A calculus of the absurd
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17.2 Russell’s paradox
Suppose that we have a set \(R\) such that
\(\seteqnumber{0}{17.}{17}\)\begin{equation} R = \{ A : A \notin A \} \end{equation}
That is, “\(R\) 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!