20.2 Russell’s paradox

Suppose that we have a set $R$ such that

That is, \say$R$ is the set of objects which are not elements of themselves.

However, is $R\in R$? Well, if $R$ is in $R$, then $R$ (by definition of $R$) is not in $R$. If $R$ is not in $R$, then $R$ (by definition of $R$) is in $R$. This is a paradox - it cannot be true.

The solution to this paradox is to be very careful when defining sets - we cannot define sets based on arbitrary criteria; we must build them out of other, well-defined and pre-existing sets!