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

294

CARDINAL ARITHMETICC

[PART III

The Arithmetical Set consists of, +c v, a x v, pY, /V, v. These formal
numbers are only interesting when p and v are also members of NC. Also
u/ and v may be complex symbols, so long as one of them at least involves a
variable. For example 23+cY is a formal number, and so is a 4c (3 +c v).
The Primary and Argumental and Arithmetical Sets of Formal Numbers
are derived from the corresponding sets of variable formal numbers, by
adding to them the constant formal numbers obtained by substituting
constants for the variables occurring in the expressions for the members
of the variable set in question.
In the formal numbers of the arithmetical set as written above,,/ and v
are called the first components. Thus every formal number of this set has
two first components. The first components (if any) of the first components
are also called components of the original formal number, and so on; so that
components of components are components of the original symbol.
A formal number of the arithmetical set, whose components are all formal
numbers, either constant or variable but not belonging to the argumental set,
is called a pure arithmetical formal number. These are the formal numbers
which it is important in arithmetic to secure from assuming the value A
owing to lowness of type.
The logical investigation of *100 and *101, where typically ambiguous
formal numbers are used, is directly concerned in investigating the premisses
necessary to secure various propositions from fluctuating truth-values owing
to the intrusion of null-values among the cardinals. The convention, necessary
to avoid determinations of type which we never wish to consider, is as follows,
where the terms used are explained fully in the prefatory statement:
IT. Argumental occurrences are bound to logical and attributive occurrences; and, if there are no argumental occurrences, equational occurrences
are bound to logical occurrences. This rule only applies so far as meaning
permits after the assignment of types to the real variables.
In *110, *113, *116, *119 we consider the arithmetical operations of
addition, multiplication, exponentiation, and subtraction. Also in *117 we
consider the comparison of cardinal numbers in respect to the relation of
greater and less.
There is no interest in complicating our theorems by allowing for the
cases when a pure arithmetical formal number, whose components are
ambiguous as to type, becomes equal to A owing to the low type of one of
its components. Also in the theory of greater and less the possibility of
null-values in low types has no real interest. Accordingly these are excluded
from any consideration by the definitions
*110'03'04, *113'04'05, *116-03'04, *117'02-03,