properties of well-ordered sets
The following properties are easy to see:
Every subset of a well-ordered set is well-ordered.
Now we define an important ingredient for understanding the structure of well-ordered sets.
Definition 1 (section).
Let be a well-ordered set. Then the mapping defined by is a bijective order morphism. In particular, is well-ordered.
Let and be well-ordered sets and a bijective order morphism. Then there exists a bijective order morphism such that for all
Let be a well-ordered set and such that there is an injective order morphism . Then .
Now is well-ordered, so the set defined by the elements of the sequence has a minimal element, that is and hence for some sufficiently large . Applying times to the latter equation yields , that is is the maximal element of , and thus of . ∎
Let and be well-ordered sets. Then there exists at most one bijective order morphism .
Let and be well-ordered sets such that for every section there is a bijective order morphism to a section and vice-versa, then there is a bijective order morphism .
Let and be well-ordered sets. Then there is an injective order morphism or . If cannot be chosen bijective, then it can at least be chosen such that its image is a section.
Let be the set of sections of from which there is an injective order morphism to . If is the empty set, then must be empty, since otherwise we could map the least element of to . If is not empty, we may consider the set . If , nothing remains to be shown. Otherwise the set is nonempty an hence has a least element . By construction, there is no injective order morphism from to , but there is an injective order morphism from for every element which is strictly smaller than . Now assume there is an element such that there is no injective order morphism from . Then we can similarly construct a least element for which there is no injective order morphism . Surely, is greater than all the elements from the images of the functions , but then there is a bijective order morphism from to by Theorem 5 which is a contradiction. Therefore, all sections of and itself map injectively and order-preserving to . ∎
Let be a well-ordered set and a nonempty subset. Then there is a bijective order morphism from to one of the sets in .
The set is well-ordered with respect to the order induced by . Assume a bijective order morphism as stated by the theorem does not exist. Then, by virtue of Theorem 6, there is an injective but not surjective order morphism whose image is a section . The element defines a section in which is identical to by Theorem 3. Thus is surjective which is a contradiction. ∎
- C G. Cantor, Beiträge zur Begründung der transfiniten Mengenlehre (Zweiter Artikel), Math. Ann. 49, 207–246 (1897).
|Title||properties of well-ordered sets|
|Date of creation||2013-03-22 15:23:42|
|Last modified on||2013-03-22 15:23:42|
|Last modified by||GrafZahl (9234)|