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

*102. ON CARDINAL NUMBERS OF ASSIGNED TYPES.

Summary of *102.
In this number, we shall consider a typically definite relation " Nc," i.e. we
shall consider the relation, to a class 8 which is given as of the same type as
/3, of the class J of those classes 7 which are similar to 8 and of the same
type as a. We shall then put
= Nc (a)'8,
7y Nc (a)'83,
7sm a, ) 8,
and the class of all such numbers as p for a given a and 8/ we shall call
NC (a), so that
NC9 (a) = D'Nc (ap).
The notations here introduced for giving typical (lefiniteness to "snm" and
"Nc" are those defined in *65 for any typically ambiguous relation.
By *63'0102. we have, if a is a typically ambiguous symbol,
-. a = a n t'z,
-. a (x) a r t't'x.
Thus F. a (x) = a,,. If we apply the definitions to 1, " lx" is meaningless
unless x is a class; we therefore write a Greek letter in place of x, and
we have. 1 - 1 n t 3= 1 (l' /v - t'/).
If x e,, we shall have t'x = /. v. t'x + 8. Hence
F:xe/3. D. t'l.
Similarly F: x e f... 'x 1e.
Thus F: e t,'8. )D. t'x 1,.
The converse implication also holds, so that
F: x 6 to3. - ~ t'x e 1.
Thus 1p consists of all unit classes whose sole members x either are or are
not members of /, i.e. for which " x e," is significant.
In "x e to'. ). t'x e 1s," the hypothesis renders explicit the condition of
significance; thus " t'xe 1" is always true when significant, and always
significant when e to'/l. On the interpretation of negative statements concerning types, see the note at the end of this number.
It should be noted that all the constant relations introduced in this work
are typically ambiguous. Consider e.g. A, sg, D, s, s, I, t, e, C1, R1. These