A universal set is the collection of all objects in a particular context or theory. All other sets in that framework constitute
subsets of the universal set, which is denoted as an uppercase italic letter *U*. The objects
themselves are known as elements or members of *U*.

The precise definition of *U* depends the context or theory under consideration. For
example, we might define *U* as the set of all living things on planet earth. In that case,
the set of all dogs is a subset of *U*, the set of all fish is another subset of *U*, and
the set of all trees is yet another subset of *U*. If we define *U* as the set of all
animals on planet earth, then the set of all dogs is a subset of *U*, the set of all fish is
another subset of *U*, but the set of all trees is not a subset of *U*.

Some philosophers have attempted to define *U* as the set of all objects (including all
sets, because sets are objects). This notion of *U* leads to a contradiction, because
*U*, which contains everything, must therefore contain the set of all sets that are *not*
members of themselves. In 1901, the philosopher and logician Bertrand Russell proved that this
state of affairs leads to a paradox. Today,
mathematicians and philosophers refer to this result as Russell's Paradox.

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