A set which is closed under the formation of unions, singletons, subelements, power sets, and pairs; more precisely:
1) , implies ;
2) implies ;
3) implies ;
4) implies ;
5) if and only if .
The existence of infinite universes in axiomatic set theory is equivalent to the existence of strongly inaccessible cardinals (cf. Cardinal number). A universe is a model for Zermelo–Fraenkel set theory. Universes were introduced by A. Grothendieck in the context of category theory in order to introduce the "set" of natural transformations of functors between (-) categories, and in order to admit other "large" category-theoretic constructions.
|[a1]||J. Barwise (ed.) , Handbook of mathematical logic , North-Holland (1977) ((especially the article of D.A. Martin on Descriptive set theory))|
|[a2]||P. Gabriel, "Des catégories abéliennes" Bull. Soc. Math. France , 90 (1962) pp. 323–448|
|[a3]||K. Kunen, "Set theory" , North-Holland (1980)|
Universe. Encyclopedia of Mathematics. URL: http://encyclopediaofmath.org/index.php?title=Universe&oldid=11866