properties of ranks of sets

properties of ranks of sets

A set

A

is said to be

grounded

, if

A

⊆

V

α

in the

cumulative hierarchy

for some

ordinal

α

. The smallest such

α

such that

A

⊆

V

α

is called the rank of

A

, and is denoted by

ρ

⁢

(

A

)

.

In this entry, we list derive some basic properties of groundedness and ranks of sets. Proofs of these properties require an understanding of some of the basic

properties of ordinals

.

1.

∅

is grounded, whose rank is itself. This is obvious.

2.

If

A

is grounded, so is every

x

∈

A

, and

ρ

⁢

(

x

)

<

ρ

⁢

(

A

)

.

Proof.

A

⊆

V

ρ

⁢

(

A

)

, so

x

∈

V

ρ

⁢

(

A

)

, which means

x

⊆

V

β

for some

β

<

ρ

⁢

(

A

)

. This shows that

x

is grounded. Then

ρ

⁢

(

x

)

≤

β

, and hence

ρ

⁢

(

x

)

<

ρ

⁢

(

A

)

.
∎

3.

If every

x

∈

A

is grounded, so is

A

, and

ρ

⁢

(

A

)

=

sup

⁡

{

ρ

⁢

(

x

)

+

∣

x

∈

A

}

.

Proof.

Let

B

=

{

ρ

⁢

(

x

)

+

∣

x

∈

A

}

. Then

B

is a set of ordinals, so that

β

:=

⋃

B

=

sup

⁡

B

is an ordinal. Since each

x

∈

V

ρ

⁢

(

x

)

+

, we have

x

∈

V

β

. So

A

⊆

V

β

, showing that

A

is grounded. If

α

<

β

, then for some

x

∈

A

,

α

<

ρ

⁢

(

x

)

+

, which means

x

∉

V

α

, and therefore

A

⊈

V

α

. This shows that

ρ

⁢

(

A

)

=

β

.
∎

4.

If

A

is grounded, so is

{

A

}

, and

ρ

⁢

(

{

A

}

)

=

ρ

⁢

(

A

)

+

. This is a direct

consequence

of the previous result.

5.

If

A

,

B

are grounded, so is

A

∪

B

, and

ρ

⁢

(

A

∪

B

)

=

max

⁡

(

ρ

⁢

(

A

)

,

ρ

⁢

(

B

)

)

.

Proof.

Since

A

,

B

are grounded, every element of

A

∪

B

is grounded by property 2, so that

A

∪

B

is also grounded by property 3. Then

ρ

⁢

(

A

∪

B

)

=

sup

⁡

{

ρ

⁢

(

x

)

+

∣

x

∈

A

∪

B

}

=

max

⁡

(

sup

⁡

{

ρ

⁢

(

x

)

+

∣

x

∈

A

}

,

sup

⁡

{

ρ

⁢

(

x

)

+

∣

x

∈

B

}

)

=

max

⁡

(

ρ

⁢

(

A

)

,

ρ

⁢

(

B

)

)

.
∎

6.

If

A

is grounded, so is

B

⊆

A

, and

ρ

⁢

(

B

)

≤

ρ

⁢

(

A

)

.

Proof.

Every element of

B

, as an element of the grounded set

A

, is grounded, and therefore

B

is grounded. So

ρ

⁢

(

B

)

=

sup

⁡

{

ρ

⁢

(

x

)

+

∣

x

∈

B

}

≤

sup

⁡

{

ρ

⁢

(

x

)

+

∣

x

∈

A

}

=

ρ

⁢

(

A

)

. Since

ρ

⁢

(

B

)

and

ρ

⁢

(

A

)

are both ordinals,

ρ

⁢

(

B

)

≤

ρ

⁢

(

A

)

.
∎

7.

If

A

is grounded, so is

P

⁢

(

A

)

, and

ρ

⁢

(

P

⁢

(

A

)

)

=

ρ

⁢

(

A

)

+

.

Proof.

Every subset of

A

is grounded, since

A

is by property 6. So

P

⁢

(

A

)

is grounded. Furthermore,

P

⁢

(

A

)

=

sup

⁡

{

ρ

⁢

(

x

)

+

∣

x

∈

P

⁢

(

A

)

}

. Since

ρ

⁢

(

B

)

≤

ρ

⁢

(

A

)

for any

B

∈

P

⁢

(

A

)

, and

A

∈

P

⁢

(

A

)

, we have

P

⁢

(

A

)

=

ρ

⁢

(

A

)

+

as a result.
∎

8.

If

A

is grounded, so is

⋃

A

, and

ρ

⁢

(

⋃

A

)

=

sup

⁡

{

ρ

⁢

(

x

)

∣

x

∈

A

}

.

Proof.

Since

A

is grounded, every

x

∈

A

is grounded. Let

B

=

{

ρ

⁢

(

x

)

∣

x

∈

A

}

. Then

β

:=

⋃

B

=

sup

⁡

B

is an ordinal. Since

ρ

⁢

(

x

)

≤

β

,

V

ρ

⁢

(

x

)

=

V

β

or

V

ρ

⁢

(

x

)

∈

V

β

. In either case,

V

ρ

⁢

(

x

)

⊆

V

β

, since

V

α

is a

transitive set

for any ordinal

α

. Since

x

⊆

V

ρ

⁢

(

x

)

,

x

⊆

V

β

for every

x

∈

A

. This means

⋃

A

⊆

V

β

, showing that

⋃

A

is grounded. If

α

<

β

, then

α

<

ρ

⁢

(

x

)

for some

ρ

⁢

(

x

)

≤

β

, which means

x

⊈

V

α

, or

⋃

A

⊈

V

α

as a result. Therefore

ρ

⁢

(

⋃

A

)

=

β

.
∎

9.

Every ordinal is grounded, whose rank is itself.

Proof.

If

α

=

0

, then apply property 1. If

α

is a

successor ordinal

, apply properties 4 and 5, so that

ρ

⁢

(

α

)

=

ρ

⁢

(

β

+

)

=

ρ

⁢

(

β

∪

{

β

}

)

=

max

⁡

(

ρ

⁢

(

β

)

,

ρ

⁢

(

{

β

}

)

)

=

max

⁡

(

ρ

⁢

(

β

)

,

ρ

⁢

(

β

)

+

)

=

ρ

⁢

(

β

)

+

. If

α

is a

limit ordinal

, then apply property 8 and

transfinite induction

, so that

ρ

⁢

(

α

)

=

ρ

⁢

(

⋃

α

)

=

sup

⁡

{

ρ

⁢

(

β

)

∣

β

<

α

}

=

sup

⁡

{

β

∣

β

<

α

}

=

α

.
∎

References

1

H. Enderton,

Elements of

Set Theory

, Academic Press, Orlando, FL (1977).

2

A. Levy,

Basic Set Theory

, Dover Publications Inc., (2002).

Title

properties of ranks of sets

Canonical name

PropertiesOfRanksOfSets

Date of creation

2013-03-22 18:50:31

Last modified on

2013-03-22 18:50:31

Owner

CWoo (3771)

Last modified by

CWoo (3771)

Numerical id

8

Author

CWoo (3771)

Entry type

Derivation

Classification

msc 03E99

Defines

grounded

Defines

grounded set

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