Editing Computability theory (section)

{{short description|Study of computable functions and Turing degrees}}
{{use dmy dates|date=August 2022|cs1-dates=y}}
'''Computability theory''', also known as '''recursion theory''', is a branch of [[mathematical logic]], [[computer science]], and the [[theory of computation]] that originated in the 1930s with the study of [[computable function]]s and [[Turing degree]]s. The field has since expanded to include the study of generalized [[computability]] and [[definable set|definability]]. In these areas, computability theory overlaps with [[proof theory]] and [[effective descriptive set theory]].

Basic questions addressed by computability theory include:
* What does it mean for a [[function (mathematics)|function]] on the [[natural number]]s to be computable?
* How can noncomputable functions be classified into a hierarchy based on their level of noncomputability?

Although there is considerable overlap in terms of knowledge and methods, mathematical computability theorists study the theory of relative computability, reducibility notions, and degree structures; those in the computer science field focus on the theory of [[computational complexity theory|subrecursive hierarchies]], [[formal method]]s, and [[formal language]]s. The study of which mathematical constructions can be effectively performed is sometimes called '''recursive mathematics'''.{{efn|The ''Handbook of Recursive Mathematics''<ref name="Ershov-Goncharov-Nerode-Remmel-1998"/> covers many of the known results in this field.}}