Volume 1 / Part 4 — *180. THE SUM OF TWO RELATION-NUMBERS.
Reading edition
*180. THE SUM OF TWO RELATION-NUMBERS.
Summary of *180.
In order to define the sum of two relation-numbers, we proceed (as in *110) to construct a relation whose relation-number shall be the required sum. For this purpose, we put This definition has the following merits: (1) whatever may be the types of and , is of the same type as ; (2) however the fields of and may overlap, and even if , the fields of and are mutually exclusive; (3) these two relations are respectively similar to and . Hence it is evident that, without placing any restriction upon and , we may take the relation-number of as defining the sum of the relation-numbers of and . Hence we put From this definition it follows that is null unless and are homogeneous relation-numbers, but that if they are the homogeneous relation-numbers of and , then is the relation-number of .
In order to be able to deal with typically ambiguous relation-numbers, we put, as in *110,
The principal propositions of the present number are
*180·111.
*180·3.
*180·31.
This proposition is essential, since otherwise would not be a function of and , but would depend upon the particular and .
*180·32.
*180·4.
*180·42.
*180·56.
which is the associative law.
*180·61.
*180·71.
This proposition gives the connection of ordinal and cardinal addition. It should be observed that, in virtue of *154·9, and are cardinals when and are relation-numbers.
*180·01.
*180·02.
*180·03.
*180·031.
On the purpose of the definitions *180·03·031, see the remarks on the corresponding definitions in *110 and of the Prefatory Statement.
*180·1.
*180·101.
*180·11.
*180·111.
Dem.
*180·12.
*180·13.
Dem.
*180·14.
*180·15.
Dem.
*180·151.
Dem.
*180·152.
*180·16.
*180·2.
*180·201.
*180·202.
Dem.
In the following propositions proofs are omitted, since they are exactly analogous to proofs of propositions in *110 whose numbers have the same decimal part.
*180·21.
*180·211.
*180·212.
*180·22.
*180·24.
*180·3.
*180·31.
*180·32.
*180·4.
*180·42.
*180·43.
*180·53.
Dem.
*180·531.
*180·54.
*180·541.
*180·55.
*180·551.
*180·56.
*180·561.
*180*57.
*180·6.
Observe that is an equation depending upon the peculiar properties of . We do not in general have unless and are finite ordinals.
*180·61.
*180·62.
*180·64.
*180·642.
Note that , which will be defined in *181, is , not .
The following propositions, being concerned with the relations of relation-numbers and cardinal numbers, have no analogues in *110.
*180·7.
Dem.
*180·71.
Dem.