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

*210. ON SERIES OF CLASSES GENERATED BY THE RELATION
OF INCLUSION.
Summary of *210.
In the theory of series it frequently happens that we have to deal with
a class of classes such that, of any two, one is contained in the other. I.e.
if K is the class of classes, we have
a,,/3 e. )a a C, 3. v. 3 C a.
Instances of this are afforded by the various classes of sections, to be
considered in *211. When K fulfils the above condition, the classes composing K can be arranged in a series by the relation of inclusion (combined
with inequality), i.e. by the relation
a/ (a, 38 e K.aC,3.a $3),
or, what comes to the same,
a/ (a,e K.!1 3-a).
If P is any relation such that K C Cl''P, the above relation of inclusion is
equal to
-Plc K.
(For the definition of PIl, see *170.) Thus under the above circumstances,
PIC K is a series, whatever P may be.
The importance of such relations of inclusion, as generators of series, is in
connection with the existence of maxima and minima or limits. If we put
Q = a/3 (a,,/36 e K.! /3-a),
where K satisfies the above condition, then if X C K, and if s'X e c, s'X is the
maximum or the upper limit of X with respect to Q, according as s'X is a
member of X or not. Similarly if p'X e Kc, p'X is the minimum or lower limit
of X, according as p'X is a member of X or not. Hence if K is such that the
sum of any sub-class of K is a member of K, every sub-class of K has either
a maximum or an upper limit; and if the product of every sub-class of K is
a member of K, every sub-class of K has either a minimum or a lower limit.
In order that every sub-class of K should have a minimum or a lower
limit, it is sufficient that the sum of every sub-class of K should be a member