Volume 1 / Part 4 — *161. ADDITION OF A TERM TO A RELATION.
Reading edition
*161. ADDITION OF A TERM TO A RELATION.
Summary of *161.
The addition of a term has two forms, according as it occurs at the beginning or end of the field of the relation in question. If we add first and then at the end, the result is the same as if we added (*161·22); if at the beginning, it is the same as if we added (*161·221). The propositions of the present number are all obvious, and offer no difficulties of any kind. As explained in the introduction to this section, we put Most of the propositions of this number require the hypothesis , because if , (*161·2·201). This is connected with the fact that there is no ordinal number 1. Apart from propositions already mentioned, the chief propositions of this number are the following (we omit propositions about when they are merely analogues of propositions about ):
*161·12.
*161·14.
*161·15.
*161·211.
*161·31.
*161·4.
*161·01.
*161·02.
*161·1.
*161·101.
*161·11.
*161·111.
*161·12.
*161·13.
Dem.
*161·131.
*161·14.
The hypothesis is necessary in this proposition, since without it we have .
*161·141.
*161·15.
*161·16.
The above proposition is used in the theory of connected relations (*202·412).
*161·161.
The two following propositions are frequently used.
*161·2.
*161·201.
*161·21.
Dem.
Note that is the relation which orders and and in the order , , .
*161·211.
*161·212.
*161·213.
These definitions serve merely for the avoidance of brackets.
*161·22.
Dem.
*161·221.
*161·23.
Dem.
*161·231.
*161·232.
Dem.
*161·24.
Dem.
*161·25.
Dem.
*161·26.
Dem.
The following propositions lead up to *161·33.
*161·3.
Dem.
*161·301.
*161·31.
Dem.
*161·32.
Dem.
*161·321.
*161·33.
The above proposition justifies addition of 1 or subtraction of 1 in ordinal arithmetic.
The following proposition (*161·4) is much used.
*161·4.
Dem.
*161·41.
*161·42.
*161·43.