Editing Set theory (section)

==Applications==
Many mathematical concepts can be defined precisely using only set theoretic concepts. For example, mathematical structures as diverse as [[graph (discrete mathematics)|graph]]s, [[manifolds]], [[ring (mathematics)|rings]], [[vector space]]s, and [[relational algebra]]s can all be defined as sets satisfying various (axiomatic) properties. [[equivalence relation|Equivalence]] and [[order relation]]s are ubiquitous in mathematics, and the theory of mathematical [[relation (mathematics)|relations]] can be described in set theory.<ref>{{Cite web |date=2019-11-25 |title=6.3: Equivalence Relations and Partitions |url=https://math.libretexts.org/Courses/Monroe_Community_College/MTH_220_Discrete_Math/6%3A_Relations/6.3%3A_Equivalence_Relations_and_Partitions |access-date=2022-07-27 |website=Mathematics LibreTexts |language=en |archive-date=2022-08-16 |archive-url=https://web.archive.org/web/20220816192743/https://math.libretexts.org/Courses/Monroe_Community_College/MTH_220_Discrete_Math/6:_Relations/6.3:_Equivalence_Relations_and_Partitions |url-status=live }}</ref><ref>{{cite web|url=https://web.stanford.edu/class/archive/cs/cs103/cs103.1132/lectures/06/Slides06.pdf|title=Order Relations and Functions|website=Web.stanford.edu|access-date=2022-07-29|archive-date=2022-07-27|archive-url=https://web.archive.org/web/20220727205803/https://web.stanford.edu/class/archive/cs/cs103/cs103.1132/lectures/06/Slides06.pdf|url-status=live}}</ref>

Set theory is also a promising foundational system for much of mathematics. Since the publication of the first volume of ''[[Principia Mathematica]]'', it has been claimed that most (or even all) mathematical theorems can be derived using an aptly designed set of axioms for set theory, augmented with many definitions, using [[first-order logic|first]] or [[second-order logic]]. For example, properties of the [[natural number|natural]] and [[real number]]s can be derived within set theory, as each of these number systems can be defined by representing their elements as sets of specific forms.<ref>{{citation
 | last = Mendelson | first = Elliott
 | mr = 357694
 | publisher = Academic Press
 | title = Number Systems and the Foundations of Analysis
 | zbl = 0268.26001
 | year = 1973}}</ref>

Set theory as a foundation for [[mathematical analysis]], [[topology]], [[abstract algebra]], and [[discrete mathematics]] is likewise uncontroversial; mathematicians accept (in principle) that theorems in these areas can be derived from the relevant definitions and the axioms of set theory. However, it remains that few full derivations of complex mathematical theorems from set theory have been formally verified, since such formal derivations are often much longer than the natural language proofs mathematicians commonly present. One verification project, [[Metamath]], includes human-written, computer-verified derivations of more than 12,000 theorems starting from [[ZFC]] set theory, [[first-order logic]] and [[propositional logic]].<ref>{{cite web|url=https://www.ams.org/bull/1956-62-05/S0002-9904-1956-10036-0/S0002-9904-1956-10036-0.pdf|title=A PARTITION CALCULUS IN SET THEORY |website=Ams.org|access-date=2022-07-29}}</ref>