aarjaneiro/planetmath huggingface dataset:
Beth property

Beth property

A logic is said to have the

Beth property

if whenever a

predicate

R

is implicitly

definable

by

ϕ

(i.e. if all models have at most one unique

extension

satisfying

ϕ

), then

R

is explicitly definable relative to

ϕ

(i.e. there is a

ψ

not containing

R

,such that

ϕ

⊧

∀

x

1

,

.

.

,

x

n

(

R

(

x

1

,

…

,

x

n

)

↔

ψ

(

x

1

,

…

,

x

n

)

)

).

Title

Beth property

Canonical

name

BethProperty

Date of creation

2013-03-22 13:49:40

Last modified on

2013-03-22 13:49:40

Owner

Aatu (2569)

Last modified by

Aatu (2569)

Numerical id

7

Author

Aatu (2569)

Entry type

Definition

Classification

msc 03B99

Defines

Beth property

Defines

Beth definability property

Generated on Thu Feb 8 19:53:31 2018 by

LaTeXML
compactness

compactness

A logic is said to be

(

κ

,

λ

)

-compact, if the following holds

If

Φ

is a set of

sentences

of

cardinality

less than or equal to

κ

and all subsets of

Φ

of cardinality less than

λ

are

consistent

, then

Φ

is consistent.

For example,

first order logic

is

(

ω

,

ω

)

-compact, for if all finite subsets of some class of sentences are consistent, so is the class itself.

Title

compactness

Canonical

name

Compactness

Date of creation

2013-03-22 13:49:34

Last modified on

2013-03-22 13:49:34

Owner

Aatu (2569)

Last modified by

Aatu (2569)

Numerical id

5

Author

Aatu (2569)

Entry type

Definition

Classification

msc 03B99

Defines

compactness

Generated on Thu Feb 8 19:53:33 2018 by

LaTeXML
consistent

consistent

If

T

is a theory of

ℒ

then it is

consistent

iff there is some model

ℳ

of

ℒ

such that

ℳ

⊨

T

. If a theory is not consistent then it is inconsistent.

A slightly different definition is sometimes used, that

T

is consistent iff

T

⊬

⊥

(that is, as long as it does not prove a

contradiction

). As long as the proof calculus used is sound and

complete

, these two definitions are

equivalent

.

Title

consistent

Canonical name

Consistent

Date of creation

2013-03-22 13:00:20

Last modified on

2013-03-22 13:00:20

Owner

Henry (455)

Last modified by

Henry (455)

Numerical id

6

Author

Henry (455)

Entry type

Definition

Classification

msc 03B99

Defines

inconsistent

Generated on Thu Feb 8 19:53:34 2018 by

LaTeXML
Hofstadter’s MIU system

Hofstadter’s MIU system

The

alphabet

of the system contains three symbols

M

,

I

,

U

. The set of

theorem

is the set of string constructed by the rules and the axiom, is denoted by

𝒯

and can be built as follows:

(axiom)

M

⁢

I

∈

𝒯

.

(i)

If

x

⁢

I

∈

𝒯

then

x

⁢

I

⁢

U

∈

𝒯

.

(ii)

If

M

⁢

x

∈

𝒯

then

M

⁢

x

⁢

x

∈

𝒯

.

(iii)

In any theorem,

I

⁢

I

⁢

I

can be replaced by

U

.

(iv)

In any theorem,

U

⁢

U

can be omitted.

example:

•

Show that

M

⁢

U

⁢

I

⁢

I

∈

𝒯

M

⁢

I

∈

𝒯

by axiom

⟹

M

⁢

I

⁢

I

∈

𝒯

by rule (ii) where

x

=

I

⟹

M

⁢

I

⁢

I

⁢

I

⁢

I

∈

𝒯

by rule (ii) where

x

=

I

⁢

I

⟹

M

⁢

I

⁢

I

⁢

I

⁢

I

⁢

I

⁢

I

⁢

I

⁢

I

∈

𝒯

by rule (ii) where

x

=

I

⁢

I

⁢

I

⁢

I

⟹

M

⁢

I

⁢

I

⁢

I

⁢

I

⁢

I

⁢

I

⁢

I

⁢

I

⁢

U

∈

𝒯

by rule (i) where

x

=

M

⁢

I

⁢

I

⁢

I

⁢

I

⁢

I

⁢

I

⁢

I

⟹

M

⁢

I

⁢

I

⁢

I

⁢

I

⁢

I

⁢

U

⁢

U

∈

𝒯

by rule (iii)

⟹

M

⁢

I

⁢

I

⁢

I

⁢

I

⁢

I

∈

𝒯

by rule (iv)

⟹

M

⁢

U

⁢

I

⁢

I

∈

𝒯

by rule (iii)

•

Is

M

⁢

U

a theorem?

No. Why? Because the number of

I

’s of a theorem is never a multiple of 3. We will show this by structural induction.

base case

:

The statement is true for the base case. Since the axiom has one

I

. Therefore not a multiple of 3.

induction hypothesis:

Suppose true for

premise

of all rule.

induction

step:

By induction hypothesis we assume the premise of each rule to be true and show that the application of the rule keeps the staement true.

Rule 1:

Applying rule 1 does not add any

I

’s to the

formula

. Therefore the statement is true for rule 1 by induction hypothesis.

Rule 2:

Applying rule 2 doubles the amount of

I

’s of the formula but since the initial amount of

I

’s was not a multiple of 3 by induction hypothesis. Doubling that amount does not make it a multiple of 3

(i.e. if

n

≢

0

⁢

mod

⁡

3

then

2

⁢

n

≢

0

⁢

mod

⁡

3

)

. Therefore the statement is true for rule 2.

Rule 3:

Applying rule 3 replaces

I

⁢

I

⁢

I

by

U

. Since the initial amount of

I

’s was not a multiple of 3 by induction hypothesis. Removing

I

⁢

I

⁢

I

will not make the number of

I

’s in the formula be a multiple of 3. Therefore the statement is true for rule 3.

Rule 4:

Applying rule 4 removes

U

⁢

U

and does not change the amount of

I

’s. Since the initial amount of

I

’s was not a multiple of 3 by induction hypothesis. Therefore the statement is true for rule 4.

Therefore all theorems do not have a multiple of 3

I

’s.

[

HD

]

References

HD

Hofstader, R. Douglas: Gödel, Escher, Bach: an Eternal Golden Braid. Basic Books, Inc., New York, 1979.

Title

Hofstadter’s

MIU system

Canonical

name

HofstadtersMIUSystem

Date of creation

2013-03-22 13:57:48

Last modified on

2013-03-22 13:57:48

Owner

Daume (40)

Last modified by

Daume (40)

Numerical id

8

Author

Daume (40)

Entry type

Definition

Classification

msc 03B99

Synonym

MIU system

Generated on Thu Feb 8 19:53:58 2018 by

LaTeXML
interpolation property

interpolation property

A logic is said to have the

interpolation property

if whenever

ϕ

⁢

(

R

,

S

)

→

ψ

⁢

(

R

,

T

)

holds, then there is a

sentence

θ

⁢

(

R

)

, so that both

ϕ

⁢

(

R

,

S

)

→

θ

⁢

(

R

)

and

θ

⁢

(

R

)

→

ψ

⁢

(

R

,

T

)

hold, where

R

,

S

and

T

are some sets of symbols that

occur in

the

formulas

,

R

being the set of symbols common to both

ϕ

and

ψ

.

The interpolation property holds for

first order logic

. The interpolation property is related to

Beth definability property

and Robinson’s

consistency

property

. Also, a natural generalisation is the

concept

Δ

-closed logic.

Title

interpolation property

Canonical

name

InterpolationProperty

Date of creation

2013-03-22 13:49:36

Last modified on

2013-03-22 13:49:36

Owner

CWoo (3771)

Last modified by

CWoo (3771)

Numerical id

7

Author

CWoo (3771)

Entry type

Definition

Classification

msc 03B99

Defines

interpolation property

Generated on Thu Feb 8 19:54:04 2018 by

LaTeXML
metalanguage

metalanguage

A remedy for Berry’s

Paradox

and related paradoxes is to
separate the

language

used to formulate a particular mathematical
theory from the language used for its discourse.

The language used to formulate a mathematical theory is called the

object language

to contrast it from the

metalanguage

used for the discourse.

The most widely used object language is the

first-order logic

. The
metalanguage could be English or other natural languages plus
mathematical symbols such as

⇒

.

Examples

1.

The object language speaks of

(

¬

⁢

A

n

)

, but we speak
of

⟨

(

,

¬

,

A

n

,

)

⟩

in the metalanguage.
[Recall that a

formula

is some

finite sequence

of the symbols.
Cf.

First Order Logic

or

Propositional Logic

.]

2.

In

induction

proofs, one might encounter “the first
symbol in the formula

φ

is

(

;” we know that the first
symbol is indeed

(

and not

⟨

because

⟨

is a
symbol in our metalanguage. Similarly, “the third symbol is

A

n

” and not

,

because

,

is a symbol in our metalanguage.

3.

⊢

and

⊧

are members of the metalanguage,

not

of object language.

4.

Parallel with the notion of metalanguage is

metatheorem

.
“

Γ

⊢

(

φ

→

ψ

)

if

Γ

∪

{

φ

}

⊢

ψ

,

Γ

⊆

ℒ

0

,

φ

,

ψ

∈

ℒ

0

” is a metatheorem.

5.

Examples from

Set Theory

. Let “Con” denote

consistency

. Then Con(ZF) and Con(ZF+AC+GCH) are metamathematical
statements; they are statements in the metalanguage.

References

1

Schechter, E.,

Handbook of Analysis and Its Foundations

, 1st ed., Academic Press, 1997.

Title

metalanguage

Canonical name

Metalanguage

Date of creation

2013-03-22 18:06:05

Last modified on

2013-03-22 18:06:05

Owner

yesitis (13730)

Last modified by

yesitis (13730)

Numerical id

6

Author

yesitis (13730)

Entry type

Definition

Classification

msc 03B99

Generated on Thu Feb 8 19:54:06 2018 by

LaTeXML
sentence

sentence

A

sentence

is a

formula

with no

free variables

.

Simple examples include:

•

∀

x

∃

y

[

x

<

y

]

•

∃

z

[

z

+

7

-

43

=

0

]

•

1

+

2

<

2

+

3

Note that the last sentence contains no

variables

.

A sentence is also called a

closed formula

. A formula that is not a sentence is called an

open formula

.

The following formula is open:

x

+

2

=

3

Remark

. In

first-order logic

, the main

difference

between a sentence and an open formula, semantically, is that a sentence has a definite truth value, whereas the truth value of an open formula may vary, depending on the

interpretations

of the free variables occurring in the formula. In the open formula above, if

x

were

1

, then the formula is true. Otherwise, it is false.

Every open formula may be converted into a sentence by placing

quantifiers

in front of it. Given a formula

φ

, the

universal closure

of

φ

is the sentence

∀

x

1

⁢

∀

x

2

⁢

⋯

⁢

∀

x

n

⁢

φ

where

{

x

1

,

…

,

x

n

}

is the set of all free variables occurring in

φ

.

The

existential closure

of a formula

φ

may be defined similarly.

For example, the universal closure of

x

+

2

=

3

is

∀

x

[

x

+

2

=

3

]

,

and its existential closure is

∃

x

[

x

+

2

=

3

]

.

Note that the first sentence is false, while the second is true.

Title

sentence

Canonical

name

Sentence

Date of creation

2013-03-22 13:00:24

Last modified on

2013-03-22 13:00:24

Owner

Henry (455)

Last modified by

Henry (455)

Numerical id

7

Author

Henry (455)

Entry type

Definition

Classification

msc 03B99

Synonym

closed formula

Defines

open formula

Defines

universal closure

Defines

existential closure

Generated on Thu Feb 8 19:54:08 2018 by

LaTeXML
Tarski’s result on the undefinability of truth

Tarski’s result on the undefinability of truth

Assume

𝐋

is a logic which is under contradictory

negation

and has the usual truth-functional

connectives

. Assume also that

𝐋

has a notion of

formula

with one

variable

and of

substitution

. Assume that

T

is a theory of

𝐋

in which we can define surrogates for formulae of

𝐋

, and in which all true

instances

of the substitution

relation

and the truth-functional connective relations are provable. We show that either

T

is

inconsistent

or

T

can’t be augmented with a truth

predicate

𝐓𝐫𝐮𝐞

for which the following T-schema holds

𝐓𝐫𝐮𝐞

(

′

ϕ

′

)

↔

ϕ

Assume that the formulae with one variable of

𝐋

have been

indexed by

some suitable set that is representable in

T

(otherwise the predicate

𝐓𝐫𝐮𝐞

would be next to useless, since if there’s no way to speak of

sentences

of a logic, there’s little hope to define a truth-predicate for it). Denote the

i

:th

element

in this indexing by

B

i

. Consider now the following open formula with one variable

𝐋𝐢𝐚𝐫

⁢

(

x

)

=

¬

⁢

𝐓𝐫𝐮𝐞

⁢

(

B

x

⁢

(

x

)

)

Now, since

𝐋𝐢𝐚𝐫

is an open formula with one

free variable

it’s indexed by some

i

. Now consider the sentence

𝐋𝐢𝐚𝐫

⁢

(

i

)

. From the T-schema we know that

𝐓𝐫𝐮𝐞

⁢

(

𝐋𝐢𝐚𝐫

⁢

(

i

)

)

↔

𝐋𝐢𝐚𝐫

⁢

(

𝐢

)

and by the

definition

of

𝐋𝐢𝐚𝐫

and the fact that

i

is the of

𝐋𝐢𝐚𝐫

⁢

(

x

)

we have

𝐓𝐫𝐮𝐞

⁢

(

𝐋𝐢𝐚𝐫

⁢

(

i

)

)

↔

¬

⁢

𝐓𝐫𝐮𝐞

⁢

(

𝐋𝐢𝐚𝐫

⁢

(

𝐢

)

)

which clearly is absurd. Thus there can’t be an of

T

with a predicate

𝐓𝐫𝐮𝐭𝐡

for which the T-schema holds.

We have made several

assumptions

on the logic

𝐋

which are crucial in order for this proof to go through. The most important is that

𝐋

is

closed under

contradictory negation. There are logics which allow truth-predicates, but these are not usually closed under contradictory negation (so that it’s possible that

𝐓𝐫𝐮𝐞

⁢

(

𝐋𝐢𝐚𝐫

⁢

(

i

)

)

is neither true nor false). These logics usually have stronger notions of negation, so that a sentence

¬

⁢

P

says

more

than just that

P

is not true, and the

proposition

that

P

is simply not true is not expressible.

An example of a logic for which Tarski’s undefinability result does not hold is the so-called

Independence Friendly logic

, the

semantics

of which is based on game theory and which allows various generalised

quantifiers

(the Henkin

branching quantifier

, etc.) to be used.

Title

Tarski’s result on the undefinability of truth

Canonical

name

TarskisResultOnTheUndefinabilityOfTruth

Date of creation

2013-03-22 13:49:19

Last modified on

2013-03-22 13:49:19

Owner

mathcam (2727)

Last modified by

mathcam (2727)

Numerical id

14

Author

mathcam (2727)

Entry type

Theorem

Classification

msc 03B99

Related topic

IFLogic

Generated on Thu Feb 8 19:54:22 2018 by

LaTeXML
congruence

congruence

Let

Σ

be a fixed

signature

, and

𝔄

a

structure

for

Σ

. A

congruence

∼

on

𝔄

is an

equivalence relation

such that for every

natural number

n

and

n

-ary function symbol

F

of

Σ

, if

a

i

∼

a

i

′

then

F

𝔄

⁢

(

a

1

,

…

⁢

a

n

)

∼

F

𝔄

⁢

(

a

1

′

,

…

⁢

a

n

′

)

.

Title

congruence

Canonical name

Congruence12

Date of creation

2013-03-22 13:46:38

Last modified on

2013-03-22 13:46:38

Owner

almann (2526)

Last modified by

almann (2526)

Numerical id

9

Author

almann (2526)

Entry type

Definition

Classification

msc 03C05

Classification

msc 03C07

Related topic

CongruenceRelationOnAnAlgebraicSystem

Generated on Thu Feb 8 19:54:24 2018 by

LaTeXML
homomorphic image of a Σ-structure is a Σ-structure

homomorphic image of a

Σ

-structure is a

Σ

-structure

Let

Σ

be a fixed

signature

, and

𝔄

and

𝔅

two

structures

for

Σ

. If

f

:

𝔄

→

𝔅

is a

homomorphism

, then

im

⁡

(

f

)

is a structure for

Σ

.

Title

homomorphic image of a

Σ

-structure is a

Σ

-structure

Canonical name

HomomorphicImageOfASigmastructureIsASigmastructure

Date of creation

2013-03-22 13:46:44

Last modified on

2013-03-22 13:46:44

Owner

almann (2526)

Last modified by

almann (2526)

Numerical id

6

Author

almann (2526)

Entry type

Theorem

Classification

msc 03C05

Classification

msc 03C07

Generated on Thu Feb 8 19:54:26 2018 by

LaTeXML
kernel

kernel

Let

Σ

be a fixed

signature

, and

𝔄

and

𝔅

be two

structures

for

Σ

. Given a

homomorphism

f

:

𝔄

→

𝔅

, the

kernel

of

f

is the

relation

ker

⁡

(

f

)

on

A

defined by

⟨

a

,

a

′

⟩

∈

ker

⁡

(

f

)

⇔

f

⁢

(

a

)

=

f

⁢

(

a

′

)

.

So defined, the kernel of

f

is a

congruence

on

𝔄

. If

Σ

has a constant symbol 0, then the kernel of

f

is often defined to be the

preimage

of

0

𝔅

under

f

. Under this definition, if

{

0

𝔅

}

is a

substructure

of

𝔅

, then the kernel of

f

is a substructure of

𝔄

.

Title

kernel

Canonical name

Kernel1

Date of creation

2013-03-22 13:46:34

Last modified on

2013-03-22 13:46:34

Owner

almann (2526)

Last modified by

almann (2526)

Numerical id

11

Author

almann (2526)

Entry type

Definition

Classification

msc 03C05

Classification

msc 03C07

Related topic

Kernel

Related topic

KernelOfAGroupHomomorphism

Related topic

KernelOfALinearTransformation

Generated on Thu Feb 8 19:54:28 2018 by

LaTeXML
kernel of a homomorphism is a congruence

kernel of a homomorphism is a congruence

Let

Σ

be a fixed

signature

, and

𝔄

and

𝔅

two

structures

for

Σ

. If

f

:

𝔄

→

𝔅

is a

homomorphism

, then

ker

⁡

(

f

)

is a

congruence

on

𝔄

.

Proof.

If

F

is an

n

-ary function symbol of

Σ

, and

f

⁢

(

a

i

)

=

f

⁢

(

a

i

′

)

, then

f

⁢

(

F

𝔄

⁢

(

a

1

,

…

,

a

n

)

)

=

F

𝔅

⁢

(

f

⁢

(

a

1

)

,

…

,

f

⁢

(

a

n

)

)

=

F

𝔅

⁢

(

f

⁢

(

a

1

′

)

,

…

,

f

⁢

(

a

n

′

)

)

=

f

⁢

(

F

𝔄

⁢

(

a

1

′

,

…

,

a

n

′

)

)

.

∎

Title

kernel of a homomorphism is a congruence

Canonical name

KernelOfAHomomorphismIsACongruence

Date of creation

2013-03-22 13:48:03

Last modified on

2013-03-22 13:48:03

Owner

almann (2526)

Last modified by

almann (2526)

Numerical id

8

Author

almann (2526)

Entry type

Theorem

Classification

msc 03C05

Classification

msc 03C07

Related topic

KernelOfAHomomorphismBetweenAlgebraicSystems

Generated on Thu Feb 8 19:54:30 2018 by

LaTeXML
quotient structure

quotient structure

Let

Σ

be a fixed

signature

,

𝔄

a

structure

for

Σ

, and

∼

a

congruence

on

𝔄

. The

quotient structure

of

𝔄

by

∼

, denoted

𝔄

/

∼

, is defined as follows:

1.

The

universe

of

𝔄

/

∼

is the set

{

[

[

a

]

]

∣

a

∈

𝔄

}

.

2.

For each constant symbol

c

of

Σ

,

c

𝔄

⁣

/

∼

=

[

[

c

𝔄

]

]

.

3.

For every

natural number

n

and every

n

-ary function symbol

F

of

Σ

,

F

𝔄

⁣

/

∼

⁢

(

[

[

a

1

]

]

,

…

⁢

[

[

a

n

]

]

)

=

[

[

F

𝔄

⁢

(

a

1

,

…

⁢

a

n

)

]

]

.

4.

For every natural number

n

and every

n

-ary relation symbol

R

of

Σ

,

R

𝔄

⁣

/

∼

⁢

(

[

[

a

1

]

]

,

…

,

[

[

a

n

]

]

)

if and only if for some

a

i

′

∼

a

i

we have

R

𝔄

⁢

(

a

1

′

,

…

,

a

n

′

)

.

Title

quotient structure

Canonical name

QuotientStructure

Date of creation

2013-03-22 13:46:41

Last modified on

2013-03-22 13:46:41

Owner

almann (2526)

Last modified by

almann (2526)

Numerical id

10

Author

almann (2526)

Entry type

Definition

Classification

msc 03C05

Classification

msc 03C07

Generated on Thu Feb 8 19:54:33 2018 by

LaTeXML
alphabet

alphabet

An

alphabet

Σ

is a nonempty

finite set

such that every string formed by elements of

Σ

can be decomposed uniquely into elements of

Σ

.

For example,

{

b

,

l

⁢

o

,

g

,

b

⁢

l

,

o

⁢

g

}

is not a valid alphabet because the string

b

⁢

l

⁢

o

⁢

g

can be broken up in two ways:

b lo g

and

bl og

.

{

ℂ

⁢

a

,

n

¨

⁢

a

,

d

,

a

}

is

a valid alphabet, because there
is only one way to fully break up any given string formed from it.

If

Σ

is our alphabet and

n

∈

ℤ

+

,
we define the following as the

powers of

Σ

:

•

Σ

0

=

λ

, where

λ

stands for the

empty string

.

•

Σ

n

=

{

x

⁢

y

|

x

∈

Σ

,

y

∈

Σ

n

-

1

}

(

x

⁢

y

is the

juxtaposition

of

x

and

y

)

So,

Σ

n

is the set of all strings formed from

Σ

of length

n

.

Title

alphabet

Canonical name

Alphabet

Date of creation

2013-03-22 12:15:58

Last modified on

2013-03-22 12:15:58

Owner

mathcam (2727)

Last modified by

mathcam (2727)

Numerical id

7

Author

mathcam (2727)

Entry type

Definition

Classification

msc 03C07

Synonym

powers of an alphabet

Related topic

KleeneStar

Related topic

Substring

Related topic

Language

Related topic

HuffmanCoding

Related topic

Word

Generated on Thu Feb 8 19:54:35 2018 by

LaTeXML
definable

definable

0.1

Definable sets and functions

0.1.1

Definability In Model Theory

Let

ℒ

be a

first order language

. Let

M

be an

ℒ

-

structure

. Denote

x

1

,

…

,

x

n

by

x

→

and

y

1

,

…

,

y

m

by

y

→

, and suppose

ϕ

⁢

(

x

→

,

y

→

)

is a

formula

from

ℒ

, and

b

1

,

…

,

b

m

is some

sequence

from

M

.

Then we write

ϕ

⁢

(

M

n

,

b

→

)

to denote

{

a

→

∈

M

n

:

M

⊧

ϕ

⁢

(

a

→

,

b

→

)

}

.
We say that

ϕ

⁢

(

M

n

,

b

→

)

is

b

→

-

definable

.
More generally if

S

is some set and

B

⊆

M

, and there is some

b

→

from

B

so that

S

is

b

→

-definable then we say that

S

is

B

-definable.

In particular we say that a set

S

is

∅

-definable or zero definable iff it is the solution set of some formula without parameters.

Let

f

be a

function

, then we say

f

is

B

-definable iff the graph of

f

(i.e.

{

(

x

,

y

)

:

f

⁢

(

x

)

=

y

}

) is a

B

-

definable set

.

If

S

is

B

-definable then any

automorphism

of

M

that fixes

B

pointwise

, fixes

S

setwise.

A set or function is

definable

iff it is

B

-definable for some parameters

B

.

Some authors use the term definable to mean what we have called

∅

-definable here.
If this is the convention of a paper, then the term

parameter definable

will refer to sets that are definable over some parameters.

Sometimes in

model theory

it is not actually very important what

language

one is using, but merely what the definable sets are, or what the definability

relation

is.

0.1.2

Definability of functions in Proof Theory

In

proof theory

, given a theory

T

in the language

ℒ

, for a function

f

:

M

→

M

to be

definable in the theory

T

, we have two conditions:

(i) There is a formula in the language

ℒ

s.t.

f

is definable over the model

M

, as in the above definition; i.e., its graph is definable in the language

ℒ

over the model

M

, by some formula

ϕ

⁢

(

x

→

,

y

)

.

(ii) The theory

T

proves

that

f

is indeed a function, that is

T

⊢

∀

x

→

∃

!

y

.

ϕ

(

x

→

,

y

)

.

For example: the

graph

of

exponentiation

function

x

y

=

z

is definable by the language of the theory

I

⁢

Δ

0

(a subsystem of PA, with

induction axiom

restricted to bounded formulas only), however the function itself is

not

definable in this theory.

Title

definable

Canonical name

Definable

Date of creation

2013-03-22 13:26:52

Last modified on

2013-03-22 13:26:52

Owner

CWoo (3771)

Last modified by

CWoo (3771)

Numerical id

17

Author

CWoo (3771)

Entry type

Definition

Classification

msc 03C07

Classification

msc 03F03

Defines

definable set

Defines

definable function

Defines

definable relation

Generated on Thu Feb 8 19:54:37 2018 by

LaTeXML
definable type

definable type

Let

M

be a

first order

structure

.
Let

A

and

B

be sets of

parameters

from

M

.
Let

p

be a

complete

n

-type over

B

.
Then we say that

p

is an

A

-

definable

type iff
for every

formula

ψ

⁢

(

x

¯

,

y

¯

)

with ln

(

x

¯

)

=

n

,
there is some formula

d

⁢

ψ

⁢

(

y

¯

,

z

¯

)

and some parameters

a

¯

from

A

so that
for any

b

¯

from

B

we have

ψ

⁢

(

x

¯

,

b

¯

)

∈

p

iff

M

⊧

d

⁢

ψ

⁢

(

b

¯

,

a

¯

)

.

Note that if

p

is a type over the model

M

then this condition is

equivalent

to showing that

{

b

¯

∈

M

:

ψ

⁢

(

x

¯

,

b

¯

)

∈

M

}

is an

A

-

definable set

.

For

p

a type over

B

, we say

p

is

definable

if it is

B

-definable.

If

p

is definable, we call

d

⁢

ψ

the

defining formula

for

ψ

, and the

function

ψ

↦

d

⁢

ψ

a

defining scheme

for

p

.

Title

definable type

Canonical

name

DefinableType

Date of creation

2013-03-22 13:29:26

Last modified on

2013-03-22 13:29:26

Owner

Timmy (1414)

Last modified by

Timmy (1414)

Numerical id

4

Author

Timmy (1414)

Entry type

Definition

Classification

msc 03C07

Classification

msc 03C45

Related topic

Type2

Defines

definable type

Defines

defining scheme

Generated on Thu Feb 8 19:54:39 2018 by

LaTeXML
downward Lowenheim-Skolem theorem

downward Lowenheim-Skolem theorem

Let

L

be a

first order language

, let

𝒜

be an

L

-

structure

and let

K

⊆

dom

⁡

(

𝒜

)

. Then there is an

L

-structure

ℬ

such that

K

⊆

ℬ

and

|

ℬ

|

≤

Max

⁡

(

|

K

|

,

|

L

|

)

and

ℬ

is elementarily embedded in

𝒜

.

Title

downward Lowenheim-Skolem theorem

Canonical name

DownwardLowenheimSkolemTheorem

Date of creation

2013-03-22 13:00:42

Last modified on

2013-03-22 13:00:42

Owner

Evandar (27)

Last modified by

Evandar (27)

Numerical id

5

Author

Evandar (27)

Entry type

Theorem

Classification

msc 03C07

Generated on Thu Feb 8 19:54:41 2018 by

LaTeXML