which is itself a progression; but a couple of progressions gives a series which is twice as long as a progression. It is therefore necessary to distinguish between and . Usage is variable; we shall use for a couple of progressions and for a progression of couples, and this decision of course governs our general interpretation of "" when and are relation-numbers: "" will have to stand for a suitably constructed sum of a relations each having terms. We can proceed indefinitely with the process of thinning out the inductive numbers. For example, we can place first the odd numbers, then their doubles, then the doubles of these, and so on. We thus obtain the series of which the number is , since it is a progression of progressions. Any one of the progressions in this new series can of course be [Pg 91] thinned out as we thinned out our original progression. We can proceed to , , ... , 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 . Thus a series whose ordinal number is has a field whose cardinal number is . We can proceed from and to and by a process exactly analogous to that by which we advanced from and to and . And there is nothing to prevent us from advancing indefinitely in this way to new cardinals and new ordinals. It is not known whether is equal to any of the cardinals in the series of Alephs. It is not even known whether it is comparable with them in magnitude; for aught we know, it may be neither equal to nor greater nor less than any one of the Alephs. This question is connected with the multiplicative axiom, of which we shall treat later. All the series we have been considering so far in this chapter have been what is called "well-ordered." A well-ordered series is one which has a beginning, and has consecutive terms, and has a term next after any selection of its terms, provided there are any terms after the selection. This excludes, on the one hand, compact series, in which there are terms between any two, and on the other hand series which have no beginning, or in which there are subordinate parts having no beginning. The series of negative integers in order of magnitude, having no beginning, but ending with -1, is not well-ordered; but taken in the reverse order, beginning with -1, it is well-ordered, being in fact a progression. The definition is: [Pg 92] A "well-ordered" series is one in which every sub-class (except, of course, the null-class) has a first term.