aarjaneiro/planetmath huggingface dataset:
example of transfinite induction

example of transfinite induction

Suppose we are interested in showing the

property

Φ

⁢

(

α

)

holds for all

ordinals

α

using

transfinite induction

. The proof basically involves three steps:

1.

(first ordinal) show that

Φ

⁢

(

0

)

holds;

2.

(

successor ordinal

) if

Φ

⁢

(

β

)

holds, then

Φ

⁢

(

S

⁢

β

)

holds;

3.

(

limit ordinal

) if

Φ

⁢

(

γ

)

holds for all

γ

<

β

and

β

=

sup

⁡

{

γ

∣

γ

<

β

}

, then

Φ

⁢

(

β

)

holds.

Below is an example of a proof using transfinite induction.

Proposition 1

.

0

+

α

=

α

for any ordinal

α

.

Proof.

Let

Φ

⁢

(

α

)

be the property:

0

+

α

=

α

. We follow the three steps outlined above.

1.

Since

0

+

0

=

0

by

definition

,

Φ

⁢

(

0

)

holds.

2.

Suppose

0

+

β

=

β

. By definition

0

+

S

⁢

β

=

S

⁢

(

0

+

β

)

, which is equal to

S

⁢

β

by the

induction hypothesis

, so

Φ

⁢

(

S

⁢

β

)

holds.

3.

Suppose

β

=

sup

⁡

{

γ

∣

γ

<

β

}

and

0

+

γ

=

γ

for all

γ

<

β

. Then

0

+

β

=

0

+

sup

⁡

{

γ

∣

γ

<

β

}

=

sup

⁡

{

0

+

γ

∣

γ

<

β

}

.

The second equality follows from definition. Furthermore, the last

expression

above is equal to

sup

⁡

{

γ

∣

γ

<

β

}

=

β

by the induction hypothesis. So

Φ

⁢

(

β

)

holds.

Therefore

Φ

⁢

(

α

)

holds for every ordinal

α

, which is the statement of the

theorem

, completing the proof.
∎

Title

example of transfinite induction

Canonical

name

ExampleOfTransfiniteInduction

Date of creation

2013-03-22 17:51:12

Last modified on

2013-03-22 17:51:12

Owner

CWoo (3771)

Last modified by

CWoo (3771)

Numerical id

6

Author

CWoo (3771)

Entry type

Example

Classification

msc 03B10

Generated on Thu Feb 8 19:26:56 2018 by

LaTeXML
example of well-founded induction

example of well-founded induction

This proof of the

fundamental theorem of arithmetic

(every

natural number

has a

prime factorization

) affords an example of proof by

well-founded induction

over a

well-founded relation

that is not a linear order.

First note that the division

relation

is obviously well-founded. The

|

-

minimal elements

are the

prime numbers

. We detail the two steps of the proof :

1.

If

n

is prime, then

n

is its own factorization into primes, so the assertion is true for the

|

-minimal elements.

2.

If

n

is not prime, then

n

has a non-trivial factorization (by definition of not being prime), i.e.

n

=

m

⁢

ℓ

, where

m

,

n

≠

1

. By

induction

,

m

and

ℓ

have prime factorizations, the

product

of which is a prime factorization of

n

.

Title

example of well-founded induction

Canonical name

ExampleOfWellfoundedInduction

Date of creation

2013-03-22 12:42:23

Last modified on

2013-03-22 12:42:23

Owner

CWoo (3771)

Last modified by

CWoo (3771)

Numerical id

9

Author

CWoo (3771)

Entry type

Example

Classification

msc 03B10

Generated on Thu Feb 8 19:27:01 2018 by

LaTeXML
first order logic

first order logic

A logic is

first order

if it has exactly one type. Usually the term refers specifically to the logic with

connectives

¬

,

∨

,

∧

,

→

, and

↔

and the

quantifiers

∀

and

∃

, all given the usual

semantics

:

•

¬

⁢

ϕ

is true iff

ϕ

is not true

•

ϕ

∨

ψ

is true if either

ϕ

is true or

ψ

is true

•

∀

x

⁢

ϕ

⁢

(

x

)

is true iff

ϕ

x

t

is true for every object

t

(where

ϕ

x

t

is the result of replacing every unbound

occurrence

of

x

in

ϕ

with

t

)

•

ϕ

∧

ψ

is the same as

¬

⁡

(

¬

⁢

ϕ

∨

¬

⁢

ψ

)

•

ϕ

→

ψ

is the same as

(

¬

⁢

ϕ

)

∨

ψ

•

ϕ

↔

ψ

is the same as

(

ϕ

→

ψ

)

∧

(

ψ

→

ϕ

)

•

∃

x

⁢

ϕ

⁢

(

x

)

is the same as

¬

⁢

∀

x

⁢

¬

⁢

ϕ

⁢

(

x

)

However

languages

with slightly different quantifiers and connectives are sometimes still called first order as long as there is only one type.

Title

first order logic

Canonical

name

FirstOrderLogic

Date of creation

2013-03-22 13:00:06

Last modified on

2013-03-22 13:00:06

Owner

Henry (455)

Last modified by

Henry (455)

Numerical id

7

Author

Henry (455)

Entry type

Definition

Classification

msc 03B10

Synonym

classical first order logic

Synonym

FO

Generated on Thu Feb 8 19:27:03 2018 by

LaTeXML
Gödel numbering

Gödel numbering

A

Gödel numbering

is any way of assigning numbers to the

formulas

of a

language

. This is often useful in allowing

sentences

of a language to be self-referential. The number associated with a formula

ϕ

is called its

Gödel number

and is denoted

⌜

⁢

ϕ

⁢

⌝

.

More formally, if

ℒ

is a language and

𝒢

is a

surjective

partial function

from the terms of

ℒ

to the formulas over

ℒ

then

𝒢

is a Gödel numbering.

⌜

⁢

ϕ

⁢

⌝

may be any term

t

such that

𝒢

⁢

(

t

)

=

ϕ

. Note that

𝒢

is not defined within

ℒ

(there is no formula or object of

ℒ

representing

𝒢

), however

properties

of it (such as being in the domain of

𝒢

, being a

subformula

, and so on) are.

Athough anything meeting the properties above is a Gödel numbering, depending on the specific language and usage, any of the following properties may also be desired (and can often be found if more effort is put into the numbering):

•

If

ϕ

is a subformula of

ψ

then

⌜

⁢

ϕ

⁢

⌝

<

⌜

⁢

ψ

⁢

⌝

•

For every number

n

, there is some

ϕ

such that

⌜

⁢

ϕ

⁢

⌝

=

n

•

𝒢

is

injective

Title

Gödel numbering

Canonical

name

GodelNumbering

Date of creation

2013-03-22 12:58:21

Last modified on

2013-03-22 12:58:21

Owner

Henry (455)

Last modified by

Henry (455)

Numerical id

8

Author

Henry (455)

Entry type

Definition

Classification

msc 03B10

Related topic

BeyondFormalism

Defines

Gödel number

Generated on Thu Feb 8 19:27:05 2018 by

LaTeXML
Herbrand structure

Herbrand structure

For a

language

ℒ

, define the

Herbrand universe

to be the set of closed terms (alternatively ground terms) of

L

.

A

structure

𝔐

for

ℒ

is a

Herbrand structure

if the domain of

𝔐

is the Herbrand universe of

ℒ

. This fixes the domain of

𝔐

, and so each Herbrand structure can be identified with its

interpretation

, leading to the alternative nomenclature of

Herbrand interpretation

.

A

Herbrand model

of a theory

T

is a Herbrand structure which is a model of

T

.

Title

Herbrand structure

Canonical name

HerbrandStructure

Date of creation

2013-03-22 14:33:13

Last modified on

2013-03-22 14:33:13

Owner

iwnbap (1760)

Last modified by

iwnbap (1760)

Numerical id

4

Author

iwnbap (1760)

Entry type

Definition

Classification

msc 03B10

Synonym

HerbrandModel HerbrandInterpretation

Related topic

HerbrandsTheoremFirstOrderLogic

Defines

HerbrandModel HerbrandInterpretation HerbrandUniverse

Generated on Thu Feb 8 19:27:10 2018 by

LaTeXML
Herbrand’s theorem (first order logic)

Herbrand’s theorem (first order logic)

Let

T

be a

first-order theory

consisting of

open formulas

only. Then:

1.

If

T

is

satisfiable

, it has a Herbrand model

2.

If

T

is not satisfiable, there is a finite subset of the set of ground

instances

of

formulas

of

T

which is

unsatisfiable

.

Title

Herbrand’s

theorem

(

first order logic

)

Canonical name

HerbrandsTheoremfirstOrderLogic

Date of creation

2013-03-22 14:33:10

Last modified on

2013-03-22 14:33:10

Owner

iwnbap (1760)

Last modified by

iwnbap (1760)

Numerical id

6

Author

iwnbap (1760)

Entry type

Theorem

Classification

msc 03B10

Related topic

HerbrandStructure

Generated on Thu Feb 8 19:27:12 2018 by

LaTeXML
Lindström’s theorem

Lindström’s theorem

One of the very first results of the study of model theoretic logics is a characterization theorem due to Per Lindström. He showed that the

classical first order logic

is the strongest logic having the following properties

•

Being

closed under

contradictory

negation

•

Compactness

•

Löwenheim-Skolem

theorem

also, he showed that first order logic can be characterised as the strongest logic for which the following hold

•

Completeness (r.e. axiomatisability)

•

Löwenheim-Skolem theorem

The notion of “strength” used here is as follows. A logic

𝐋

′

is stronger than

𝐋

or as strong if every class of

structures

definable

in

𝐋

is also definable in

𝐋

′

.

Title

Lindström’s theorem

Canonical name

LindstromsTheorem

Date of creation

2013-03-22 13:49:30

Last modified on

2013-03-22 13:49:30

Owner

mathcam (2727)

Last modified by

mathcam (2727)

Numerical id

8

Author

mathcam (2727)

Entry type

Theorem

Classification

msc 03B10

Generated on Thu Feb 8 19:27:13 2018 by

LaTeXML
prenex form

prenex form

A

formula

in

first order logic

is said to be in

prenex form

if all

quantifiers

occur in

the front of the formula, before any occurrences of

predicates

and

connectives

. Schematically, a

proposition

in prenex form will appear as follows

(

Q

1

⁢

x

1

)

⁢

(

Q

1

⁢

x

1

)

⁢

…

⁢

⟨

matrix

⟩

where each “

Q

” stands for either “

∀

” or “

∃

” and “

⟨

matrix

⟩

” is constructed from predicates and connectives. For example, the proposition

(

∀

x

)

(

∃

y

)

(

∀

z

)

(

x

>

z

→

y

>

z

)

is in prenex form whilst the statement

(

∀

x

)

(

x

>

0

→

(

∃

y

)

(

x

=

y

2

)

)

is not in prenex form, but is

equivalent

to the statement

(

∀

x

)

(

∃

y

)

(

x

>

0

→

x

=

y

2

)

which is in prenex form.

This requirement that the statement be expressed in prenex form puts no real

restriction

on the statements which we may consider because it is possible to systematically express any statement in prenex form by a systematic use of the several equivalences:

Title

prenex form

Canonical name

PrenexForm

Date of creation

2013-03-22 15:06:55

Last modified on

2013-03-22 15:06:55

Owner

rspuzio (6075)

Last modified by

rspuzio (6075)

Numerical id

7

Author

rspuzio (6075)

Entry type

Definition

Classification

msc 03B10

Classification

msc 03C07

Generated on Thu Feb 8 19:27:15 2018 by

LaTeXML
Presburger arithmetic

Presburger arithmetic

Presburger arithmetic

is a weakened form of

arithmetic

which includes the

structure

ℕ

, the

constant

0

, the unary function

S

, the binary function

+

, and the

binary relation

<

. Essentially, it is

Peano arithmetic

without

multiplication

.

The axioms are:

1.

0

≠

S

⁢

x

2.

S

⁢

x

=

S

⁢

y

→

x

=

y

3.

x

+

0

=

x

4.

x

+

S

⁢

y

=

S

⁢

(

x

+

y

)

5.

For each

first order

formula

P

⁢

(

x

)

,

P

(

0

)

∧

∀

x

[

P

(

x

)

→

P

(

x

+

1

)

]

→

∀

x

P

(

x

)

Presburger arithmetic is

decidable

, but is consequently very limited in what it can express.

Title

Presburger arithmetic

Canonical

name

PresburgerArithmetic

Date of creation

2013-03-22 12:51:11

Last modified on

2013-03-22 12:51:11

Owner

Henry (455)

Last modified by

Henry (455)

Numerical id

8

Author

Henry (455)

Entry type

Definition

Classification

msc 03B10

Related topic

PeanoArithmetic

Generated on Thu Feb 8 19:27:17 2018 by

LaTeXML
proof of compactness theorem for first order logic

proof of compactness theorem for first order logic

The

theorem

states that if a set of

sentences

of a

first-order language

L

is

inconsistent

, then some finite subset of it is inconsistent. Suppose

Δ

⊆

L

is inconsistent. Then by definition

Δ

⊢

⟂

, i.e. there is a formal proof of “false” using only

assumptions

from

Δ

. Formal proofs are finite objects, so let

Γ

collect all the

formulas

of

Δ

that are used in the proof.

Title

proof of compactness theorem for first order logic

Canonical name

ProofOfCompactnessTheoremForFirstOrderLogic

Date of creation

2013-03-22 12:44:02

Last modified on

2013-03-22 12:44:02

Owner

CWoo (3771)

Last modified by

CWoo (3771)

Numerical id

4

Author

CWoo (3771)

Entry type

Proof

Classification

msc 03B10

Classification

msc 03C07

Generated on Thu Feb 8 19:27:22 2018 by

LaTeXML
proof of principle of transfinite induction

proof of principle of transfinite induction

To prove the

transfinite induction

theorem

, we note that the class of

ordinals

is

well-ordered

by

∈

. So suppose for some

Φ

, there are ordinals

α

such that

Φ

⁢

(

α

)

is not true. Suppose further that

Φ

satisfies

the

hypothesis

, i.e.

∀

α

(

∀

β

<

α

(

Φ

(

β

)

)

⇒

Φ

(

α

)

)

. We will reach a

contradiction

.

The class

C

=

{

α

:

¬

⁢

Φ

⁢

(

α

)

}

is not empty. Note that it may be a

proper class

, but this is not important. Let

γ

=

min

⁡

(

C

)

be the

∈

-

minimal element

of

C

. Then by

assumption

, for every

λ

<

γ

,

Φ

⁢

(

λ

)

is true. Thus, by hypothesis,

Φ

⁢

(

γ

)

is true, contradiction.

Title

proof of principle of transfinite induction

Canonical

name

ProofOfPrincipleOfTransfiniteInduction

Date of creation

2013-03-22 12:29:06

Last modified on

2013-03-22 12:29:06

Owner

jihemme (316)

Last modified by

jihemme (316)

Numerical id

11

Author

jihemme (316)

Entry type

Proof

Classification

msc 03B10

Generated on Thu Feb 8 19:27:25 2018 by

LaTeXML
proof of the well-founded induction principle

proof of the well-founded induction principle

This proof is very

similar

to the proof of the

transfinite induction

theorem

. Suppose

Φ

is defined for a

well-founded

set

(

S

,

R

)

, and suppose

Φ

is not true for every

a

∈

S

. Assume further that

Φ

satisfies

requirements 1 and 2 of the statement. Since

R

is a well-founded relation, the set

{

a

∈

S

:

¬

⁢

Φ

⁢

(

a

)

}

has an

R

minimal element

r

. This

element

is either an

R

minimal element of

S

itself, in which case condition 1 is violated, or it has

R

predessors. In this case, we have by minimality

Φ

⁢

(

s

)

for every

s

such that

s

⁢

R

⁢

r

, and by condition 2,

Φ

⁢

(

r

)

is true,

contradiction

.

Title

proof of the

well-founded induction

principle

Canonical name

ProofOfTheWellfoundedInductionPrinciple

Date of creation

2013-03-22 12:42:20

Last modified on

2013-03-22 12:42:20

Owner

jihemme (316)

Last modified by

jihemme (316)

Numerical id

7

Author

jihemme (316)

Entry type

Proof

Classification

msc 03B10

Generated on Thu Feb 8 19:27:30 2018 by

LaTeXML
R-minimal element

R-minimal element

Let

A

be a set and

R

be a relation on

A

. Suppose that

B

is a subset of

A

. An

element

a

∈

B

is said to be

R

-

minimal

in

B

if and only if there is no

x

∈

B

such that

x

⁢

R

⁢

a

. An

R

-

minimal element

in

A

is simply called

R

-minimal

.

From this definition, it is evident that if

A

has an

R

-minimal element, then

R

is not

reflexive

. However, the definition of

R

-minimality is sometimes adjusted slightly so as to allow

reflexivity

:

a

∈

B

is

R

-minimal (in

B

) iff the only

x

∈

B

such that

x

⁢

R

⁢

a

is when

x

=

a

.

Remark

. Using the second definition, it is easy to see that when

R

is a

partial order

, then an element

a

is

R

-minimal iff it is minimal.

Title

R-minimal element

Canonical name

RminimalElement

Date of creation

2013-03-22 12:42:43

Last modified on

2013-03-22 12:42:43

Owner

CWoo (3771)

Last modified by

CWoo (3771)

Numerical id

11

Author

CWoo (3771)

Entry type

Definition

Classification

msc 03B10

Synonym

R-minimal

Synonym

R

-minimal

Related topic

WellFoundedRelation

Generated on Thu Feb 8 19:27:48 2018 by

LaTeXML
semantic properties of substitutability

semantic properties of substitutability

In

addition

, we also have the following semantic properties on

substitutability

:

1.

If

x

,

y

are respectively free for

z

in

A

, and do not

occur free

in

A

, then

⊧

∃

x

A

[

x

/

z

]

↔

∃

y

A

[

y

/

z

]

and

⊧

∀

x

A

[

x

/

z

]

↔

∀

y

A

[

y

/

z

]

2.

(

substitution theorem

)

–

⊧

(

t

1

=

t

2

)

→

(

s

[

t

1

/

x

]

=

s

[

t

2

/

x

]

)

–

⊧

(

t

1

=

t

2

)

→

(

A

[

t

1

/

x

]

=

A

[

t

2

/

x

]

)

3.

Every wff

A

is

equivalent

to a wff

A

′

in the sense

⊧

A

↔

A

′

such that

A

′

contains no variables that are both free and bound.

Title

semantic properties of substitutability

Canonical name

SemanticPropertiesOfSubstitutability

Date of creation

2013-03-22 19:36:00

Last modified on

2013-03-22 19:36:00

Owner

CWoo (3771)

Last modified by

CWoo (3771)

Numerical id

6

Author

CWoo (3771)

Entry type

Result

Classification

msc 03B10

Classification

msc 03B05

\@unrecurse

Generated on Thu Feb 8 19:27:58 2018 by

LaTeXML
some valid schemas of first order logic

some valid schemas of first order logic

In this entry, we record some valid schemas of

first order logic

FO

(

Σ

)

based on the

signature

Σ

:

1.

∀

x

⁢

A

→

∀

y

⁢

A

⁢

[

y

/

x

]

if

y

does not

occur free

, but

free for

x

, in

A

2.

∀

y

⁢

A

⁢

[

y

/

x

]

→

∀

x

⁢

A

if

y

does not

occur in

A

3.

∀

x

⁢

A

↔

A

if

x

is not free in

A

4.

∃

x

⁢

A

↔

A

if

x

is not free in

A

5.

∀

x

⁢

∀

y

⁢

A

→

∀

y

⁢

∀

x

⁢

A

6.

∃

x

⁢

∃

y

⁢

A

→

∃

y

⁢

∃

x

⁢

A

7.

∀

x

(

A

→

B

)

→

(

∀

x

A

→

∀

x

B

)

8.

(

∀

x

A

→

∀

x

B

)

→

∀

x

(

A

→

B

)

9.

∀

x

⁢

(

A

∧

B

)

↔

∀

x

⁢

A

∧

∀

x

⁢

B

10.

∃

x

⁢

(

A

∨

B

)

↔

∃

x

⁢

A

∨

∃

x

⁢

B

11.

∀

x

⁢

(

A

∨

B

)

↔

(

∀

x

⁢

A

)

∨

B

if

x

is not free in

B

12.

∃

x

⁢

(

A

∧

B

)

↔

(

∃

x

⁢

A

)

∧

B

if

x

is not free in

B

where

A

,

B

are well-formed formulas (wff’s) of FO

(

Σ

)

.

Title

some valid schemas of first order logic

Canonical name

SomeValidSchemasOfFirstOrderLogic

Date of creation

2013-03-22 19:32:28

Last modified on

2013-03-22 19:32:28

Owner

CWoo (3771)

Last modified by

CWoo (3771)

Numerical id

11

Author

CWoo (3771)

Entry type

Definition

Classification

msc 03B10

Generated on Thu Feb 8 19:28:08 2018 by

LaTeXML
syntactic compactness theorem for first order logic

syntactic compactness theorem for first order logic

Let

L

be a

first-order language

, and

Δ

⊆

L

be a set of

sentences

. If

Δ

is

inconsistent

, then some finite

Γ

⊆

Δ

is inconsistent.

Title

syntactic compactness theorem for first order logic

Canonical name

SyntacticCompactnessTheoremForFirstOrderLogic

Date of creation

2013-03-22 12:43:59

Last modified on

2013-03-22 12:43:59

Owner

jihemme (316)

Last modified by

jihemme (316)

Numerical id

5

Author

jihemme (316)

Entry type

Theorem

Classification

msc 03B10

Classification

msc 03C07

Related topic

UpwardsSkolemLowenheimTheorem

Related topic

ProofOfUpwardsSkolemLowenheimTheorem

Related topic

GettingModelsIModelsConstructedFromConstants

Generated on Thu Feb 8 19:28:25 2018 by

LaTeXML
theory

theory

If

L

is a

logical language

for some logic

ℒ

, a set

T

of

formulas

with no

free variables

is called a

theory

(of

ℒ

). If

ℒ

is a

first-order logic

, then

T

is called a

first-order theory

.

We write

T

⊨

ϕ

for any formula

ϕ

if every model

ℳ

of

ℒ

such that

M

⊨

T

,

M

⊨

ϕ

.

We write

T

⊢

ϕ

is for there is a proof of

ϕ

from

T

.

Remark

. Let

S

be an

L

-

structure

for some

signature

L

. The

theory of

S

is the set of formulas satisfied by

S

:

{

φ

∣

S

⊧

φ

}

,

and is denoted by

Th

⁡

(

S

)

.

Title

theory

Canonical name

Theory

Date of creation

2013-03-22 13:00:12

Last modified on

2013-03-22 13:00:12

Owner

CWoo (3771)

Last modified by

CWoo (3771)

Numerical id

8

Author

CWoo (3771)

Entry type

Definition

Classification

msc 03B10

Classification

msc 03B05

Generated on Thu Feb 8 19:28:38 2018 by

LaTeXML
transfinite induction

transfinite induction

Suppose

Φ

⁢

(

α

)

is a

property

defined for every

ordinal

α

, the principle of

transfinite induction

states that in the case where for every

α

, if the fact that

Φ

⁢

(

β

)

is true for every

β

<

α

implies that

Φ

⁢

(

α

)

is true, then

Φ

⁢

(

α

)

is true for every ordinal

α

. Formally :

∀

α

(

∀

β

(

β

<

α

⇒

Φ

(

β

)

)

⇒

Φ

(

α

)

)

⇒

∀

α

(

Φ

(

α

)

)

The

principle of transfinite induction

is very

similar

to the

principle of finite induction

, except that it is stated in terms of the whole class of the ordinals.

Title

transfinite induction

Canonical

name

TransfiniteInduction

Date of creation

2013-03-22 12:29:03

Last modified on

2013-03-22 12:29:03

Owner

jihemme (316)

Last modified by

jihemme (316)

Numerical id

10

Author

jihemme (316)

Entry type

Theorem

Classification

msc 03B10

Synonym

principle of transfinite induction

Related topic

PrincipleOfFiniteInduction

Related topic

Induction

Related topic

TransfiniteRecursion

Generated on Thu Feb 8 19:28:40 2018 by

LaTeXML
universal relation

universal relation

If

Φ

is a class of

n

-ary

relations

with

x

→

as the only

free variables

, an

n

+

1

-ary

formula

ψ

is

universal

for

Φ

if for any

ϕ

∈

Φ

there is some

e

such that

ψ

⁢

(

e

,

x

→

)

↔

ϕ

⁢

(

x

→

)

. In other words,

ψ

can simulate any element of

Φ

.

Similarly, if

Φ

is a class of

function

of

x

→

, a formula

ψ

is universal for

Φ

if for any

ϕ

∈

Φ

there is some

e

such that

ψ

⁢

(

e

,

x

→

)

=

ϕ

⁢

(

x

→

)

.

Title

universal relation

Canonical

name

UniversalRelation

Date of creation

2013-03-22 12:58:18

Last modified on

2013-03-22 12:58:18

Owner

Henry (455)

Last modified by

Henry (455)

Numerical id

6

Author

Henry (455)

Entry type

Definition

Classification

msc 03B10

Synonym

universal

Defines

universal function

Generated on Thu Feb 8 19:28:42 2018 by

LaTeXML
universal relations exist for each level of the arithmetical hierarchy

universal relations exist for each level of the arithmetical hierarchy

Let

L

∈

{

Σ

n

,

Δ

n

,

Π

n

}

and take any

k

∈

ℕ

. Then there is a

k

+

1

-ary

relation

U

∈

L

such that

U

is

universal

for the

k

-ary relations in

L

.

Proof

First we prove the case where

L

=

Δ

1

, the recursive relations. We use the example of a Gödel numbering.

Define

T

to be a

k

+

2

-ary relation such that

T

⁢

(

e

,

x

→

,

a

)

if:

•

e

=

⌜

⁢

ϕ

⁢

⌝

•

a

is a

deduction

of either

ϕ

⁢

(

x

→

)

or

¬

⁢

ϕ

⁢

(

x

→

)

Since deductions are (

http://planetmath.org/DeductionsAreDelta1

)

Δ

1

, it follows that

T

is

Δ

1

. Then define

U

′

⁢

(

e

,

x

→

)

to be the least

a

such that

T

⁢

(

e

,

x

→

,

a

)

and

U

⁢

(

e

,

x

→

)

↔

(

U

′

⁢

(

e

,

x

→

)

)

len

⁡

(

U

′

⁢

(

e

,

x

→

)

)

=

e

. This is again

Δ

1

since the

Δ

1

functions

are

closed under

minimization

(

http://planetmath.org/Delta_1Bootstrapping

).

If

f

is any

k

-

a

⁢

r

⁢

y

Δ

1

function then

f

⁢

(

x

→

)

=

U

⁢

(

⌜

⁢

f

⁢

⌝

,

x

→

)

.

Now take

L

to be the

k

-ary relatons in either

Σ

n

or

Π

n

. Call the universal relation for

k

+

n

-ary

Δ

1

relations

U

Δ

. Then any

ϕ

∈

L

is

equivalent

to a relation in the form

Q

⁢

y

1

⁢

Q

′

⁢

y

2

⁢

⋯

⁢

Q

*

⁢

y

n

⁢

ψ

⁢

(

x

→

,

y

→

)

where

g

∈

Δ

1

, and so

U

⁢

(

x

→

)

=

Q

⁢

y

1

⁢

Q

′

⁢

y

2

⁢

⋯

⁢

Q

*

⁢

y

n

⁢

U

Δ

⁢

(

⌜

⁢

ψ

⁢

⌝

,

x

→

,

y

→

)

. Then

U

is universal for

L

.

Finally, if

L

is the

k

-ary

Δ

n

relations and

ϕ

∈

L

then

ϕ

is equivalent to relations of the form

∃

y

1

⁢

∀

y

2

⁢

⋯

⁢

Q

⁢

y

n

⁢

ψ

⁢

(

x

→

,

y

→

)

and

∀

z

1

⁢

∃

z

2

⁢

⋯

⁢

Q

⁢

z

n

⁢

η

⁢

(

x

→

,

z

→

)

. If the

k

-ary universal relations for

Σ

n

and

Π

n

are

U

Σ

and

U

Π

respectively then

ϕ

⁢

(

x

→

)

↔

U

Σ

⁢

(

⌜

⁢

ψ

⁢

⌝

,

x

→

)

∧

U

Π

⁢

(

⌜

⁢

η

⁢

⌝

,

x

→

)

.

Title

universal relations exist for each level of the

arithmetical hierarchy

Canonical name

UniversalRelationsExistForEachLevelOfTheArithmeticalHierarchy

Date of creation

2013-03-22 12:58:40

Last modified on

2013-03-22 12:58:40

Owner

Henry (455)

Last modified by

Henry (455)

Numerical id

5

Author

Henry (455)

Entry type

Theorem

Classification

msc 03B10

Generated on Thu Feb 8 19:28:44 2018 by

LaTeXML