Further, the process of co-ordination can be extended to a third and a fourth group and so on, with the result that what has been done in one of the co-ordinated groups must happen in all. If hereby the third group is co-ordinated with the second, the effects are quite the same as if it were co-ordinated directly with the first instead of indirectly through the second. And the same is true for the fourth and the fifth groups, etc. Thus, co-ordination can be extended to any number of groups we please, and each single group proves to be co-ordinated with every other.
Finally, a group can be co-ordinated with itself, each of its members corresponding to a certain definite other member. It is not impossible that individual members should correspond to themselves, in which case the group has double members, or double points. The limit-case is identity, in which every member corresponds to itself. This last case cannot supply any special knowledge in itself, but may be applied profitably to throw light on those observations for which it represents the extreme possibility.
28. Comparison.
If we have two groups A and B, and if we co-ordinate their members severally, three cases may arise. Either group A is exhausted while there are members remaining in B, or B is exhausted before A, or, finally, both groups allow of a mutual co-ordination of all their members. In the first case A is called, in the broader sense of the word, smaller than B, in the second B is called smaller than A, in the third the two groups are said to be of equal magnitude. The expression, "B is greater than A," is equivalent to the expression, "A is smaller than B," and inversely.
It is to be noted that the relations mentioned above are true, whether the members are considered as individually different from one another or whether the difference of the members is disregarded, and they are treated as alike. This comes from the fact that every definite co-ordination of a group can be translated into every other possible co-ordination by exchanging two members at a time in pairs. Since in this process one member is each time substituted for another, and a gap therefore can never occur in its place, the group in the new arrangement can be co-ordinated with the other group as successfully as in the old arrangement. At the same time we learn from this that in every co-ordination of a group with itself, independently of the arrangement of its members, it must prove equal to itself.
By carrying out the co-ordination proof is further supplied of the following propositions:
| { | greater than | } | |
| If group A is | equal to | group B | |
| smaller than | |||
| { | greater than | } | |
| and group B is | equal to | group C | |
| smaller than | |||
| { | greater than | } | |
| then group A is | equal to | group C | |
| smaller than |
From this it follows that any collection of finite groups whatsoever, of which no one is equal to the other, can always be so arranged that the series should begin with the smallest and end with the greatest, and that a larger should always follow a smaller. This order would be unequivocal, that is, there is only one series of the given groups which has this peculiarity. As we shall soon see, the series of integers is the purest type of a series so arranged.
In comparing two infinitely large groups by co-ordination, it may be said on the one hand that never will one group be exhausted while the other still contains members. Accordingly, it is possible to designate two unlimited or infinite groups (or as many such groups as we please) as equal to each other. On the other hand, the statement that in both groups each member of the one is co-ordinated with a member of the other has no definite meaning on account of the infinitely large number of members. The definition of equality is therefore not completely fulfilled, and we must not loosely apply a principle valid for finite groups to infinite groups. This consideration, which may assume very different forms according to circumstances, explains the "paradoxes of the infinite," that is, the contradictions which arise when concepts of a definite content are applied to cases possessing in part a different content. If we wish to attempt such an application, we must in each instance make a special investigation as to the manner in which the relations on their part change by the change of those contents (or premises). As a general rule we must expect that the former relations will not remain valid in these circumstances without any change at all.