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 setMathworldPlanetmath X, where every chain has an upper bound. According to the maximum principle there exists a maximal totally orderedPlanetmathPlanetmath subset YX. This then has an upper bound, x. If x is not the largest element in Y then {x}Y would be a totally ordered set in which Y would be properly contained, contradicting the definition. Thus x is a maximal elementMathworldPlanetmath in X.

Zorn’s lemma implies the well-ordering theorem.

Let X be any non-empty set, and let 𝒜 be the collectionMathworldPlanetmath of pairs (A,), where AX and is a well-ordering on A. Define a relationMathworldPlanetmathPlanetmath , on 𝒜 so that for all x,y𝒜:xy iff x equals an initial of y. It is easy to see that this defines a partial orderMathworldPlanetmath relation on 𝒜 (it inherits reflexibility, anti symmetryPlanetmathPlanetmath and transitivity from one set being an initial and thus a subset of the other).

For each chain C𝒜, define C=(R,) where R is the union of all the sets A for all (A,)C, and is the union of all the relations for all (A,)C. It follows that C is an upper bound for C in 𝒜.

According to Zorn’s lemma, 𝒜 now has a maximal element, (M,M). We postulateMathworldPlanetmath that M contains all members of X, for if this were not true we could for any aX-M construct (M*,*) where M*=M{a} and * is extended so Sa(M*)=M. Clearly * then defines a well-order on M*, and (M*,*) would be larger than (M,M) contrary to the definition.

Since M contains all the members of X and M is a well-ordering of M, it is also a well-ordering on X as required.

The well-ordering theorem implies Hausdorff’s maximum principle.

Let (X,) be a partially ordered set, and let be a well-ordering on X. We define the function ϕ by transfinite recursion over (X,) so that

ϕ(a)={{a}if {a}b<aϕ(b) is totally ordered under .otherwise..

It follows that xXϕ(x) is a maximal totally ordered subset of X 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