boolean algebra B is effectively dense if for each x C B we can effectively determine an F(x) < x such that x 4: 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 ? (the lattice of computably enumerable sets under inclusion) which are not Boolean algebras. We derive a similar result for theories of certain initial intervals [o, a] of subrecursive degree structures, where a is the degree of a set of relatively small complexity, for instance a set in exponential time.

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