Principia mathematica, by Alfred North Whitehead ... and Bertrand Russell.

*103. HOMOGENEOUS CARDINALS.

Summary of *103.
In this number, we shall consider cardinals generated by a homogeneous
relation of similarity. A "homogeneous" cardinal is to mean all the classes
similar to some class a and of the same type as a. The "homogeneous
cardinal of a" will be defined as Nc'a n t'a; we shall denote it by " Noc'a."
Then the class of homogeneous cardinals is the class of all such cardinals
as "N0c'a," i.e. it is D'N0c; this we shall denote by "NoC." The symbol
"Noc'a" is typically definite as soon as a is assigned; "NoC," on the contrary, is typically ambiguous: it must be a Cls3, but otherwise its type may
vary indefinitely. Homogeneous cardinals have, however, many properties
which do not require that the ambiguity of " NoC" should be determined,
and few which do require this. They are important also as being the
simplest kind of cardinals, and as being a kind to which other kinds can
usually be reduced.
The chief advantage of homogeneous cardinals is that they are never null
(*103'13'22). This enables us to avoid by their means the explicit exclusion
of exceptional cases; thus throughout Section B we shall use homogeneous
cardinals in defining the arithmetical operations: the arithmetical sum of
Nc'a and Nc',, for example, will be defined by means of Noc'a and Noc'/, in
order to exclude such a determination of the typical ambiguity of Nc'a and
Nc', as would make either of them null. It is true that not only homogeneous cardinals, but also ascending cardinals (cf. 104), are never null. But
homogeneous cardinals are much the simplest kind of cardinals that are never
null, and are therefore the most convenient.
The fact that no homogeneous cardinal is null is derived from
*103-12. F. a e N0c'a
Other important propositions in this number are the following:
*103-2.: pe e NoC. -. (3a). p = Nc'a n t'a.. ([a). / = Noc'a
*103'26. F:. e NC. ): a e p.. No'a =
The above proposition is used constantly.