Editing Set theory (section)

== Formalized set theory<!--'Axiomatic set theory' redirects here--> ==
{{anchor|Axiomatic set theory}}
Elementary set theory can be studied informally and intuitively, and so can be taught in primary schools using [[Venn diagram]]s. The intuitive approach tacitly assumes that a set may be formed from the class of all objects satisfying any particular defining condition. This assumption gives rise to paradoxes, the simplest and best known of which are [[Russell's paradox]] and the [[Burali-Forti paradox]]. '''Axiomatic set theory'''<!--boldface per WP:R#PLA--> was originally devised to rid set theory of such paradoxes.{{NoteTag|In his 1925 paper ""An Axiomatization of Set Theory", [[John von Neumann]] observed that "set theory in its first, "naive" version, due to Cantor, led to contradictions. These are the well-known [[antinomy|antinomies]] of the set of all sets that do not contain themselves (Russell), of the set of all transfinite ordinal numbers (Burali-Forti), and the set of all finitely definable real numbers (Richard)." He goes on to observe that two "tendencies" were attempting to "rehabilitate" set theory. Of the first effort, exemplified by [[Bertrand Russell]], [[Julius König]], [[Hermann Weyl]] and [[L. E. J. Brouwer]], von Neumann called the "overall effect of their activity . . . devastating". With regards to the axiomatic method employed by second group composed of Zermelo, Fraenkel and Schoenflies, von Neumann worried that "We see only that the known modes of inference leading to the antinomies fail, but who knows where there are not others?" and he set to the task, "in the spirit of the second group", to "produce, by means of a finite number of purely formal operations . . . all the sets that we want to see formed" but not allow for the antinomies. (All quotes from von Neumann 1925 reprinted in van Heijenoort, Jean (1967, third printing 1976), ''From Frege to Gödel: A Source Book in Mathematical Logic, 1879–1931'', Harvard University Press, Cambridge MA, {{ISBN|0-674-32449-8}} (pbk). A synopsis of the history, written by van Heijenoort, can be found in the comments that precede von Neumann's 1925 paper.}}

The most widely studied systems of axiomatic set theory imply that all sets form a [[cumulative hierarchy]].{{efn|This is the converse for ZFC; V is a model of ZFC.}} Such systems come in two flavors, those whose [[ontology]] consists of:
*''Sets alone''. This includes the most common axiomatic set theory, [[Zermelo–Fraenkel set theory|'''Z'''ermelo–'''F'''raenkel set theory]] with the [[Axiom of choice|axiom of '''c'''hoice]] (ZFC). Fragments of '''ZFC''' include:
** [[Zermelo set theory]], which replaces the [[axiom schema of replacement]] with that of [[axiom schema of separation|separation]];
** [[General set theory]], a small fragment of Zermelo set theory sufficient for the [[Peano axioms]] and [[finite set]]s;
** [[Kripke–Platek set theory]], which omits the axioms of infinity, [[axiom of power set|powerset]], and choice, and weakens the axiom schemata of [[axiom schema of separation|separation]] and [[axiom schema of replacement|replacement]].
*''Sets and [[proper class]]es''. These include [[Von Neumann–Bernays–Gödel set theory]], which has the same [[Strength (mathematical logic)|strength]] as [[ZFC]] for theorems about sets alone, and [[Morse–Kelley set theory]] and [[Tarski–Grothendieck set theory]], both of which are stronger than ZFC.
The above systems can be modified to allow ''[[urelement]]s'', objects that can be members of sets but that are not themselves sets and do not have any members.

The ''[[New Foundations]]'' systems of '''NFU''' (allowing [[urelement]]s) and '''NF''' (lacking them), associate with [[Willard Van Orman Quine]], are not based on a cumulative hierarchy. NF and NFU include a "set of everything", relative to which every set has a complement. In these systems urelements matter, because NF, but not NFU, produces sets for which the [[axiom of choice]] does not hold. Despite NF's ontology not reflecting the traditional cumulative hierarchy and violating well-foundedness, [[Thomas Forster (mathematician)|Thomas Forster]] has argued that it does reflect an [[iterative conception of set]].<ref>{{cite journal | last=Forster |first=T. E. |date=2008 | title = The iterative conception of set | journal = The Review of Symbolic Logic | volume = 1 |  pages = 97–110 |doi=10.1017/S1755020308080064 |s2cid=15231169 |url=https://www.dpmms.cam.ac.uk/~tf/iterativeconception.pdf}}</ref>

Systems of [[constructive set theory]], such as CST, CZF, and IZF, embed their set axioms in [[intuitionistic logic|intuitionistic]] instead of [[classical logic]]. Yet other systems accept classical logic but feature a nonstandard membership relation. These include [[Rough set|rough set theory]] and [[fuzzy set theory]], in which the value of an [[atomic formula]] embodying the membership relation is not simply '''True''' or '''False'''. The [[Boolean-valued model]]s of [[ZFC]] are a related subject.

An enrichment of ZFC called [[internal set theory]] was proposed by [[Edward Nelson]] in 1977.<ref>{{cite journal |last1=Nelson |first1=Edward |title=Internal Set Theory: a New Approach to Nonstandard Analysis |journal=Bulletin of the American Mathematical Society |date=November 1977 |volume=83 |issue=6 |page=1165 |doi=10.1090/S0002-9904-1977-14398-X |doi-access=free }}</ref>