Principia mathematica, vol. 1 (of 3)

The following example may illustrate the scope of *95·65. Let , be the generating relations of two well-ordered series, neither of which has a last term. Put . Then is the relation of immediately preceding in the -series, and is the relation of immediately preceding in the -series. We shall have Also, except in certain exceptional cases, , are the first derivatives of the two series (including the first terms of the two series). states that, starting from any term of the series and going backwards, a finite number of steps will bring us to a member of the first derivative, which is true. Hence, by *95·65, neglecting certain exceptional cases, we arrive at the result that if the first derivatives of two well-ordered series have the same cardinal number of terms, then the series themselves have the same cardinal number of terms. This proposition can of course be proved otherwise; the above is merely mentioned as an illustration of the results of *95·65.