Editing Set (mathematics) (section)

==Context==
Before the end of the 19th century, sets were not studied specifically, and were not clearly distinguished from [[sequence (mathematics)|sequence]]s. Most mathematicians considered [[infinity (mathematics)|infinity]] as [[potential infinity|potential]]{{Mdash}}meaning that it is the result of an endless process{{mdash}}and were reluctant to consider [[infinite set]]s, that is sets whose number of members is not a [[natural number]]. Specifically, a [[line (geometry)|line]] was not considered as the set of its points, but as a [[locus (mathematics)|locus]] where points may be located. 

The mathematical study of infinite sets began with [[Georg Cantor]] (1845–1918). This provided some counterintuitive facts and paradoxes. For example, the [[number line]] has an [[cardinal number|infinite number]] of elements that is strictly larger than the infinite number of [[natural number]]s, and any [[line segment]] has the same number of elements as the whole space. Also, [[Russell's paradox]] implies that the phrase "the set of all sets" is self-contradictory.

Together with other counterintuitive results, this led to the [[foundational crisis of mathematics]], which was eventually resolved with the general adoption of [[Zermelo–Fraenkel set theory]] as a robust foundation of [[set theory]] and all mathematics. 

Meanwhile, sets started to be widely used in all mathematics. In particular, [[algebraic structure]]s and [[mathematical space]]s are typically defined in terms of sets. Also, many older mathematical results are restated in terms of sets. For example, [[Euclid's theorem]] is often stated as "the ''set'' of the [[prime number]]s is infinite". This wide use of sets in mathematics was prophesied by [[David Hilbert]] when saying: "No one will drive us from the [[Cantor's paradise|paradise which Cantor created for us]]."<ref>{{Citation
   | last = Hilbert
   | first = David
   | title = Über das Unendliche
   | year = 1926
   | author-link = David Hilbert
   | periodical = Mathematische Annalen
   | volume = 95
   | pages = 161&ndash;190
  |doi=10.1007/BF01206605|jfm=51.0044.02| s2cid = 121888793
 }}
: "''Aus dem Paradies, das Cantor uns geschaffen, soll uns niemand vertreiben können.''"
: Translated in {{Citation| title = On the infinite
   | first = Jean
   | last = Van Heijenoort
   | author-link = Jean Van Heijenoort
   | publisher = Harvard University Press
  }}</ref>

Generally, the common usage of sets in mathematics does not require the full power of Zermelo–Fraenkel set theory. In mathematical practice, sets can be manipulated independently of the [[mathematical logic|logical framework]] of this theory.

The object of this article is to summarize the manipulation rules and properties of sets that are commonly used in mathematics, without reference to any logical framework. For the branch of mathematics that studies sets, see [[Set theory]]; for an informal presentation of the corresponding logical framework, see [[Naive set theory]]; for a more formal presentation, see [[Axiomatic set theory]] and [[Zermelo–Fraenkel set theory]].