equivalence of Hausdorff’s maximum principle, Zorn’s lemma and the well-ordering theorem

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

X

, where every chain has an

upper bound

. According to the

maximum principle

there exists a maximal

totally ordered

subset

Y

⊆

X

. 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 element

in

X

.

Zorn’s lemma implies the well-ordering theorem.

Let

X

be any non-empty set, and let

𝒜

be the

collection

of pairs

(

A

,

≤

)

, where

A

⊆

X

and

≤

is a

well-ordering

on

A

. Define a

relation

⪯

, on

𝒜

so that for all

x

,

y

∈

𝒜

:

x

⪯

y

iff

x

equals an initial of

y

. It is easy to see that this defines a

partial order

relation on

𝒜

(it inherits reflexibility, anti

symmetry

and

transitivity

from one set being an initial and thus a subset of
the other).

For each chain

C

⊆

𝒜

, 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

postulate

that

M

contains all
members of

X

, for if this were not true we could for any

a

∈

X

-

M

construct

(

M

*

,

≤

*

)

where

M

*

=

M

∪

{

a

}

and

≤

*

is extended so

S

a

⁢

(

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

<

a

ϕ

⁢

(

b

)

⁢

is totally ordered under

⪯

.

∅

otherwise

.

.

It follows that

⋃

x

∈

X

ϕ

⁢

(

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

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