If we start counting on from i, we never come back to i. This is the no-cycles rule or property of counting on.
This is connected to order.
If we count on from i to j by one or more then i is less than j.
If we including counting on by zero, then counting on from i to j shows that i is less than or equal to j.
Counting on is the basis of order. This again highlights that counting on by one and successor are the same.
Counting on encounters numbers along the way and these are less than the numbers encountered later in such a path.
The initial segments are found by counting on from 0.