Editing Computability theory (section)

===Reverse mathematics===
{{Main|Reverse mathematics}}

The program of ''[[reverse mathematics]]'' asks which set-existence axioms are necessary to prove particular theorems of mathematics in subsystems of [[second-order arithmetic]]. This study was initiated by [[Harvey Friedman (mathematician)|Harvey Friedman]] and was studied in detail by [[Steve Simpson (mathematician)|Stephen Simpson]] and others; in 1999, Simpson<ref name="Simpson_1999"/> gave a detailed discussion of the program. The set-existence axioms in question correspond informally to axioms saying that the [[powerset]] of the natural numbers is closed under various reducibility notions. The weakest such axiom studied in reverse mathematics is ''recursive comprehension'', which states that the powerset of the naturals is closed under Turing reducibility.