We want to define successor chains from 2 to 5 or 9 to 17 or i to j. Will the following work or is something left out?
We consider two numbers i and j and we don’t yet have a definition of which is first. We consider container sets as follows.
- Contains i and j.
- Contains successor of each element except possibly of one element.
We don’t even say that i and j are one of the elements without a successor. Nor do we say it is the same element that has no successor from one container to the other.
We now attempt to define the chain as the intersection of all minimal containers. Does this work?
All the minimal containers contain i and j. So that is in the closure, even if i and j are the same.
Suppose i was 2 and j was 12 and a container set did not contain 4. For example all naturals except 4.
This set now has a single element in the set without a successor in the set. When we take intersections, 4 won’t be in the intersection. So this fails.
A container set must meet following two conditions.
- Contains i and j.
- Contains successor of each element in container except of j.
Now suppose i is 12 and j is 2. The container set will have to contain all elements from 12 on. It won’t contain 3 and it doesn’t have to contain 4 through 11.
So we will get as the set, 2 and the tail starting from and including 12. The set will fail to contain 11 as well as fail to contain 1 and 3.
We can use the above definition and then check the result and if it has these problems, we can reject it. Then we say to switch i and j.
Now what happens if i is 2 and j is 12? Then we know one set that contains this is 2 through 12 including the endpoints.
Since 2 is in the set, and the only element without a successor is 12, we know it will contain 2 through 12.
When we take intersections this will be common to every container set.
At this point, the set has only one of i and j without a predecessor.
When we did it the “wrong way” we got both i and j did not have a predecessor. That was after the intersections.
Can we put the condition in before the intersection? For example, that one of i and j has on predecessor in a container?
Do we require it is always the same one?
Can we come up with an indicator of which is first i and j?
Can we grow the set from i by induction? This is like growing initial segments from 0 but this time with an offset i.