proof of downward Lowenheim-Skolem theorem

proof of downward Lowenheim-Skolem theorem

We present here a proof of the Downward Lowenheim Skolem

Theorem

. The idea is to construct a

submodel

that meets the requirements of the DLS Theorem: we take

K

and close it under a procedure of choosing appropriate witnesses for the existential formulas satisfied by

𝒜

. Choosing the appropriate witnesses is done with the help of the so-called

Skolem functions

and thus rests upon the

choice function

.

Proof

: First of all we introduce a usefull tool for the proof.
Lemma: (

Tarski’s Lemma

) If

𝒜

and

ℬ

are

L

-

structures

with the domain of

𝒜

being a subset of the domain of

ℬ

then

𝒜

is an

elementary substructure

of

ℬ

if for every

L

-

formula

ϕ

⁢

(

x

,

y

)

with

a

∈

A

and

b

∈

B

, we have

ℬ

⊧

ϕ

⁢

(

a

,

b

)

¡-¿ For some

a

′

∈

A

we have

𝒜

⊧

ϕ

⁢

(

a

,

a

′

)

Proof

. Let’s supposes the biconditionnal holds, then we need to show that

𝒜

is a substructure of

ℬ

. This means that we need to show that every formula that is true in

𝒜

is true in

ℬ

. The proof is straighforward with an

induction

on the

complexity

of formulas (

connectives

,

negation

,

quantifiers

).

Now, fix a point

p

∈

d

⁢

o

⁢

m

⁢

(

𝒜

)

. For each

L

-formula

ϕ

⁢

(

x

,

y

)

, define the

Skolem function

of

ϕ

,

g

ϕ

:

A

n

→

A

(with

A

=

d

⁢

o

⁢

m

⁢

(

𝒜

)

) by:

p

∈

A

n

→

some

p

′

∈

A

such that

𝒜

⊧

ϕ

⁢

(

p

,

p

′

)

if such a

p

′

exists and

p

otherwise.
The set of

Skolem functions

has a

cardinality

equal to that of

L

.

The purpose here is to construct a model

ℬ

whose domain

B

is

closed under

the skolem functions. This means that the domain of

𝒜

contains all the witnesses we’ve appropriately choosen. If we take an existential formula

∃

y

⁢

ϕ

⁢

(

x

,

y

)

and

b

∈

B

k

and if we apply the Skolem function to

b

we will have a witness for

∃

x

⁢

ϕ

⁢

(

b

,

x

)

. In other words, this means that

𝒜

⊧

∃

x

⁢

ϕ

⁢

(

b

,

x

)

→

𝒜

⊧

ϕ

⁢

(

b

,

g

ϕ

⁢

(

b

)

)

. By construction

g

ϕ

⁢

(

b

)

is in

B

and thus

ℬ

meets the requirements of Tarski’s Lemma. We can find an elementary substructure of

𝒜

.

Let’s take

K

of above and set

K

0

:=

K

and

K

i

+

1

is the set of the

g

ϕ

⁢

(

p

)

,

p

∈

K

i

∧

g

ϕ

(with

g

ϕ

a Skolem function). Let

B

:=

⋃

K

i

. Then

B

is closed under Skolem functions. And we have

|

K

|

≤

ω

.

|

L

|

+

|

K

|

=

|

L

|

+

|

K

|

. This comes from the fact that

|

L

|

+

|

K

i

|

=

|

SF

|

+

|

K

i

|

=

∑

k

∈

ℕ

|

(

S

⁢

F

)

k

|

+

|

K

i

|

=

∑

k

∈

ℕ

|

(

S

⁢

F

)

k

|

*

|

K

i

|

but we have

|

K

i

+

1

|

≤

∑

k

∈

ℕ

|

(

S

⁢

F

)

k

|

*

|

K

i

|

.
We need now to provide

interpretations

of

relations

,

predicates

,

functions

and constants so it can fit

𝒜

.

We have because

B

is closed under

L

-terms and for an

L

-

function symbol

f

, the Skolem function of the

L

-formula

f

⁢

(

x

)

=

y

takes the value

f

𝒜

⁢

(

p

)

at p:

for any n-ary relation symbol

P

:

P

ℬ

=

P

𝒜

⁢

⋂

B

n

for an m-ary function symbol

f

and

p

∈

B

m

,

p

′

∈

B

we have

f

𝒜

⁢

(

p

)

=

p

′

¡-¿

f

ℬ

⁢

(

p

)

=

p

′

.

for a constant symbol

c

, we have

c

𝒜

=

c

ℬ

We have constructed a substructure

ℬ

of

𝒜

.

Title

proof of downward Lowenheim-Skolem theorem

Canonical name

ProofOfDownwardLowenheimSkolemTheorem

Date of creation

2013-03-22 18:18:59

Last modified on

2013-03-22 18:18:59

Owner

GodelsTheorem (21277)

Last modified by

GodelsTheorem (21277)

Numerical id

14

Author

GodelsTheorem (21277)

Entry type

Proof

Classification

msc 03C07

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