We prove that each interval of the lattice [Escr] of c.e. sets under inclusion is either a boolean algebra or has an undecidable theory. This solves an open problem of Maass and Stob [11]. We develop a method to prove undecidability by interpreting ideal lattices, which can also be applied to degree structures from complexity theory. We also answer a question left open in [7] by giving an example of a non-definable subclass of [Escr]* which has an arithmetical index set and is invariant under automorphisms.