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In [[set theory]], a class is essentially a set which we do not call a set for logical or semantic reasons. For example, if <math>A=\{a,b,c\}</math> is a set, then <math>B=\{\{a\},\{b\},\{b,c\}\}</math> is a class consisting of some subsets of <math>A</math>. In this example though, <math>B</math> can also be called a set.

To understand why one would make such a distinction, consider Russell's Paradox:
"Define <math>T</math> to be the set of all sets which do not contain themselves. Is it true or not that <math>T\in T</math>?" If <math>T\in T</math>, then <math>T</math> must not contain itself; that is, <math>T\not\in T</math>. If <math>T\not\in T</math>, then it must be because <math>T \in T</math>. Either way there is a contradiction. One resolution to Russell's paradox changes the language every so slightly: "Define <math>T</math> to be the class of all sets which do not contain themselves. Is it true or not that <math>T\in T</math>?" Indeed, <math>T\not\in T</math> for <math>T</math> 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.

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[[Category:Mathematics]]