*4. One-to-one correspondence and enumeration.* If a one-to-one
correspondence has been set up between the objects of one set and the
objects of another set, then the inference may usually be drawn that they
have the same number of elements. If, however, there is an infinite number
of individuals in each of the two sets, the notion of counting is
necessarily ruled out. It may be possible, nevertheless, to set up a
one-to-one correspondence between the elements of two sets even when the
number is infinite. Thus, it is easy to set up such a correspondence
between the points of a line an inch long and the points of a line two
inches long. For let the lines (Fig. 1) be _AB_ and _A’B’_. Join _AA’_ and
_BB’_, and let these joining lines meet in _S_. For every point _C_ on
_AB_ a point _C’_ may be found on _A’B’_ by joining _C_ to _S_ and noting
the point _C’_ where _CS_ meets _A’B’_. Similarly, a point _C_ may be
found on _AB_ for any point _C’_ on _A’B’_. The correspondence is clearly
one-to-one, but it would be absurd to infer from this that there were just
as many points on _AB_ as on _A’B’_. In fact, it would be just as
reasonable to infer that there were twice as many points on _A’B’_ as on
_AB_. For if we bend _A’B’_ into a circle with center at _S_ (Fig. 2), we
see that for every point _C_ on _AB_ there are two points on _A’B’_. Thus
it is seen that the notion of one-to-one correspondence is more extensive
than the notion of counting, and includes the notion of counting only when
applied to finite assemblages.




*5. Correspondence between a part and the whole of an infinite
assemblage.* In the discussion of the last paragraph the remarkable fact