Editing Universal set (section)

===Universal objects that are not sets===
{{main|Universe (mathematics)}}
The idea of a universal set seems intuitively desirable in the [[Zermelo–Fraenkel set theory]], particularly because most versions of this theory do allow the use of quantifiers over all sets (see [[universal quantifier]]).  One way of allowing an object that behaves similarly to a universal set, without creating paradoxes, is to describe {{mvar|V}} and similar large collections as [[Class (set theory)|proper classes]] rather than as sets. Russell's paradox does not apply in these theories because the axiom of comprehension operates on sets, not on classes.

The [[category of sets]] can also be considered to be a universal object that is, again, not itself a set. It has all sets as elements, and also includes arrows for all functions from one set to another. 
Again, it does not contain itself, because it is not itself a set.