*10. Infinity-to-one correspondence.* It might be inferred that any
infinite assemblage could be put into one-to-one correspondence with any
other. Such is not the case, however, if the correspondence is to be
continuous, between the points on a line and the points on a plane.
Consider two lines which lie in different planes, and take _m_ points on
one and _n_ points on the other. The number of lines joining the _m_
points of one to the _n_ points jof the other is clearly _mn_. If we
symbolize the totality of points on a line by [infinity], then a
reasonable symbol for the totality of lines drawn to cut two lines would
be [infinity]2. Clearly, for every point on one line there are [infinity]
lines cutting across the other, so that the correspondence might be called
[infinity]-to-one. Thus the assemblage of lines cutting across two lines
is of higher order than the assemblage of points on a line; and as we have
called the point-row an assemblage of the first order, the system of lines
cutting across two lines ought to be called of the second order.




*11. Infinitudes of different orders.* Now it is easy to set up a
one-to-one correspondence between the points in a plane and the system of
lines cutting across two lines which lie in different planes. In fact,
each line of the system of lines meets the plane in one point, and each
point in the plane determines one and only one line cutting across the two
given lines—namely, the line of intersection of the two planes determined
by the given point with each of the given lines. The assemblage of points
in the plane is thus of the same order as that of the lines cutting across
two lines which lie in different planes, and ought therefore to be spoken
of as of the second order. We express all these results as follows: