equivalence of Hausdorff’s maximum principle, Zorn’s lemma and the well-ordering theorem
Hausdorff’s maximum principle implies Zorn’s lemma.
Consider a partially ordered set , where every chain has an upper bound. According to the maximum principle there exists a maximal totally ordered subset . This then has an upper bound, . If is not the largest element in then would be a totally ordered set in which would be properly contained, contradicting the definition. Thus is a maximal element in .
Zorn’s lemma implies the well-ordering theorem.
Let be any non-empty set, and let be the collection of pairs , where and is a well-ordering on . Define a relation , on so that for all iff equals an initial of . It is easy to see that this defines a partial order relation on (it inherits reflexibility, anti symmetry and transitivity from one set being an initial and thus a subset of the other).
For each chain , define where R is the union of all the sets for all , and is the union of all the relations for all . It follows that is an upper bound for in .
According to Zorn’s lemma, now has a maximal element, . We postulate that contains all members of , for if this were not true we could for any construct where and is extended so . Clearly then defines a well-order on , and would be larger than contrary to the definition.
Since contains all the members of and is a well-ordering of , it is also a well-ordering on as required.
The well-ordering theorem implies Hausdorff’s maximum principle.
It follows that is a maximal totally ordered subset of as required.
|Title||equivalence of Hausdorff’s maximum principle, Zorn’s lemma and the well-ordering theorem|
|Date of creation||2013-03-22 13:04:45|
|Last modified on||2013-03-22 13:04:45|
|Last modified by||mathcam (2727)|
|Synonym||proof ofZorn’s lemma|
|Synonym||proof of Hausdorff’s maximum principle|
|Synonym||proof of the maximum principle|