A calculus of the absurd

Chapter 7 Sets and numbers

7.1 Sets

Sometimes it is helpful to talk about collections of things. Some things have common attributes, which means that any reasoning we apply to one object with this attribute can be applied to any other object with these attributes. Sets are about abstraction - we can focus less on the individual particularities of different objects, and instead focus more on their commonalities.

7.1.1 Russell’s paradox

Suppose that we have a set \(R\) such that

\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!