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. ?