constructing well ordered sets
A set well-orderable if there is a well-ordering on . For example, the empty set , is well-orderable, the well-ordering itself is . Similarly, any singleton has as the unique well-ordering, and is therefore well-orderable. More generally, every finite set is well-orderable. To construct a well-ordering on a finite set, pick any element from the set and designate the element to be the smallest element of the set. Then, pick an element from the remaining collection, and make this element the next smallest element. Keep doing this until every element has been picked and assigned. Obviously, this process will not go on forever. If there are elements in the set, there are well-orderings on the set (since every well-ordering is a linear ordering).
By the axiom of choice, every set is well-orderable. However, this does not provide us with an explicit way to construct the well-order. For example, how does one well-order , the set of real numbers? Nevertheless, given two well-ordered sets , one can explicitly construct well-orderings on the following sets, without AC.
We may assume , since , and any subset of an well-ordered set is well-ordered. Define a binary relation on as follows: for ,
So is a (strict) linear ordering on . Furthermore, extends both and . If , then the least element of with respect to is the least element of with respect to if , or the least element of with respect to , if . Hence is a well-ordered set. ∎
We may assume that are both non-empty, since the empty set is well-orderable. Define on as follows:
Again, it is easy to see that is a strict linear ordering on . Next, let . Define . Let be the least element of . Define . Let be the least element of . Then . For any distinct from , since , either or . In the first case, . In the second case, , either or . In the first case, , and the second case is impossible for otherwise . Therefore is a well-ordering on . ∎
The ordering constructed in Proposition 2 is called the Hebrew lexicographic ordering. It is similar to the usual lexicographic ordering, except that the second coordinates are ordered first before the first coordinates. The reason for choosing this ordering is to make the ordering on compatible with the ordering on , the disjoint union of with itself. Had we chosen the lexicographic ordering instead, on, say, , then every initial segment of it would be finite, whereas the initial segment of on is infinite.
One would hope that for well-ordered sets , the set is well-orderable without resorting to AC. However, this is not possible, and the best one can do is the following: if is linearly ordered, and well-ordered, then one can linearly order without AC:
Let . For , let . If , it has a least element . Define on as follows, for any :
If , then so that or . To show that is transitive, suppose and . Let and . Then:
if , then , which implies and .
if , then , which implies and .
In either case, . Now, suppose . Then . As a result, . ∎
|Title||constructing well ordered sets|
|Date of creation||2013-03-22 18:49:12|
|Last modified on||2013-03-22 18:49:12|
|Last modified by||CWoo (3771)|
|Defines||Hebrew lexicographic ordering|