 Cantor’s derived sets are of particular interest (for the context of this idea in real analysis, see e.g., Dauben 1979, Hallett 1984, Lavine 1994, Kanamori 1996, Ferreirós 1999). Cantor took as given the “conceptual sphere” of the real numbers, and he considered arbitrary subsets \(P\), which he called ‘point sets’. A real number \(r\) is called a limit point of \(P\), when all neighbourhoods of \(r\) contain points of \(P\). This can only happen if \(P\) is infinite. With that concept, due to Weierstrass, Cantor went on to define the derived set \(P'\) of \(P\), as the set of all the limit points of \(P\). In general \(P'\) may be infinite and have its own limit points (see Cantor’s paper in Ewald [1996, vol. 2, 840ff], esp. p. 848). Thus one can iterate the operation and obtain further derived sets \(P''\), \(P'''\)… \(P^{(n)}\) … It is easy to give examples of a set \(P\) that will give rise to non-empty derived sets \(P^{(n)}\) for all finite \(n\). (A rather trivial example is \(P = \mathbf{Q}_{[0,1]}\), the set of rational numbers in the unit interval; in this case \(P' = [0,1] = P''\).) Thus one can define \(P^{(\infty)}\) as the intersection of all \(P^{(n)}\) for finite \(n\). This was Cantor’s first encounter with transfinite iterations.