Iterating through an infinite ordered set

[Ookke]

If we want to do a specific operation for each element in a set in specific order, is there some limitations for that?

Here is an example that seems to lead a bit strange conclusion:

Let S be an infinite totally ordered set without maximal element and A an empty set to begin with.

For each element x in S, do the following operation:
If set A contains element larger than x, do nothing.
Else, select y > x and add it into A.
Do these operations in descending order, i.e. if x < y, process y before processing x.

Now it seems that each iteration for x makes sure that A will contain an element larger than x, yet no iteration will actually add anything to A: For each x being iterated, some y > x must have been iterated earlier, resulting that there already must be element larger than x in A. ?

 

[wofsy]

Right - it seems that the machine will be caught in an infinite loop looking for the largest element.

 

[g_edgar]

For "step-by-step" operations of this kind, the usual requirement on the ordered set would be "well ordered" (or in your case, reverse well ordered).