+-- {: .rightHandSide} +-- {: .toc .clickDown tabindex="0"} ### Context #### Foundations +-- {: .hide} [[!include foundations - contents]] =-- =-- =-- # Three-sorted set theory * table of contents {: toc} ## Overview Three-sorted set theories are [[set theories]] with three different sorts, representing [[sets]], [[elements]], and some third sort, usually [[functions]] or [[relations]]. Three-sorted set theories are usually [[structural set theories]]. In common with other [[simply sorted set theory]], the membership $\in$ is a [[relation]], in contrast to [[dependently sorted set theory]], where membership is a typing judgment. In contrast to [[unsorted set theory]] and [[two-sorted set theory]], sets and elements are different, and sets and elements are different from whatever relates sets and elements to each other ([[functions]] in [[categorical set theory]], [[relations]] in [[allegorical set theory]]). ## Examples * [[three-sorted SEAR]] * [[three-sorted structural ZFC]] * Similarly, there should be three-sorted presentations of [[ETCS with elements]], although it has not been written out yet. ## See also * [[unsorted set theory]] * [[two-sorted set theory]] * [[simply sorted set theory]] * [[dependently sorted set theory]]