Why are we not all called Smith?

In Section 15.1 we saw that the question of whether the Turing bombes would work depended on whether an effectively random process in which ‘mother statements’ give rise to a certain random number of ‘daughter statements’ which in turn give rise to ‘granddaughter statements’ and so on would die away quickly or give rise to a family containing all (or almost all) possible statements. The bombe problem is complicated by several special features and, though I hope that the discussion of the next two sections will shed some light on it, I shall concentrate on simpler but related problems.

Sir Thomas Browne wrote ‘Generations pass while some trees stand and old families last not three oaks’. (The quotation is taken from Kendall's fascinating papers [114] and [115].) Less concisely, Galton wrote: