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re 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