An Undecidable Linear Order That Is n-Decidable for All n

A linear order is n-decidable if its universe is and the relations determined byn formulas are uniformly computable. This means that there is a computable procedure which, when applied to an formula ϕ(¯ x) and a se- quence ¯ a of elements of the linear order, will determine whether or not ϕ(¯ a) is true in the structure. A linear order is decidable if the relations determined by all formulas are uniformly computable. These definitions suggest two questions. Are there, for each n, n-decidable lin- ear orders that are not (n + 1)-decidable? Are there linear orders that are n- decidable for all n but not decidable? The former was answered in t he positive by Moses in 1993. Here we answer the latter, also positively.