Editing Naive set theory (section)

===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.