  Thus, Frege has a definition of precedes which applies to the ordered pairs \(\langle 0,1\rangle\), \(\langle 1,2\rangle \), \(\langle 2,3\rangle\), … . Frege then defined the ancestral of this relation, namely, \(x\) is an ancestor of \(y\) in the predecessor-series, or \(\mathit{Precedes}^*\). Though the exact definition will not be given here, we note that it has the following consequence: from the facts that \(10\) precedes \(11\) and \(11\) precedes \(12\), it follows that \(10\) precedes\(^*\) \(12\) in the predecessor-series. Note, however, that although \(10\) precedes\(^*\) \(12\), \(10\) does not precede \(12\), for the notion of precedes is that of immediately precedes. Note also that by defining the ancestral of the precedence relation, Frege had in effect defined \(x\) &lt; \(y\) relative to the predecessor-series.