  In order to define the concept of natural number, Frege first defined, for every \(2\)-place relation \(R\), the general concept ‘\(x\) is an ancestor of \(y\) in the \(R\)-series’. This new relation is called ‘the ancestral of the relation \(R\)’. The ancestral of the relation \(R\) was first defined in Frege’s Begriffsschrift (1879, §26, Proposition 76; 1884, §79). The intuitive idea is easily grasped if we consider the relation \(x\) is the father of \(y\). Suppose that \(a\) is the father of \(b\), that \(b\) is the father of \(c\), and that \(c\) is the father of \(d\). Then Frege’s definition of ‘\(x\) is an ancestor of \(y\) in the fatherhood-series’ ensures that \(a\) is an ancestor of \(b\), \(c\), and \(d\) in this series, that \(b\) is an ancestor of \(c\) and \(d\) in this series, and that \(c\) is an ancestor of \(d\) in this series.