In order to make matters clear, it will be best to begin with examples. Let us first consider various different kinds of series which can be made out of the inductive numbers arranged on various plans. We start with the series
which, as we have already seen, represents the smallest of infinite serial numbers, the sort that Cantor calls
. Let us proceed to thin out this series by repeatedly performing the operation of removing to the end the first even number that occurs. We thus obtain in succession the various series:
and so on. If we imagine this process carried on as long as possible, we finally reach the series
in which we have first all the odd numbers and then all the even numbers.
The serial numbers of these various series are