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''universe''

A certain set, fixed within the framework of a given fundamental theory and containing as members all objects considered in this theory. For example, in elementary arithmetic a universal set is the set of all integers. The concept of a universal set plays a basic role in [[Set theory|set theory]]. Here the objects of study are sets, so the universal set is the collection of all sets; however, this is not itself a set, i.e. it cannot be considered as an object in set theory. From this arises the paradox connected with the notion of the set of all sets (cf. Cantor's [[Antinomy|Antinomy]]).

The set of all sets forms an object of study in the theory of sets and classes. In this theory one considers along with sets, (proper) classes — objects which cannot be members of other sets or classes.

====References====
<table><TR><TD valign="top">[1]</TD> <TD valign="top">  S.C. Kleene,   "Mathematical logic" , Wiley  (1967)</TD></TR><TR><TD valign="top">[2]</TD> <TD valign="top">  A.A. Fraenkel,   Y. Bar-Hillel,   "Foundations of set theory" , North-Holland  (1958)</TD></TR></table>



====Comments====
Cf. also [[Types, theory of|Types, theory of]]; [[Universe|Universe]].