. But there is no theoretical reason why a method of selection such as this should always be possible, and the case of the socks, with a little goodwill on the part of the reader, may serve to show how a selection might be impossible.
It is to be observed that, if it were impossible to select one out of each pair of socks, it would follow that the socks could not be arranged in a progression, and therefore that there were not
of them. This case illustrates that, if
is an infinite number, one set of
pairs may not contain the same number of terms as another set of
pairs; for, given