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.