Editing Naive set theory (section)

{{short description|Informal set theories}}
{{otheruses4|the mathematical topic|the book of the same name|Naive Set Theory (book)}}

'''Naive set theory''' is any of several theories of sets used in the discussion of the [[foundations of mathematics]].{{refn|Jeff Miller writes that ''naive set theory'' (as opposed to axiomatic set theory) was used occasionally in the 1940s and became an established term in the 1950s. It appears in Hermann Weyl's review of {{cite journal |editor=P. A. Schilpp |year=1946 |title=The Philosophy of Bertrand Russell |journal=American Mathematical Monthly |volume=53 |issue=4 |page=210|postscript=,}} and in a review by Laszlo Kalmar ({{cite journal |author=Laszlo Kalmar |year=1946 |title=The Paradox of Kleene and Rosser |journal=Journal of Symbolic Logic |volume=11 |issue=4 |page=136}}).<ref>{{cite web |title=Earliest Known Uses of Some of the Words of Mathematics (S) |date=April 14, 2020 |url=http://jeff560.tripod.com/s.html}}</ref> The term was later popularized in a book by [[Paul Halmos]].{{sfn|Halmos|1960|loc=''Naive Set Theory''}} }}
Unlike [[Set theory#Axiomatic set theory|axiomatic set theories]], which are defined using [[Mathematical_logic#Formal_logical_systems|formal logic]], naive set theory is defined informally, in [[natural language]]. It describes the aspects of [[Set (mathematics)|mathematical sets]] familiar in [[discrete mathematics]] (for example [[Venn diagram]]s and symbolic reasoning about their [[Boolean algebra (logic)|Boolean algebra]]), and suffices for the everyday use of set theory concepts in contemporary mathematics.<ref>{{citation
 | last = Mac Lane | first = Saunders
 | contribution = Categorical algebra and set-theoretic foundations
 | mr = 0282791
 | pages = 231–240
 | publisher = Amer. Math. Soc. |place=Providence, RI
 | title = Axiomatic Set Theory (Proc. Sympos. Pure Math., Vol. XIII, Part I, Univ. California, Los Angeles, Calif., 1967)
 | year = 1971}}. "The working mathematicians usually thought in terms of a naive set theory (probably one more or less equivalent to ZF) ... a practical requirement [of any new foundational system] could be that this system could be used "naively" by mathematicians not sophisticated in foundational research" ([https://books.google.com/books?id=TVi2AwAAQBAJ&pg=PA236 p.&nbsp;236]).</ref>

Sets are of great importance in mathematics; in modern formal treatments, most mathematical objects ([[number]]s, [[relation (mathematics)|relations]], [[function (mathematics)|functions]], etc.) are defined in terms of sets. Naive set theory suffices for many purposes, while also serving as a stepping stone towards more formal treatments.