constructing well ordered sets

constructing well ordered sets

A set

A

well-orderable

if there is a

well-ordering

on

A

. For example, the

empty set

∅

, is well-orderable, the well-ordering itself is

∅

. Similarly, any

singleton

has

∅

as the unique well-ordering, and is therefore well-orderable. More generally, every

finite set

is well-orderable. To

construct

a well-ordering on a finite set, pick any

element

from the set and designate the element to be the smallest element of the set. Then, pick an element from the remaining

collection

, and make this element the next smallest element. Keep doing this until every element has been picked and assigned. Obviously, this process will not go on forever. If there are

n

elements in the set, there are

n

!

well-orderings on the set (since every well-ordering is a

linear ordering

).

By the

axiom of choice

, every set is well-orderable. However, this does not provide us with an explicit way to construct the well-order. For example, how does one well-order

ℝ

, the set of real numbers? Nevertheless, given two well-ordered sets

(

A

,

<

A

)

,

(

B

,

<

B

)

, one can explicitly construct well-orderings on the following sets, without AC.

Proposition 1

.

A

∪

B

is well-orderable.

Proof.

We may assume

A

∩

B

=

∅

, since

A

∩

B

=

A

∩

(

A

-

B

)

, and any subset of an well-ordered set is well-ordered. Define a

binary relation

<

(

A

,

B

)

on

A

∪

B

as follows: for

x

,

y

∈

A

∪

B

,

x

<

(

A

,

B

)

y

iff

{

x

,

y

∈

A

⁢

and

⁢

x

<

A

y

,

or

x

,

y

∈

B

⁢

and

⁢

x

<

B

y

,

or

x

∈

A

⁢

and

⁢

y

∈

B

.

So

<

(

A

,

B

)

is a (strict) linear ordering on

A

∪

B

. Furthermore,

<

(

A

,

B

)

extends both

<

A

and

<

B

. If

X

⊆

A

∪

B

, then the least element of

X

with respect to

<

(

A

,

B

)

is the least element of

X

with respect to

<

B

if

X

⊆

B

, or the least element of

X

∩

A

with respect to

<

A

, if

X

∩

A

≠

∅

. Hence

(

A

∪

B

,

<

(

A

,

B

)

)

is a well-ordered set.
∎

Proposition 2

.

A

×

B

is well-orderable.

Proof.

We may assume that

A

,

B

are both non-empty, since the empty set is well-orderable. Define

<

A

×

B

on

A

×

B

as follows:

(

x

1

,

x

2

)

<

A

×

B

(

y

1

,

y

2

)

iff

{

y

1

<

B

y

2

,

or

y

1

=

y

2

⁢

and

⁢

x

1

<

A

x

2

.

Again, it is easy to see that

<

A

×

B

is a strict linear ordering on

A

×

B

. Next, let

X

⊆

A

×

B

. Define

X

⁢

[

A

]

:=

{

y

∈

B

∣

(

x

,

y

)

∈

X

⁢

for some

⁢

x

∈

A

}

. Let

b

be the least element of

X

⁢

[

A

]

. Define

X

-

1

⁢

(

b

)

:=

{

x

∈

A

∣

(

x

,

b

)

∈

X

}

. Let

a

be the least element of

X

-

1

⁢

(

b

)

. Then

(

a

,

b

)

∈

X

. For any

(

c

,

d

)

∈

X

distinct from

(

a

,

b

)

, since

d

∈

X

⁢

[

A

]

, either

b

<

B

d

or

b

=

d

. In the first case,

(

a

,

b

)

<

A

×

B

(

c

,

d

)

. In the second case,

c

∈

X

-

1

⁢

(

b

)

, either

a

<

B

c

or

a

=

c

. In the first case,

(

a

,

b

)

<

A

×

B

(

c

,

d

)

, and the second case is impossible for otherwise

(

a

,

b

)

=

(

c

,

d

)

. Therefore

<

A

×

B

)

is a well-ordering on

A

×

B

.
∎

Remarks

.

•

The

ordering

constructed in

Proposition

2 is called the

Hebrew lexicographic ordering

. It is

similar

to the usual

lexicographic ordering

, except that the second coordinates are ordered first before the first coordinates. The reason for choosing this ordering is to make the ordering on

A

×

2

compatible

with the ordering on

A

⁢

∐

A

, the

disjoint union

of

A

with itself. Had we chosen the lexicographic ordering instead, on, say,

ω

×

2

, then every

initial segment

of it would be finite, whereas the initial segment of

(

0

,

1

)

on

ω

⁢

∐

ω

=

(

ω

×

{

0

}

)

∪

(

ω

×

{

1

}

)

is

infinite

.

•

The above constructions parallel the

ordinal arithmetic

: if

α

,

β

are

ordinals

such that

(

A

,

<

A

)

≅

α

and

(

B

,

<

B

)

≅

β

, then

(a)

(

A

∪

B

,

<

(

A

,

B

)

)

≅

α

+

β

, provided that

A

∩

B

=

∅

,

(b)

(

A

×

B

,

<

A

×

B

)

≅

α

⋅

β

,

where

≅

denotes order isomorphism between well-ordered sets.

•

One would hope that for well-ordered sets

A

,

B

, the set

B

A

is well-orderable without resorting to AC. However, this is not possible, and the best one can do is the following: if

B

is linearly ordered, and

A

well-ordered, then one can linearly order

B

A

without AC:

Proof.

Let

C

:=

B

A

. For

f

,

g

:

A

→

B

, let

[

f

,

g

]

:=

{

a

∈

A

∣

f

⁢

(

a

)

<

B

g

⁢

(

a

)

}

. If

[

f

,

g

]

≠

∅

, it has a least element

min

⁡

[

f

,

g

]

. Define

<

C

on

C

as follows, for any

f

,

g

∈

C

:

f

<

C

g

iff

[

f

,

g

]

≠

∅

⁢

and

⁢

f

⁢

(

a

)

=

g

⁢

(

a

)

⁢

for all

⁢

a

<

A

min

⁡

[

f

,

g

]

.

If

f

≠

g

, then

[

f

,

g

]

∪

[

g

,

f

]

≠

∅

so that

f

<

C

g

or

g

<

C

f

.
To show that

<

C

is

transitive

, suppose

f

<

C

g

and

g

<

C

h

. Let

b

=

min

⁡

[

f

,

g

]

and

c

=

min

⁡

[

g

,

h

]

. Then:

–

if

b

≤

A

c

, then

f

⁢

(

b

)

<

B

g

⁢

(

b

)

≤

B

h

⁢

(

b

)

, which implies

[

f

,

h

]

≠

∅

and

min

⁡

[

f

,

h

]

≤

A

b

.

–

if

c

<

A

b

, then

f

⁢

(

c

)

=

g

⁢

(

c

)

<

B

h

⁢

(

c

)

, which implies

[

f

,

h

]

≠

∅

and

min

⁡

[

f

,

h

]

≤

A

c

.

In either case,

min

⁡

[

f

,

h

]

≤

A

min

⁡

(

b

,

c

)

. Now, suppose

a

<

min

⁡

[

f

,

h

]

. Then

f

⁢

(

a

)

=

g

⁢

(

a

)

=

h

⁢

(

a

)

. As a result,

f

<

C

h

. ∎

•

In fact, it can be shown that the following are all

equivalent

to AC in ZF:

(a)

the

well-ordering principle

,

(b)

if

A

,

B

are well-orderable, so is

B

A

,

(c)

if

A

is well-orderable, so is

P

⁢

(

A

)

, the

powerset

of

A

.

Title

constructing well ordered sets

Canonical

name

ConstructingWellOrderedSets

Date of creation

2013-03-22 18:49:12

Last modified on

2013-03-22 18:49:12

Owner

CWoo (3771)

Last modified by

CWoo (3771)

Numerical id

9

Author

CWoo (3771)

Entry type

Example

Classification

msc 03E25

Classification

msc 06A05

Defines

well-orderable

Defines

Hebrew lexicographic ordering

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