Iterating through an infinite ordered set

In summary, the conversation is discussing the limitations of performing a specific operation for each element in a set in a specific order. An example is given where the set is infinite and without a maximal element, and the operation is to add elements to an empty set in descending order. It is concluded that this type of operation requires the set to be "well ordered" or "reverse well ordered".
  • #1
Ookke
172
0
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. ?
 
Physics news on Phys.org
  • #2
Right - it seems that the machine will be caught in an infinite loop looking for the largest element.
 
  • #3
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).
 

FAQ: Iterating through an infinite ordered set

What is an infinite ordered set?

An infinite ordered set is a collection of items that are arranged in a specific order and can continue infinitely without reaching an end. This means that there is no largest or smallest element in the set and it can continue on forever.

How is iteration through an infinite ordered set possible?

Iteration through an infinite ordered set is possible using mathematical techniques such as limits and approximations. These techniques allow us to approach the elements of the set in a systematic way without reaching an actual endpoint.

What is the difference between iterating through a finite and infinite ordered set?

The main difference between iterating through a finite and infinite ordered set is that with a finite set, we can reach an end point and know exactly how many elements there are. With an infinite set, there is no end point and the number of elements is infinite.

Can we ever reach the end of an infinite ordered set?

No, it is not possible to reach the end of an infinite ordered set. As the name suggests, an infinite ordered set has an infinite number of elements, meaning there is always one more element to iterate through.

What is the purpose of iterating through an infinite ordered set?

The purpose of iterating through an infinite ordered set is to analyze and understand the behavior of the set as a whole. By examining the elements of the set in a systematic way, we can make conclusions and predictions about the set's properties and patterns.

Similar threads

Replies
5
Views
2K
Replies
6
Views
1K
Replies
1
Views
1K
Replies
14
Views
2K
Replies
14
Views
3K
Replies
3
Views
2K
Back
Top