Of the usual formal laws of addition, multiplication, and exponentiation, all are obeyed by transfinite cardinals, but only some are obeyed by transfinite ordinals, and those that are obeyed by them are obeyed by all relation-numbers. By the "usual formal laws" we mean the following:—
I. The commutative law:
II. The associative law:
III. The distributive law:
When the commutative law does not hold, the above form of the distributive law must be distinguished from
As we shall see immediately, one form may be true and the other false.