Editing Naive set theory (section)

==Method==
A ''naive theory'' in the sense of "naive set theory" is a non-formalized theory, that is, a theory that uses [[natural language]] to describe sets and operations on sets. Such theory treats sets as platonic absolute objects. The words ''and'', ''or'', ''if ... then'', ''not'', ''for some'', ''for every'' are treated as in ordinary mathematics. As a matter of convenience, use of naive set theory and its formalism prevails even in higher mathematics &ndash; including in more formal settings of set theory itself.

The first development of [[set theory]] was a naive set theory. It was created at the end of the 19th&nbsp;century by [[Georg Cantor]] as part of his study of [[infinite set]]s{{sfn|Cantor|1874}} and developed as a formal but inconsistent system <ref>https://plato.stanford.edu/entries/frege/</ref> by [[Gottlob Frege]] in his ''Grundgesetze der Arithmetik''.

Naive set theory may refer to several very distinct notions. It may refer to
* Informal presentation of an axiomatic set theory, e.g. as in ''[[Naive Set Theory (book)|Naive Set Theory]]'' by [[Paul Halmos]].
* Early or later versions of [[Georg Cantor]]'s theory and other informal systems.
* Decidedly inconsistent theories (whether axiomatic or not), such as a theory of [[Gottlob Frege]]<ref>{{harvnb|Frege|1893}} In Volume 2, Jena 1903. pp. 253-261 Frege discusses the antionomy in the afterword.</ref> that yielded [[Russell's paradox]], and theories of [[Giuseppe Peano]]<ref>{{harvnb|Peano|1889}} Axiom 52. chap. IV produces antinomies.</ref> and [[Richard Dedekind]].

===Paradoxes===
The assumption that any property may be used to form a set, without restriction, leads to [[paradoxes of set theory|paradoxes]].  One common example is [[Russell's paradox]]: there is no set consisting of "all sets that do not contain themselves". Thus consistent systems of (either naive or formal) set theory must include some limitations on the principles which can be used to form sets.

===Cantor's theory===
Some believe that [[Georg Cantor]]'s set theory was not actually implicated in the set-theoretic paradoxes (see Frápolli 1991). One difficulty in determining this with certainty is that Cantor did not provide an axiomatization of his system. By 1899, Cantor was aware of some of the paradoxes following from unrestricted interpretation of his theory, for instance [[Cantor's paradox]]<ref name=Letter_to_Hilbert>Letter from Cantor to [[David Hilbert]] on September 26, 1897, {{harvnb|Meschkowski|Nilson|1991}} p. 388.</ref> and the [[Burali-Forti paradox]],<ref>Letter from Cantor to [[Richard Dedekind]] on August 3, 1899, {{harvnb|Meschkowski|Nilson|1991}} p. 408.</ref> and did not believe that they discredited his theory.<ref name=Letters_to_Dedekind>Letters from Cantor to [[Richard Dedekind]] on August 3, 1899 and on August 30, 1899, {{harvnb|Zermelo|1932}} p. 448 (System aller denkbaren Klassen) and {{harvnb|Meschkowski|Nilson|1991}} p. 407. (There is no set of all sets.)</ref> Cantor's paradox can actually be derived from the above (false) assumption—that any property {{math|''P''(''x'')}} may be used to form a set—using for {{math|''P''(''x'')}} "{{mvar|x}} is a [[cardinal number]]". Frege explicitly axiomatized a theory in which a formalized version of naive set theory can be interpreted, and it is ''this'' formal theory which [[Bertrand Russell]] actually addressed when he presented his paradox, not necessarily a theory Cantor{{--}}who, as mentioned, was aware of several paradoxes{{--}}presumably had in mind.

===Axiomatic theories===
Axiomatic set theory was developed in response to these early attempts to understand sets, with the goal of determining precisely what operations were allowed and when.

===Consistency===
A naive set theory is not ''necessarily'' inconsistent, if it correctly specifies the sets allowed to be considered. This can be done by the means of definitions, which are implicit axioms. It is possible to state all the axioms explicitly, as in the case of Halmos' ''Naive Set Theory'', which is actually an informal presentation of the usual axiomatic [[Zermelo–Fraenkel set theory]]. It is "naive" in that the language and notations are those of ordinary informal mathematics, and in that it does not deal with consistency or completeness of the axiom system.

Likewise, an axiomatic set theory is not necessarily consistent: not necessarily free of paradoxes. It follows from [[Gödel's incompleteness theorems]] that a sufficiently complicated [[first-order logic]] system (which includes most common axiomatic set theories) cannot be proved consistent<ref>More precisely, cannot prove certain sentences (within the system) whose natural interpretation asserts the theory's own consistency.</ref> from within the theory itself &ndash; unless it is actually inconsistent. However, the common axiomatic systems are generally believed to be consistent; by their axioms they do exclude ''some'' paradoxes, like [[Russell's paradox]]. Based on [[Gödel's incompleteness theorems|Gödel's theorem]], it is just not known &ndash; and never can be &ndash; if there are ''no'' paradoxes at all in these theories or in any sufficiently complicated first-order set theory, again, unless such theories are actually inconsistent. It should be mentioned, however, that results in [[proof theory|proof theoretical]] [[ordinal analysis]] are sometimes interpreted as [[Gentzen's consistency proof|consistency proofs]].

The term ''naive set theory'' is still today also used in some literature<ref>F. R. Drake, ''Set Theory: An Introduction to Large Cardinals'' (1974). ISBN 0 444 10535 2.</ref> to refer to the set theories studied by Frege and Cantor, rather than to the informal counterparts of modern axiomatic set theory.

===Utility===
The choice between an axiomatic approach and other approaches is largely a matter of convenience. In everyday mathematics the best choice may be informal use of axiomatic set theory.  References to particular axioms typically then occur only when demanded by tradition, e.g. the [[axiom of choice]] is often mentioned when used. Likewise, formal proofs occur only when warranted by exceptional circumstances. This informal usage of axiomatic set theory can have (depending on notation) precisely the ''appearance'' of naive set theory as outlined below. It is considerably easier to read and write (in the formulation of most statements, proofs, and lines of discussion) and is less error-prone than a strictly formal approach.