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

A linear order is n-decidable if its universe is N and the relations determined by n formulas are uniformly computable. This means that there is a computable procedure which, when applied to a n formula φ(x̄) and a sequence ā of elements of the linear order, will determine whether or not φ(ā) 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 linear orders that are not (n + 1)-decidable? Are there linear orders that are ndecidable 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.