aarjaneiro/planetmath huggingface dataset:
subformula

subformula

Let

L

be a

first order language

and suppose

φ

is a

formula

of

L

. A

subformula

of

φ

is defined as any of the following:

1.

φ

is a subformula of

φ

;

2.

if

¬

⁢

ψ

is a subformula of

φ

for some

L

-formula

ψ

, then so is

ψ

;

3.

if

α

∧

β

is a subformula of

φ

for some

L

-formulas

α

,

β

, then so are

α

and

β

;

4.

if

∃

x

⁢

(

ψ

)

is a subformula of

φ

for some

L

-formula

ψ

, then so is

ψ

⁢

[

t

/

x

]

for any

t

free for

x

in

ψ

.

Remark

. And if the

language

contains a modal connective, say

□

, then we also have

5.

if

□

⁢

α

is a subformula of

φ

for some

L

-formula

α

, then so is

α

.

The phrase “

t

is free for

x

in

ψ

” means that after substituting the term

t

for the

variable

x

in the formula

ψ

, no

free variables

in

t

will become

bound variables

in

ψ

⁢

[

t

/

x

]

.

For example, if

φ

=

α

∨

β

, then

α

and

β

are subformulas of

φ

. This is so because

α

∨

β

=

¬

⁡

(

¬

⁢

α

∧

¬

⁢

β

)

, so that

¬

⁢

α

∧

¬

⁢

β

is a subformula of

φ

by applications of 1 followed by 2 above. By 3 above,

¬

⁢

α

and

¬

⁢

β

are subformulas of

φ

. Therefore, by 2 again,

α

and

β

are subformulas of

φ

.

For another example, if

φ

=

∃

x

(

∃

y

(

x

2

+

y

2

=

1

)

)

, then

∃

y

(

t

2

+

y

2

=

1

)

is a subformula of

φ

as long as

t

is a term that does not contain the variable

y

. Therefore, if

t

=

y

+

2

, then

∃

y

(

(

y

+

2

)

2

+

y

2

=

1

)

is not a subformula of

φ

. In fact, if

y

∈

ℝ

, the equation

(

y

+

2

)

2

+

y

2

=

1

is never true.

Finally, it is easy to see (by

induction

) that if

α

is a subformula of

ψ

and

ψ

is a subformula of

φ

, then

α

is a subformula of

φ

. “Being a subformula of” is a

reflexive

transitive relation

on

L

-formulas.

Remark

. There is also the notion of a

literal subformula

of a formula

φ

. A formula

ψ

is a literal subformula of

φ

if it is a subformula of

φ

obtained in any one of the first three ways above, or if

∃

x

⁢

(

ψ

)

is a literal subformula of

φ

.

Note that any literal subformula of

φ

is a subformula of

φ

, for if

φ

=

∃

x

⁢

(

ψ

)

, then

x

occurs free in

ψ

and

ψ

=

ψ

⁢

[

x

/

x

]

.

In the second example above,

∃

y

(

x

2

+

y

2

=

1

)

and

x

2

+

y

2

=

1

are both literal subformulas of

φ

=

∃

x

(

∃

y

(

x

2

+

y

2

=

1

)

)

.

Title

subformula

Canonical

name

Subformula

Date of creation

2013-03-22 12:42:52

Last modified on

2013-03-22 12:42:52

Owner

CWoo (3771)

Last modified by

CWoo (3771)

Numerical id

13

Author

CWoo (3771)

Entry type

Definition

Classification

msc 03C07

Classification

msc 03B10

Related topic

Substitutability

Defines

literal subformula

Generated on Thu Feb 8 19:55:50 2018 by

LaTeXML
substructure

substructure

Let

Σ

be a fixed

signature

, and

𝔄

and

𝔅

structures

for

Σ

. We say

𝔄

is a

substructure

of

𝔅

, denoted

𝔄

⊆

𝔅

, if for all

x

∈

𝔄

we have

x

∈

𝔅

, and the

inclusion map

i

:

𝔄

→

𝔅

:

x

↦

x

is an

embedding

.

When

𝔄

is a substructure of

𝔅

, we also say that

𝔅

is an

extension

of

𝔄

.

A

submodel

𝔄

of a model

𝔅

of a (first-order)

language

ℒ

if

𝔄

is a model of

ℒ

and

𝔄

is a substructure of

𝔅

.

Title

substructure

Canonical name

Substructure

Date of creation

2013-03-22 13:50:32

Last modified on

2013-03-22 13:50:32

Owner

almann (2526)

Last modified by

almann (2526)

Numerical id

6

Author

almann (2526)

Entry type

Definition

Classification

msc 03C07

Synonym

submodel

Related topic

StructuresAndSatisfaction

Defines

extension

Generated on Thu Feb 8 19:55:52 2018 by

LaTeXML
type

type

Let

L

be a

first order language

.
Let

M

be an

http://planetmath.org/node/3384

L

-

structure

.
Let

B

⊆

M

, and let

a

∈

M

n

.
Then we define the

type of

a

over

B

to be the set of

L

-

formulas

ϕ

⁢

(

x

,

b

¯

)

with

parameters

b

¯

from

B

so that

M

⊧

ϕ

⁢

(

a

,

b

¯

)

.
A

collection

of

L

-formulas is a

complete

n

-type over

B

iff it is of the above form for some

B

,

M

and

a

∈

M

n

.

We call any

consistent

collection of formulas

p

in

n

variables

with parameters from

B

a

partial

n

-type over

B

. (See

criterion for consistency of sets of formulas

.)

Note that a complete

n

-type

p

over

B

is consistent so is in particular a

partial type

over

B

. Also

p

is maximal in the sense that for every formula

ψ

⁢

(

x

,

b

¯

)

over

B

we have either

ψ

⁢

(

x

,

b

¯

)

∈

p

or

¬

⁢

ψ

⁢

(

x

,

b

¯

)

∈

p

.
In fact, for every collection of formulas

p

in

n

variables the following are

equivalent

:

•

p

is the type of some

sequence

of

n

elements

a

over

B

in some model

N

≡

M

•

p

is a maximal consistent set of formulas.

For

n

∈

ω

we define

S

n

⁢

(

B

)

to be the set of complete

n

-types over

B

.

Some authors define a collection of formulas

p

to be a

n

-type

iff

p

is a partial

n

-type. Others define

p

to be a

type

iff

p

is a complete

n

-type.

A type (resp. partial type/complete type) is any

n

-type (resp. partial type/complete type) for some

n

∈

ω

.

Title

type

Canonical

name

Type

Date of creation

2013-03-22 13:22:45

Last modified on

2013-03-22 13:22:45

Owner

ratboy (4018)

Last modified by

ratboy (4018)

Numerical id

6

Author

ratboy (4018)

Entry type

Definition

Classification

msc 03C07

Related topic

Formula

Related topic

DefinableType

Related topic

TermsAndFormulas

Defines

type

Defines

complete type

Defines

partial type

Generated on Thu Feb 8 19:55:54 2018 by

LaTeXML
upward Lowenheim-Skolem theorem

upward Lowenheim-Skolem theorem

Let

L

be a

first-order language

and let

𝒜

be an

infinite

L

-

structure

. Then if

κ

is a

cardinal

with

κ

≥

Max

⁡

(

|

𝒜

|

,

|

L

|

)

then there is an

L

-structure

ℬ

such that

|

ℬ

|

=

κ

and

𝒜

≼

ℬ

.

Title

upward Lowenheim-Skolem theorem

Canonical name

UpwardLowenheimSkolemTheorem

Date of creation

2013-03-22 13:00:39

Last modified on

2013-03-22 13:00:39

Owner

Evandar (27)

Last modified by

Evandar (27)

Numerical id

6

Author

Evandar (27)

Entry type

Theorem

Classification

msc 03C07

Generated on Thu Feb 8 19:55:56 2018 by

LaTeXML
quantifier free

quantifier free

Let

L

be a

first order language

.
A

formula

ψ

is

quantifier free

iff it contains no

quantifiers

.

Let

T

be a

complete

L

-theory. Let

S

⊆

L

. Then

S

is an

elimination set

for

T

iff
for every

ψ

⁢

(

x

¯

)

∈

L

there is some

ϕ

⁢

(

x

¯

)

∈

S

so that

T

⊢

∀

x

¯

⁢

(

ψ

⁢

(

x

¯

)

)

↔

ϕ

⁢

(

x

¯

)

.

In particular,

T

has

quantifier elimination

iff the set of

quantifier free formulas

is an elimination set for

T

.
In other

T

has

quantifier elimination

iff
for every

ψ

⁢

(

x

¯

)

∈

L

there is some quantifier free

ϕ

⁢

(

x

¯

)

∈

L

so that

T

⊢

∀

x

¯

⁢

(

ψ

⁢

(

x

¯

)

)

↔

ϕ

⁢

(

x

¯

)

.

Title

quantifier free

Canonical name

QuantifierFree

Date of creation

2013-03-22 13:27:49

Last modified on

2013-03-22 13:27:49

Owner

mathcam (2727)

Last modified by

mathcam (2727)

Numerical id

7

Author

mathcam (2727)

Entry type

Definition

Classification

msc 03C10

Classification

msc 03C07

Classification

msc 03B10

Related topic

Quantifier

Related topic

LogicalLanguage

Defines

quantifier free formula

Defines

quantifier elimination

Defines

elimination set

Generated on Thu Feb 8 19:55:58 2018 by

LaTeXML
countably categorical structures

countably categorical structures

A

countably infinite

structure

is called

countably

categorical

(also called

ω

-categorical

, or

ℵ

0

-categorical

)
if all

countable

models of its

first-order theory

are

isomorphic

.

Ryll-Nardzewski, Engeler, and Svenonius proved that a
countable structure is

ω

-categorical if and only if
it has an

oligomorphic automorphism group

.

Title

countably categorical structures

Canonical name

CountablyCategoricalStructures

Date of creation

2013-03-22 15:15:38

Last modified on

2013-03-22 15:15:38

Owner

amador (8479)

Last modified by

amador (8479)

Numerical id

5

Author

amador (8479)

Entry type

Derivation

Classification

msc 03C35

Synonym

ℵ

0

-categorical

Synonym

ω

-categorical

Related topic

oligomorphicPermutationGroup

Related topic

OligomorphicPermutationGroup

Generated on Thu Feb 8 19:56:02 2018 by

LaTeXML
oligomorphic permutation group

oligomorphic permutation group

A

permutation group

acting on a

countably
infinite

set is called

oligomorphic

,
if it has finitely many orbits of

n

-tuples,
for all

n

≥

1

.

Ryll-Nardzewski, Engeler, and Svenonius proved that
a countably infinite first-order

structure

has an

oligomorphic
automorphism group

if and only if the structure is

ω

-categorical.

Title

oligomorphic permutation group

Canonical

name

OligomorphicPermutationGroup

Date of creation

2013-03-22 15:15:36

Last modified on

2013-03-22 15:15:36

Owner

amador (8479)

Last modified by

amador (8479)

Numerical id

6

Author

amador (8479)

Entry type

Definition

Classification

msc 03C35

Related topic

CountablyCategoricalStructures

Defines

oligomorphic automorphism group

Generated on Thu Feb 8 19:56:03 2018 by

LaTeXML
proof of Vaught’s test

proof of Vaught’s test

Let

φ

be an

L

-

sentence

, and let

𝒜

be the unique model of S of

cardinality

κ

. Suppose

𝒜

⊨

φ

. Then if

ℬ

is any model of

S

then by the upward (

http://planetmath.org/UpwardLowenheimSkolemTheorem

) and

downward Lowenheim-Skolem theorems

, there is a model

𝒞

of

S

which is

elementarily equivalent

to

ℬ

such that

|

𝒞

|

=

κ

. Then

𝒞

is

isomorphic

to

𝒜

, and so

𝒞

⊨

φ

, and

ℬ

⊨

φ

. So

ℬ

⊨

φ

for all models

ℬ

of

S

, so

S

⊨

φ

.

Similarly, if

𝒜

⊨

¬

⁢

φ

then

S

⊨

¬

⁢

φ

. So

S

is

complete

(

http://planetmath.org/Complete6

).

□

Title

proof of Vaught’s test

Canonical name

ProofOfVaughtsTest

Date of creation

2013-03-22 13:00:44

Last modified on

2013-03-22 13:00:44

Owner

Evandar (27)

Last modified by

Evandar (27)

Numerical id

4

Author

Evandar (27)

Entry type

Proof

Classification

msc 03C35

Generated on Thu Feb 8 19:56:05 2018 by

LaTeXML
Vaught’s test

Vaught’s test

Let

L

be a

first order language

, and let

S

be a set of

L

-

sentences

with no finite models which is

κ

-categorical for some

κ

≥

|

L

|

. Then

S

is

complete

(

http://planetmath.org/FirstOrderTheories

).

Title

Vaught’s test

Canonical name

VaughtsTest

Date of creation

2013-03-22 13:00:37

Last modified on

2013-03-22 13:00:37

Owner

Evandar (27)

Last modified by

Evandar (27)

Numerical id

7

Author

Evandar (27)

Entry type

Theorem

Classification

msc 03C35

Related topic

KappaCategorical

Generated on Thu Feb 8 19:56:07 2018 by

LaTeXML
κ-categorical

κ

-categorical

Let

L

be a

first order language

and let

S

be a set of

L

-

sentences

. If

κ

is a

cardinal

, then

S

is said to be

if

S

has a model of

cardinality

κ

and any two such models are

isomorphic

.

In other words,

S

is

categorical

iff it has a unique model of cardinality

κ

, to within

isomorphism

.

Title

κ

-categorical

Canonical name

kappacategorical

Date of creation

2013-03-22 13:00:33

Last modified on

2013-03-22 13:00:33

Owner

Evandar (27)

Last modified by

Evandar (27)

Numerical id

4

Author

Evandar (27)

Entry type

Definition

Classification

msc 03C35

Related topic

VaughtsTest

Related topic

ExampleOfUniversalStructure

Generated on Thu Feb 8 19:56:09 2018 by

LaTeXML
example of strongly minimal

example of strongly minimal

Let

L

R

be the

language of rings

.
In other words

L

R

has two

constant symbols

0

,

1

, one unary symbol

-

, and two binary function symbols

+

,

⋅

satisfying the axioms (

identities

) of a ring. Let

T

be the

L

R

-theory that includes the

field axioms

and for each

n

the

formula

∀

x

0

,

x

1

,

…

,

x

n

∃

y

(

¬

(

⋀

1

≤

i

≤

n

x

i

=

0

)

→

∑

0

≤

i

≤

n

x

i

y

i

=

0

)

which expresses that every degree

n

polynomial

which is non constant has a root. Then any model of

T

is an algebraically closed field.

One can show that this is a

complete theory

and has

quantifier elimination

(Tarski).
Thus every

B

-

definable

subset of any

K

⊧

T

is definable by a

quantifier free formula

in

L

R

⁢

(

B

)

with one

free variable

y

.
A quantifier free formula is a

Boolean

combination

of atomic formulas.
Each of these is of the form

∑

i

≤

n

b

i

⁢

y

i

=

0

which defines a

finite set

.
Thus every definable subset of

K

is a finite or cofinite set.
Thus

K

and

T

are

strongly minimal

Title

example of strongly minimal

Canonical name

ExampleOfStronglyMinimal

Date of creation

2013-03-22 13:27:46

Last modified on

2013-03-22 13:27:46

Owner

CWoo (3771)

Last modified by

CWoo (3771)

Numerical id

7

Author

CWoo (3771)

Entry type

Example

Classification

msc 03C45

Classification

msc 03C10

Classification

msc 03C07

Related topic

AlgebraicallyClosed

Defines

language of rings

Generated on Thu Feb 8 19:56:11 2018 by

LaTeXML
homogeneous

homogeneous

Let

L

be a

first order language

. Let

M

be an

L

-

structure

.
Then we say

M

is

homogeneous

if the following holds:

if

σ

is an

isomorphism

between finite

substructures

of

M

,
then

σ

extends to an

automorphism

of

M

.

Title

homogeneous

Canonical name

Homogeneous

Date of creation

2013-03-22 13:23:13

Last modified on

2013-03-22 13:23:13

Owner

mathcam (2727)

Last modified by

mathcam (2727)

Numerical id

5

Author

mathcam (2727)

Entry type

Definition

Classification

msc 03C50

Related topic

ExampleOfUniversalStructure

Related topic

RandomGraph

Generated on Thu Feb 8 19:56:16 2018 by

LaTeXML
universal structure

universal structure

Let

L

be a

first order language

, and let

R

be an

elementary class

of

L

-

structures

.
Let

κ

be a

cardinal

.

R

κ

be the set of structures from

R

with

cardinality

less than or equal to

κ

.

Let

M

∈

R

κ

.
Suppose that for every

N

∈

R

κ

there is an

embedding

of

N

into

M

.
Then we say

M

is

universal

.

Title

universal structure

Canonical name

UniversalStructure

Date of creation

2013-03-22 13:23:09

Last modified on

2013-03-22 13:23:09

Owner

Timmy (1414)

Last modified by

Timmy (1414)

Numerical id

4

Author

Timmy (1414)

Entry type

Definition

Classification

msc 03C50

Classification

msc 03C52

Defines

universal

Generated on Thu Feb 8 19:56:18 2018 by

LaTeXML
amalgamation property

amalgamation property

A class of

L

-

structures

S

has the

amalgamation property

if and only if
whenever

A

,

B

1

,

B

2

∈

S

and

f

i

:

A

→

B

i

are

elementary embeddings

for

i

∈

{

1

,

2

}

then there is some

C

∈

S

and some elementary embeddings

g

i

:

B

i

→

C

for

i

∈

{

1

,

2

}

so that

g

1

⁢

(

f

1

⁢

(

x

)

)

=

g

2

⁢

(

f

2

⁢

(

x

)

)

for all

x

∈

A

. That is, the following diagram commutes.

\xymatrix

⁢

&

⁢

A

⁢

\ar

⁢

[

d

⁢

l

]

f

1

⁢

\ar

⁢

[

d

⁢

r

]

f

2

⁢

&

⁢

B

1

⁢

\ar

⁢

[

d

⁢

r

]

g

1

⁢

&

⁢

&

⁢

B

2

⁢

\ar

⁢

[

d

⁢

l

]

g

2

⁢

&

⁢

C

⁢

&

Compare this with the

free product with amalgamated subgroup

for groups and
the definition of pushout there.

Title

amalgamation property

Canonical name

AmalgamationProperty

Date of creation

2013-03-22 13:25:01

Last modified on

2013-03-22 13:25:01

Owner

Mathprof (13753)

Last modified by

Mathprof (13753)

Numerical id

8

Author

Mathprof (13753)

Entry type

Definition

Classification

msc 03C52

Related topic

FreeProductWithAmalgamatedSubgroup

Related topic

Confluence

Related topic

JointEmbeddingProperty

Defines

amalgamation property

Generated on Thu Feb 8 19:56:20 2018 by

LaTeXML
axiomatizable class

axiomatizable class

Let

L

be a

first order language

and

T

a theory in

L

. Recall that a model

M

is an

L

-

structure

such that

M

satisfies

every

sentence

in

T

. We say that the structure

M

is a model of

T

. Let us write

Mod

⁡

(

T

)

the class of all

L

-structures that are models of

T

.

Definition

. A class

K

of

L

-structures is said to be

axiomatizable

if there is a theory

T

such that

K

=

Mod

⁡

(

T

)

. Furthermore,

K

is a

finitely axiomatizable

or

elemenary class

if

T

is finite.

For example, the class of groups is

elementary

(and hence axiomatizable), because the set of group axioms is finite. However, the class of

infinite groups

is axiomatizable but not elementary. Similarly, the class of

R

-modules is elementary iff

R

is finite. The class of

locally finite groups

is an example of a non-axiomatizable class.

Remarks

.

•

K

is an elementary class iff there is a sentence

φ

such that

K

=

Mod

⁡

(

{

φ

}

)

, for sentences

φ

1

,

…

,

φ

n

can be combined to form

φ

1

∧

⋯

∧

φ

n

, which is also a sentence since it has no

free variables

.

•

A class is axiomatizable iff it is an

intersection

of elementary classes. As such elementary class is sometimes abbreviated EC, and axiomatizable class EC

Δ

, where

Δ

means is another symbol for intersection.

•

A caution to the reader: some authors call an elementary class an axiomatizable class that is defined here.

Title

axiomatizable class

Canonical name

AxiomatizableClass

Date of creation

2013-03-22 17:34:38

Last modified on

2013-03-22 17:34:38

Owner

CWoo (3771)

Last modified by

CWoo (3771)

Numerical id

7

Author

CWoo (3771)

Entry type

Definition

Classification

msc 03C52

Synonym

axiomatisable class

Synonym

finitely axiomatizable

Synonym

finitely axiomatisable

Synonym

EC

Synonym

EC

Δ

Related topic

Supercategories3

Related topic

AxiomaticAndCategoricalFoundationsOfMathematicsII2

Defines

elementary class

Generated on Thu Feb 8 19:56:30 2018 by

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joint embedding property

joint embedding property

Let

K

be a class of models (

structures

) of a given

signature

. We say that

K

has the

joint embedding property

(abbreviated

JEP

) iff for any models

A

and

B

in

K

there exists a model

C

in

K

such that both

A

and

B

are embeddable in

C

.

[

1

,

2

]

0.0.1

Examples

Examples include

[

2

]

:

•

The class of all groups.

•

The class of all monoids.

•

The class of all non-trivial

Boolean algebras

.

As is the case with the above examples, classes having the joint embedding property often

satisfy

an even stronger condition - for every indexed family of models in the class there is a model in the class into which each member of the family can be embedded. This is known as the

strong joint embedding property

(abbreviated

SJEP

).

[

3

]

In general any

factor embeddable class

closed under

products

will have the strong joint embedding property.

[

2

]

0.0.2

Characterizations

Elementary classes

with the joint embedding property may be characterized syntactically and semantically:

Let

T

be a

first order theory

in a

language

L

and let

K

be the class of models of

T

then:

1.

K

has the joint embedding property iff for all

universal sentences

ϕ

and

ψ

in

L

,

T

⊢

ϕ

∨

ψ

implies either

T

⊢

ϕ

or

T

⊢

ψ

.

[

1

]

2.

If

T

is

consistent

, then

K

has the joint embedding property iff

T

has an

ultra-universal model

.

[

2

]

References

1

Abraham Robinson:

Forcing

in

model theory

, Actes du Congrès International des Mathématiciens (Nice, 1970) Gauthier-Villars, Paris, 1971, pp. 245-250

2

Colin Naturman, Henry Rose:

Ultra-universal models

, Quaestiones Mathematicae, 15(2), 1992, 189-195

3

Colin Naturman:

Interior Algebras and Topology

, Ph.D. thesis, University of Cape Town Department of Mathematics, 1991

Title

joint embedding property

Canonical name

JointEmbeddingProperty

Date of creation

2013-03-22 19:36:14

Last modified on

2013-03-22 19:36:14

Owner

Naturman (26369)

Last modified by

Naturman (26369)

Numerical id

25

Author

Naturman (26369)

Entry type

Definition

Classification

msc 03C52

Synonym

JEP

Synonym

SJEP

Related topic

ultrauniversal

Related topic

amalgamationproperty

Related topic

UltraUniversal

Related topic

FactorEmbeddable

Defines

joint embedding property

Defines

strong joint embedding property

Generated on Thu Feb 8 19:56:34 2018 by

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o-minimality

o-minimality

Let

M

be an ordered

structure

. An

interval

in

M

is any subset of

M

that can be expressed in one of the following forms:

•

{

x

:

a

<

x

<

b

}

for some

a

,

b

from

M

•

{

x

:

x

>

a

}

for some

a

from

M

•

{

x

:

x

<

a

}

for some

a

from

M

Then we define

M

to be

o-minimal

iff every

definable

subset of

M

is a finite union of intervals and points. This is a property of the theory of

M

i.e. if

M

≡

N

and

M

is o-minimal, then

N

is o-minimal.
Note that

M

being o-minimal is

equivalent

to every definable subset of

M

being

quantifier free

definable in the

language

with just the ordering. Compare this with strong minimality.

The

model theory

of o-minimal structures is well understood, for an excellent account see Lou van den Dries, Tame topology and o-minimal structures, CUP 1998.
In particular, although this condition is merely on definable subsets of

M

it gives very good information about definable subsets of

M

n

for

n

∈

ω

.

Title

o-minimality

Canonical name

Ominimality

Date of creation

2013-03-22 13:23:01

Last modified on

2013-03-22 13:23:01

Owner

Timmy (1414)

Last modified by

Timmy (1414)

Numerical id

7

Author

Timmy (1414)

Entry type

Definition

Classification

msc 03C64

Classification

msc 14P10

Related topic

StronglyMinimal

Defines

o-minimal

Generated on Thu Feb 8 19:56:39 2018 by

LaTeXML
generalized quantifier

generalized quantifier

Generalized quantifiers

are an abstract way of defining

quantifiers

.

The underlying principle is that

formulas

quantified by a generalized quantifier are true if the set of

elements

satisfying those formulas belong in some

relation

associated with the quantifier.

Every generalized quantifier has an arity, which is the number of formulas it takes as

arguments

, and a type, which for an

n

-ary quantifier is a tuple of length

n

. The tuple

represents

the number of quantified

variables

for each argument.

The most common quantifiers are those of type

⟨

1

⟩

, including

∀

and

∃

. If

Q

is a quantifier of type

⟨

1

⟩

,

M

is the

universe

of a model, and

Q

M

is the relation associated with

Q

in that model, then

Q

⁢

x

⁢

ϕ

⁢

(

x

)

↔

{

x

∈

M

∣

ϕ

⁢

(

x

)

}

∈

Q

M

.

So

∀

M

=

{

M

}

, since the quantified formula is only true when all elements

satisfy

it. On the other hand

∃

M

=

P

⁢

(

M

)

-

{

∅

}

.

In general, the

monadic

quantifiers are those of type

⟨

1

,

…

,

1

⟩

and if

Q

is an

n

-ary monadic quantifier then

Q

M

⊆

P

⁢

(

M

)

n

. Härtig’s quantifier, for

instance

, is

⟨

1

,

1

⟩

, and

I

M

=

{

⟨

X

,

Y

⟩

∣

X

,

Y

⊆

M

∧

|

X

|

=

|

Y

|

}

.

A quantifier

Q

is

polyadic

if it is of type

⟨

n

1

,

…

,

n

n

⟩

where each

n

i

∈

ℕ

. Then:

Q

M

⊆

∏

i

P

⁢

(

M

n

i

)

These can get quite elaborate;

W

⁢

x

⁢

y

⁢

ϕ

⁢

(

x

,

y

)

is a

⟨

2

⟩

quantifier where

X

∈

W

M

↔

X

is a

well-ordering

. That is, it is true if the set of pairs making

ϕ

true is a well-ordering.

Title

generalized quantifier

Canonical

name

GeneralizedQuantifier

Date of creation

2013-03-22 12:59:57

Last modified on

2013-03-22 12:59:57

Owner

Henry (455)

Last modified by

Henry (455)

Numerical id

5

Author

Henry (455)

Entry type

Definition

Classification

msc 03C80

Classification

msc 03B15

Classification

msc 03B10

Related topic

quantifier

Related topic

Quantifier

Defines

monadic

Defines

polyadic

Generated on Thu Feb 8 19:56:41 2018 by

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