*1. Definition of one-to-one correspondence.* Given any two sets of
individuals, if it is possible to set up such a correspondence between the
two sets that to any individual in one set corresponds one and only one
individual in the other, then the two sets are said to be in _one-to-one
correspondence_ with each other. This notion, simple as it is, is of
fundamental importance in all branches of science. The process of counting
is nothing but a setting up of a one-to-one correspondence between the
objects to be counted and certain words, ’one,’ ’two,’ ’three,’ etc., in
the mind. Many savage peoples have discovered no better method of counting
than by setting up a one-to-one correspondence between the objects to be
counted and their fingers. The scientist who busies himself with naming
and classifying the objects of nature is only setting up a one-to-one
correspondence between the objects and certain words which serve, not as a
means of counting the objects, but of listing them in a convenient way.
Thus he may be able to marshal and array his material in such a way as to
bring to light relations that may exist between the objects themselves.
Indeed, the whole notion of language springs from this idea of one-to-one
correspondence.




*2. Consequences of one-to-one correspondence.* The most useful and
interesting problem that may arise in connection with any one-to-one
correspondence is to determine just what relations existing between the
individuals of one assemblage may be carried over to another assemblage in
one-to-one correspondence with it. It is a favorite error to assume that