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

192

CARDINAL ARITHMETIC

[PART III

In *122 we shall deal with progressions, i.e. with series of the type of
the series of natural numbers. In this number, we shall deal with such
series as generated by one-one relations; they will be dealt with at a later
stage (*263) as generated by transitive relations. We define a progression
as a one-one relation whose domain is the posterity of its first term, i.e.
Prog = (1 -+ 1) n R (D'R = R*B'R) Df.
According to this definition, there must be a first term B'R; (P'R will be
v t- C- 4 -R"R*,'B'R, i.e. Rpo'B'R, which is contained in R*'B'R, i.e. in D'R; since
('R C D'R, every term of the field of R has a successor, so that there is no
end to the series; since C(R = D'R = R'B'R, every term of the series can
be reached from the beginning by successive steps. These characteristics
suffice to define progressions.
In *123 we proceed to the definition and discussion of r,, the smallest
of reflexive cardinals. This is the cardinal number of any class whose terms
can be arranged in a progression; hence it is the class of domains of progressions, i.e. we may put
K= D"Prog Df.
With this definition, remembering that A is a cardinal, we can prove that
N, is a cardinal; but to prove that K, is an existent cardinal, we need the
axiom of infinity. The existence-theorem for Ko is then derived from the
inductive cardinals, which, if no one of them is null, form a progression
when arranged in order of magnitude. It should be observed that this
existence-theorem is for a higher type than that for which the axiom of
infinity is assumed. In order to get an existence-theorem for the same type,
we need the multiplicative axiom as well.
After a number on reflexive classes and cardinals (*124) and a number on
the axiom of infinity (*125), the Section ends with a number (*126) on
"typically indefinite inductive cardinals." The constant inductive cardinals
are the typically ambiguous symbols 0, 1, 2,...; thus we want to define the
class of inductive cardinals in such a way that a variable member of the class
shall be typically ambiguous. This is not possible without a sacrifice of
rigour, but in *126 it is shown how to minimize the sacrifice of rigour, and
how to obviate the resulting logical dangers. A variable whose values are
typically ambiguous is said to be " typically indefinite."
A proof that all inductive cardinals exist has often been derived from
*120'57 (below). But according to the doctrine of types, this proof is invalid,
since " p +o 1" in *120'57 is necessarily of higher type than " A."