, ...
, and so on; however far we have gone, we can always go further.
The series of all the ordinals that can be obtained in this way, i.e. all that can be obtained by thinning out a progression, is itself longer than any series that can be obtained by re-arranging the terms of a progression. (This is not difficult to prove.) The cardinal number of the class of such ordinals can be shown to be greater than
; it is the number which Cantor calls
. The ordinal number of the series of all ordinals that can be made out of an
, taken in order of magnitude, is called