Volume 1 / Part 4 — *171. THE PRINCIPLE OF FIRST DIFFERENCES (continued).
Reading edition
*171. THE PRINCIPLE OF FIRST DIFFERENCES (continued).
Summary of *171.
In this number, we shall consider a more restricted form of the principle of first differences, which is applicable when there is a definite first member of one class not belonging to the other class. In this case, if is the first differing member, the part of which precedes is to be the same as the part of which precedes . If belongs to and not to , we put before ; in the converse case, we put before . In case , itself is not to be counted among its own predecessors; thus the predecessors of are to be . Hence the relation in question will hold between two sub-classes ( and ) of when there is a such that or, what comes to the same thing (owing to ), This relation between and we denote by "," where "" stands for "difference."
Thus our definition is
On the analogy of , we put also
When is well-ordered, and coincide respectively with and . Their properties are closely analogous to those of and . Thus e.g. the following propositions remain true when is substituted for :
*170·17·35·36·37·38·44·5·51·52·64·67·68·69.
The only new propositions to be noted in this number are
*171·2.
*171·21.
and the following formulae suggesting an inductive identification of and in cases to which such induction is applicable:
*171·7.
*171·71.
These propositions are however superseded (at a later stage) by the proof that and coincide if is well-ordered (*251·37).
The chief property of is that its relation-number is to the power . This will be proved in *177 and *186·4
*171·01.
*171·02.
*171·1.
*171·101.
*171·102.
*171*11.
*171·12.
*171·13.
*171·14.
Dem.
*171·15.
Dem.
*171·16.
Dem.
*171·17.
Dem.
*171·18.
Dem.
*171·19.
Dem.
*171·2.
Dem.
*171·21.
Dem.
*171·22.
*171·4.
*171·41.
*171·42.
*171·43.
*171·44.
*171·5.
Dem.
*171·51.
Dem.
*171·52.
*171·64.
The proof proceeds by the same stages as the proof of *170·64.
*171·67.
*171·68.
*171·69.
*171·7.
*171·71.