Editing Reverse mathematics (section)

{{Short description|Branch of mathematical logic}}
{{redirect|Reverse Mathematics|the book by John Stillwell|Reverse Mathematics: Proofs from the Inside Out}}
{{more citations needed|date=October 2016}}
'''Reverse mathematics''' is a program in [[mathematical logic]] that seeks to determine which axioms are required to prove theorems of mathematics.  Its defining method can briefly be described as "going backwards from the [[theorem]]s to the [[axiom]]s", in contrast to the ordinary mathematical practice of deriving theorems from axioms. It can be conceptualized as sculpting out [[Necessity and sufficiency|necessary]] conditions from [[Necessity and sufficiency|sufficient]] ones.

The reverse mathematics program was foreshadowed by results in [[set theory]] such as the classical theorem that the [[axiom of choice]] and [[Zorn's lemma]] are equivalent over [[ZF set theory]]. The goal of reverse mathematics, however, is to study possible axioms of ordinary theorems of mathematics rather than possible axioms for set theory.

Reverse mathematics is usually carried out using subsystems of [[second-order arithmetic]],<ref name="Simpson2009">Simpson, Stephen G. (2009), Subsystems of second-order arithmetic, Perspectives in Logic (2nd ed.), Cambridge University Press, doi:10.1017/CBO9780511581007, ISBN 978-0-521-88439-6, MR 2517689</ref> where many of its definitions and methods are inspired by previous work in [[constructive analysis]] and [[proof theory]]. The use of second-order arithmetic also allows many techniques from [[recursion theory]] to be employed; many results in reverse mathematics have corresponding results in [[computable analysis]].  In ''higher-order'' reverse mathematics, the focus is on subsystems of [[higher-order arithmetic]], and the associated richer language.{{clarify|date=May 2021}}

The program was founded by [[Harvey Friedman (mathematician)|Harvey Friedman]]<ref>{{harvs|txt|authorlink=Harvey Friedman (mathematician)|first=Harvey|last= Friedman|year1=1975|year2=1976}}</ref><ref>H. Friedman, Some systems of second-order arithmetic and their use (1974), ''Proceedings of the International Congress of Mathematicians''</ref> and brought forward by [[Steve Simpson (mathematician)|Steve Simpson]].<ref name="Simpson2009" />