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.
Let be a partially ordered set, and let be a well-ordering on . We define the function by transfinite recursion over so that
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 |
Canonical name | EquivalenceOfHausdorffsMaximumPrincipleZornsLemmaAndTheWellorderingTheorem |
Date of creation | 2013-03-22 13:04:45 |
Last modified on | 2013-03-22 13:04:45 |
Owner | mathcam (2727) |
Last modified by | mathcam (2727) |
Numerical id | 9 |
Author | mathcam (2727) |
Entry type | Proof |
Classification | msc 03E25 |
Synonym | proof ofZorn’s lemma |
Synonym | proof of Hausdorff’s maximum principle |
Synonym | proof of the maximum principle |
Related topic | ZornsLemma |
Related topic | AxiomOfChoice |
Related topic | ZermelosWellOrderingTheorem |
Related topic | HaudorffsMaximumPrinciple |