Abstract

This paper studies the conjecture of Hirschfeldt, Miller, and Podzorov in [13] on the complexity of order-computable sets, where a set A is order-computable if there is a computable copy of the structure $(\mathbb {N}, <,A)$ in the language of linear orders together with a unary predicate. The class of order-computable sets forms a subclass of $\Delta ^{0}_{2}$ sets. Firstly, we study the complexity of computably enumerable (c.e.) order-computable sets and prove that the index set of c.e. order-computable sets is $\Sigma ^{0}_{4}$ -complete. Secondly, as a corollary of the main result on c.e. order-computable sets, we obtain that the index set of general order computable sets is $\Sigma ^{0}_{4}$ -complete within the index set of $\Delta ^{0}_{2}$ sets. Finally, we continue to study the complexity of more general $\Delta ^{0}_{2}$ sets and prove that the index set of $\Delta ^{0}_{2}$ sets is $\Pi ^{0}_{3}$ -complete.