Set theory

Set theory is any of a number of subtly different things in mathematics:

Naive set theory is the original set theory developed by mathematicians at the end of the 19th century, treating sets simply as collections of things.
Axiomatic set theory is a rigorous axiomatic theory developed in response to the discovery of serious flaws (such as Russell's paradox) in naive set theory. It treats sets as "whatever satisfies the axioms", and the notion of collections of things serves only as motivation for the axioms.
Internal set theory is an axiomatic extension of set theory that supports a logically consistent identification of illimited (enormously large) and infinitesimal elements within the real numbers.
Various versions of logic have associated sorts of sets (such as fuzzy sets in fuzzy logic).

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