permutation model

permutation model

A

permutation model

is a model of the axioms of

set theory

in which there is a non

trivial

automorphism

of the set theoretic

universe

. Such models are used to show the

consistency

of the

negation

of the

Axiom of Choice

(AC).

A typical construction of a permutation model is done here. By

Z

⁢

F

-

we denote the axioms of

Z

⁢

F

minus the

axiom of foundation

. In particular we allow sets

a

such that

a

=

{

a

}

which we will call atoms. Let

A

be an

infinite set

of atoms.

Define

V

α

⁢

(

A

)

by

induction

on

α

as follows:

V

0

⁢

(

A

)

=

A

V

α

+

1

⁢

(

A

)

=

𝒫

⁢

(

V

α

)

V

α

⁢

(

A

)

=

⋃

γ

<

α

V

γ

⁢

(

A

)

⁢

for

α

limit

Finally define

V

=

⋃

α

∈

ON

V

α

⁢

(

A

)

. Then we have

A

=

V

0

⁢

(

A

)

⊆

V

1

⁢

(

A

)

⊆

⋯

⊆

V

α

⁢

(

A

)

⁢

⋯

⊆

V

For any

x

∈

V

we can assign a rank,

rank

(

x

)

=

least

α

[

x

∈

V

α

+

1

(

A

)

]

Let

G

be the group of

permutations

of

A

. For

π

∈

G

we extend

π

to
a permutation of

V

by induction on

∈

by defining

π

⁢

(

x

)

=

{

π

⁢

(

y

)

:

y

∈

x

}

and letting

π

⁢

(

∅

)

=

∅

. Then

G

permutes

V

and fixes the well founded sets

W

⁢

F

⊆

V

.

Lemma

.

For all

x

,

y

∈

V

and any

π

∈

G

.

x

∈

y

⇔

π

⁢

(

x

)

∈

π

⁢

(

y

)

That is,

π

is an

∈

-automorphism of

V

. From this we can prove that

π

⁢

(

{

X

,

Y

}

)

=

{

π

⁢

(

X

)

,

π

⁢

(

Y

)

}

and so

π

⁢

(

(

X

,

Y

)

)

=

(

π

⁢

(

X

)

,

π

⁢

(

Y

)

)

π

⁢

(

(

X

,

Y

,

Z

)

)

=

(

π

⁢

(

X

)

,

π

⁢

(

Y

)

,

π

⁢

(

Z

)

)

Also by induction on

α

it is easy to show that

rank

⁡

(

x

)

=

rank

⁡

(

π

⁢

(

x

)

)

for all

x

∈

V

.

Let

a

1

,

⋯

,

a

n

∈

A

and define

[

a

1

,

⋯

,

a

n

]

=

{

π

∈

G

:

π

⁢

(

a

i

)

=

a

i

,

for

⁢

i

=

1

,

⋯

,

n

}

Call a set

X

∈

V

symmetric

if there exists

a

1

,

⋯

,

a

n

∈

A

such that

π

⁢

(

X

)

=

X

for all

π

∈

[

a

1

,

⋯

,

a

n

]

. Define the class

H

⁢

S

⊆

V

of hereditarily

symmetric sets

H

⁢

S

=

{

x

∈

V

:

x

⁢

is symmetric and

⁢

x

⊆

H

⁢

S

}

Call a class

N

transitive

if

∀

x

∈

N

[

x

⊆

N

]

and call

N

almost

universal

if (for sets S)

∀

S

⊆

N

[

∃

Y

∈

N

(

S

⊆

Y

)

]

H

⁢

S

is transitive and almost universal.

To show that a class

N

⊧

Z

⁢

F

-

is straightforward for most axioms of

Z

⁢

F

-

except for the

axiom of Comprehension

. To show

N

is a model of Comprehension it suffices to show that

N

is

closed under

Gödel

Operations

:

G

1

⁢

(

X

,

Y

)

=

{

X

,

Y

}

G

2

⁢

(

X

,

Y

)

=

X

∖

Y

G

3

⁢

(

X

,

Y

)

=

X

×

Y

G

4

⁢

(

X

)

=

dom

⁢

(

X

)

G

5

⁢

(

X

)

=

∈

∩

X

2

G

6

⁢

(

X

)

=

{

(

a

,

b

,

c

)

:

(

b

,

c

,

a

)

∈

X

}

G

7

⁢

(

X

)

=

{

(

a

,

b

,

c

)

:

(

c

,

b

,

a

)

∈

X

}

G

8

⁢

(

X

)

=

{

(

a

,

b

,

c

)

:

(

a

,

c

,

b

)

∈

X

}

Theorem

.

(

Z

⁢

F

) If

N

is transitive, almost universal and closed under Gödel Operations, then

N

⊧

Z

⁢

F

.

H

⁢

S

is closed under Gödel operations and so

H

⁢

S

⊧

Z

⁢

F

-

. The class

H

⁢

S

is a permutation model. The set of atoms

A

∈

H

⁢

S

and furthermore:

Lemma

.

Let

f

:

ω

→

A

be a one to one

function

. Then

f

∉

H

⁢

S

and so

A

cannot be well ordered in

H

⁢

S

.

Which proves the

theorem

:

Theorem

.

H

⁢

S

⊧

Z

⁢

F

-

+

¬

⁢

A

⁢

C

.

which

completes

the proof that

Con

⁢

(

Z

⁢

F

-

)

⟹

Con

⁢

(

Z

⁢

F

-

+

¬

⁢

A

⁢

C

)

. In particular we have that

Z

⁢

F

-

⊬

A

⁢

C

.

Title

permutation model

Canonical name

PermutationModel

Date of creation

2013-03-22 14:46:48

Last modified on

2013-03-22 14:46:48

Owner

ratboy (4018)

Last modified by

ratboy (4018)

Numerical id

13

Author

ratboy (4018)

Entry type

Definition

Classification

msc 03E25

Defines

Gödel Operations

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