Editing Equality (mathematics) (section)

=== Background ===
{{further|Set theory#History}}
[[File:Ernst Zermelo 1900s.jpg|thumb|[[Ernst Zermelo]], a contributor to modern Set theory, was the first to explicitly formalize set equality in his [[Zermelo set theory]] (now obsolete), by his ''{{lang|de|Axiom der Bestimmtheit}}''.<ref>{{citation |last=Zermelo |first=Ernst |title=Untersuchungen über die Grundlagen der Mengenlehre I |journal=Mathematische Annalen |volume=65 |issue=2 |pages=261–281 |year=1908 |url=http://gdz.sub.uni-goettingen.de/dms/load/img/?PPN=PPN235181684_0065&DMDID=DMDLOG_0018 |doi=10.1007/bf01449999 |s2cid=120085563 |author-link=Ernst Zermelo |language=de}}</ref>]]
Around the turn of the 20th century, mathematics faced several [[paradox]]es and counter-intuitive results. For example, [[Russell's paradox]] showed a contradiction of [[naive set theory]], it was shown that the [[parallel postulate]] cannot be proved, the existence of [[mathematical object]]s that cannot be computed or explicitly described, and the existence of theorems of arithmetic that cannot be proved with [[Peano arithmetic]]. The result was a [[foundational crisis of mathematics]].{{Sfn|Ferreirós|2007|p=299}}

The resolution of this crisis involved the rise of a new mathematical discipline called [[mathematical logic]], which studies [[Logic#Formal logic|formal logic]] within mathematics. Subsequent discoveries in the 20th century then stabilized the foundations of mathematics into a coherent framework valid for all mathematics. This framework is based on a systematic use of [[axiomatic method]] and on set theory, specifically [[Zermelo–Fraenkel set theory]], developed by [[Ernst Zermelo]] and [[Abraham Fraenkel]]. This set theory (and set theory in general) is now considered the most common [[foundation of mathematics]].{{Sfn|Ferreirós|2007|p=366|loc="[...] the most common axiom system was and is called the Zermelo-Fraenkel system."}}