In set theory, a class is essentially a set which we do not call a set for logical or semantic reasons. For example, if is a set, then is a class consisting of some subsets of . In this example though, can also be called a set.
To understand why one would make such a distinction, consider Russell's Paradox: "Define to be the set of all sets which do not contain themselves. Is it true or not that ?" If , then must not contain itself; that is, . If , then it must be because . Either way there is a contradiction. One resolution to Russell's paradox changes the language every so slightly: "Define to be the class of all sets which do not contain themselves. Is it true or not that ?" Indeed, for is not a set.
Compare Russell's Paradox to the Barber of Seville problem: "The barber of Seville shaves exactly those men who do not shave themselves. How can this be?" Naturally, the barber is a woman.
This article is a stub. Help us out by.