Class

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

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