Universal set

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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. 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).

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.


[1] S.C. Kleene, "Mathematical logic" , Wiley (1967)
[2] A.A. Fraenkel, Y. Bar-Hillel, "Foundations of set theory" , North-Holland (1958)


Cf. also Types, theory of; Universe.

How to Cite This Entry:
Universal set. V.E. Plisko (originator), Encyclopedia of Mathematics. URL:
This text originally appeared in Encyclopedia of Mathematics - ISBN 1402006098