Editing Universe (mathematics) (section)

{{short description|All-encompassing set or class}}

[[File:Probability_venn_event.svg|thumb|320x320px|The relationship between universe and complement]]

In [[mathematics]], and particularly in [[set theory]], [[category theory]], [[type theory]], and the [[foundations of mathematics]], a '''universe''' is a collection that contains all the entities one wishes to consider in a given situation.

In set theory, universes are often [[class (set theory)|classes]] that contain (as [[element (set theory)|elements]]) all sets for which one hopes to [[Mathematical proof|prove]] a particular [[theorem]]. These classes can serve as [[Inner model|inner models]] for various axiomatic systems such as [[Zermelo–Fraenkel set theory|ZFC]] or [[Morse–Kelley set theory]]. Universes are of critical importance to formalizing concepts in [[category theory]] inside set-theoretical foundations. For instance, the [[List of mathematical jargon#canonical|canonical]] motivating example of a category is '''[[Category of sets|Set]]''', the category of all sets, which cannot be formalized in a set theory without some notion of a universe.

In type theory, a universe is a type whose elements are types.