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 was brought to light that it is sometimes possible to set the elements of an assemblage into one-to-one correspondence with a part of those elements.  A moment’s reflection will convince one that this is never possible when there is a finite number of elements in the assemblage.—­Indeed, we may take this property as our definition of an infinite assemblage, and say that an infinite assemblage is one that may be put into one-to-one correspondence with part of itself.  This has the advantage of being a positive definition, as opposed to the usual negative definition of an infinite assemblage as one that cannot be counted.

6.  Infinitely distant point. We have illustrated above a simple method of setting the points of two lines into one-to-one correspondence.  The same illustration will serve also to show how it is possible to set the points on a line into one-to-one correspondence with the lines through a point.  Thus, for any point C on the line AB there is a line SC through S.  We must assume the line AB extended indefinitely in both directions, however, if we are to have a point on it for every line through S; and even with