In the naive set theory of the nineteenth century, the term
universal set referred to the set of all sets. If one was doing set
theory with objects that were not sets (these are sometimes called
urelements), those were included in the universal set as well.
However, Bertrand Russell and others discovered that this concept
leads to paradoxes, such as the set of all sets not members of
themselves (the universal set being a member of itself), which is a
member of itself if it is not, and not a member of itself if it is.
So axiomatic set theories were developed to hopefully avoid these
paradoxes. It was also discovered that urelements are not necessary
to do set theory that can be used as the basis of all areas of
mathematics.
In a more limited context, the term universal set or universe of
discourse is used to refer to the set of things being discussed and
studied. For example, in the area of the mathematical study of
integers (positive and negative whole numbers), the set of all
integers is the universe of discourse. This seems to be harmless in
that it does not lead to paradoxes, as far as is known.
