models constructed from constants

models constructed from constants

The

definition

of a

structure

and of the

satisfaction relation

is
nice, but it raises the following question : how do we get models in
the first place? The most basic construction for models of

first-order theory

is the construction that uses

constants

. Throughout
this entry,

L

is a fixed

first-order language

.

Let

C

be a set of

constant symbols

of

L

, and

T

be a theory in

L

. Then we say

C

is a

set of witnesses

for

T

if and only
if for every

formula

φ

with at most one

free variable

x

,
we have

T

⊢

∃

x

⁢

(

φ

)

⇒

φ

⁢

(

c

)

for some

c

∈

C

.

Lemma.

Let

T

is any consistent
set of

sentences

of

L

, and

C

is a set of new symbols
such that

|

C

|

=

|

L

|

. Let

L

′

=

L

∪

C

. Then there is a consistent
set

T

′

⊆

L

′

extending

T

and which has

C

as set of
witnesses.

Lemma.

If

T

is a

consistent

theory in

L

, and

C

is a set of witnesses
for

T

in

L

, then

T

has a model whose

elements

are the constants
in

C

.

Proof:

Let

Σ

be the

signature

for

L

. If

T

is a consistent set of
sentences of

L

, then there is a maximal consistent

T

′

⊇

T

.
Note that

T

′

and

T

have the same sets of witnesses. As every
model of

T

′

is also a model of

T

, we may assume

T

is maximal
consistent.

We let the

universe

of

𝔐

be the set of

equivalence classes

C

/

∼

, where

a

∼

b

if and only if

“

⁢

a

=

b

⁢

”

∈

T

. As

T

is
maximal consistent, this is an

equivalence relation

. We interpret the
non-logical symbols as follows :

1.

[

a

]

=

𝔐

[

b

]

if and only if

a

∼

b

;

2.

Constant symbols are interpreted in the

obvious

way, i.e. if

c

∈

Σ

is a constant symbol, then

c

𝔐

=

[

c

]

;

3.

If

R

∈

Σ

is an

n

-ary relation symbol, then

(

[

a

1

]

,

…

,

[

a

n

]

)

∈

R

𝔐

if and only if

R

⁢

(

a

1

,

…

,

a

n

)

∈

T

;

4.

If

F

∈

Σ

is an

n

-any function symbol, then

F

𝔐

⁢

(

[

a

0

]

,

…

,

[

a

n

]

)

=

[

b

]

if and only if

“

⁢

F

⁢

(

a

1

,

…

,

a

n

)

=

b

⁢

”

∈

T

.

From the fact that

T

is maximal consistent, and

∼

is an
equivalence relation, we get that the

operations

are

well-defined

(it
is not so simple, i’ll write it out later).
The proof that

𝔐

⊧

T

is a straightforward

induction

on the
complexity of the formulas of

T

.

♢

Corollary

.

(The extended completeness theorem) A set

T

of formulas of

L

is
consistent if and only if it has a model (regardless of whether or not

L

has witnesses for

T

).

Proof:

First add a set

C

of new constants to

L

, and expand

T

to

T

′

in
such a way that

C

is a set of witnesses for

T

′

. Then expand

T

′

to a maximal consistent set

T

′′

. This set has a model

𝔐

consisting of
the constants in

C

, and

𝔐

is also a model of

T

.

♢

Corollary.

(Compactness theorem) A set

T

of sentences of

L

has a model if
and only if every finite subset of

T

has a model.

Proof:

Replace “has a model” by “is consistent”, and apply the syntactic
compactness theorem.

♢

Corollary.

(Gödel’s completeness theorem)
Let

T

be a consistent set of formulas of

L

. Then
A sentence

φ

is a

theorem

of

T

if and only if it is true in
every model of

T

.

Proof:

If

φ

is not a theorem of

T

, then

¬

⁢

φ

is consistent
with

T

, so

T

∪

{

¬

⁢

φ

}

has a model

𝔐

, in which

φ

cannot be true.

♢

Corollary.

(Downward Löwenheim-Skolem theorem) If

T

⊆

L

has a model,
then it has a model of power at most

|

L

|

.

Proof:

If

T

has a model, then it is consistent. The model constructed from
constants has power at most

|

L

|

(because we must add at most

|

L

|

many new constants).

♢

Most of the treatment found in this entry can be read in more details
in Chang and Keisler’s book

Model Theory

.

Title

models constructed from constants

Canonical

name

ModelsConstructedFromConstants

Date of creation

2013-03-22 12:44:42

Last modified on

2013-03-22 12:44:42

Owner

ratboy (4018)

Last modified by

ratboy (4018)

Numerical id

17

Author

ratboy (4018)

Entry type

Definition

Classification

msc 03C07

Synonym

completeness theorem

Synonym

Gödel completeness theorem

Related topic

UpwardsSkolemLowenheimTheorem

Defines

set of witnesses

Generated on Thu Feb 8 19:55:14 2018 by

LaTeXML