Effectively inseparable Boolean algebras in lattices of sentences

Abstract

We show the non-arithmeticity of 1st order theories of lattices of Σn sentences modulo provable equivalence in a formal theory, of diagonalizable algebras of a wider class of arithmetic theories than has been previously known, and of the lattice of degrees of interpretability over PA. The first two results are applications of Nies’ theorem on the non-arithmeticity of the 1st order theory of the lattice of r.e. ideals on any effectively dense r.e. Boolean algebra. The theorem on degrees of interpretability relies on an adaptation of techniques leading to Nies’ theorem.