Volume 1 / Part 4 — *181. ON THE ADDITION OF UNITY TO A RELATION-NUMBER.
Reading edition
*181. ON THE ADDITION OF UNITY TO A RELATION-NUMBER.
Summary of *181.
The relation-number has, according to our definitions, no meaning in isolation, because our definitions are framed with a view to series, and a series cannot consist of one term. But we can add one term to a series; hence is required as an addendum. In order to get our definitions in the most manageable form, we first construct a relation, which we call , which is such that, whenever exists, has one more term in its field than ; the relation-number of this relation is then defined as . We add also a definition which is purely formal, and serves to minimize exceptions to the associative law of addition.
The definitions are closely analogous to those of *180. We put with a similar definition for . and may be of any relative types, and we have always We put with a similar definition for . We also introduce definitions analogous to *180·03·031.
The principal propositions of this number are
*181·3.
*181·31.
*181·32.
*181·33.
*181·4.
*181·42.
The following propositions are formally forms of the associative law, but they need separate proof on account of the peculiarity of .
*181·54.
*181·56.
*181·58.
*181·59.
The hypotheses in the above propositions are essential.
*181·6.
*181·62.
These propositions give the connection with cardinals.
*181·01.
*181·011.
*181·02.
*181·021.
*181·03.
*181·031.
*181·04.
Propositions concerning are omitted in what follows, since they are proved exactly as the analogous propositions concerning are proved.
*181·1.
*181·11.
Dem.
*181·12.
*181·13.
Dem.
*181·2.
*181·21.
*181*22.
Dem.
*181·24.
*181·3.
*181·31.
*181·32.
*181·33.
The above proposition is used in *253·23·571.
*181·4.
*181·42.
Dem.
*181·43.
The following propositions are concerned with the associative law when is one of the addenda.
*181·53.
Dem.
*181·54.
Dem.
*181·55.
*181·56.
Dem.
The last line in the above proof, in which *24·1 is used, is legitimate because and may be of any type whatever, and therefore the fact that is sufficient to establish in the sense wanted.
*181·561.
This definition adopts the opposite convention to that usually adopted. But it is convenient to have , and also to have as much similarity as possible between the results of adding at the beginning and end of a relation. Both reasons lead to the adoption of the above convention. (Cf. *181·57·571, below.)
*181·57.
*181·571.
*181·58.
The proof proceeds in the same way as that of *181·54.
*181·59.
The above propositions show that, except when one of the summands is zero, the associative law holds for just as if it were a relation-number.
The following propositions are concerned with relations to cardinal addition.
*181·6.
Dem.
*181·61.
*181·62.
Dem.