If the structure is considered in second-order logic instead of first-order logic, the definable sets of natural numbers in the resulting structure are classified in the analytical hierarchy. These hierarchies reveal many relationships between definability in this structure and computability theory, and are also of interest in descriptive set theory.
In a more complex framework, where second-order logic is applied instead of first-order logic, the sets of natural numbers that can be defined are organized into an analytical hierarchy. This hierarchy discloses many relationships between definability within this structure and computability principles, as well as relevance in descriptive set theory.