Gödel’s beta function

Gödel’s beta function

Gödel’s

β

function

is the tool needed to express in a formal theory

Z

assertions about

finite sequences

of

natural numbers

. (For a description of

Z

see the PM entry "beyond

formalism

: Gödel’s

incompleteness

" (

http://planetmath.org/BeyondFormalism

)).

Let

1

,

…

,

n

.

be a

sequence

of natural numbers. Then there is the

factorial

n

!

such that

n

⋅

(

n

-

1

)

⋅

…

⋅

1

=

n

!

.

This means that the factorial

n

!

is divided by each of the

elements of

1

,

…

,

n

so that we can build the derived sequence

1

∣

n

!

,

2

∣

n

!

,

…

,

n

∣

n

!

.

Definition 1

[

Gödel’s

Γ

sequence

]

If we denote

n

!

by

l

then the elements of Gödel’s

Γ

sequence are the numbers of the form

(

k

+

1

)

⋅

l

+

1

.

Its general representation is

Γ

=

1

⋅

l

+

1

,

2

⋅

l

+

1

,

…

,

n

⋅

l

+

1

,

(

n

+

1

)

⋅

l

+

1

.

Theorem

1

:

The elements of

Γ

are pairwise relatively prime

.

Proof

: (

Reductio

)

1.

We assume the

existence

of a

prime number

p

such that

p

divides

(

j

+

1

)

⋅

l

+

1

and

p

divides also

(

j

+

k

+

1

)

⋅

l

+

1

.

2.

This gives by definition the

congruence

[

(

j

+

k

+

1

)

⋅

l

+

1

]

≡

[

(

j

+

1

)

⋅

l

+

1

]

(

mod

p

)

3.

Therefore

p

divides

[

(

j

+

k

+

1

)

⋅

l

+

1

]

-

[

(

j

+

1

)

⋅

l

+

1

]

.

4.

Then

p

divides

j

⋅

l

+

k

⋅

l

+

l

+

1

-

j

⋅

l

-

l

-

1

,

that is,

p

divides

k

⋅

l

.

5.

But

p

∣

k

⋅

l

→

p

∣

k

∨

p

∣

l

.

Case I

:

Assume

p

∣

l

1.

p

∣

l

→

p

∣

l

⋅

(

j

+

1

)

.

2.

But by

hypothesis

:

p

∣

(

j

+

1

)

⋅

l

+

1

.

3.

Therefore:

(

p

∤

l

)

.

Case II

:

Assume

p

∣

k

1.

k

≤

n

≤

max

⁢

(

1

,

…

,

n

)

.

2.

But

(

∀

k

)

k

∣

l

.

3.

p

∣

k

∧

k

∣

l

→

p

∣

l

Case I.3. and Case II.3. are the desired

contradiction

.

Definition 2

:  [

Gödel’s

β

function

]

If

m

,

l

,

and

k

are natural numbers then the function

β

⁢

(

m

,

l

,

k

)

computes the rest of the division of

m

by a term

(

k

+

1

)

⋅

l

+

1

of the

Γ

-sequence

.

Therefore:

β

⁢

(

m

,

l

,

k

)

=

R

⁢

[

m

,

(

k

+

1

)

⋅

l

+

1

]

.

Theorem 2

:  [

Sequences of natural numbers are

representable

by the

β

function

.]

<

a

o

,

…

,

a

n

>

∈

ℕ

→

(

∃

m

)

(

∃

l

)

[

a

k

=

β

(

m

,

k

,

l

)

]

.

Proof

:

1.

Let

a

o

,

…

,

a

n

be a sequence of natural numbers.

2.

Then there is a number

l

such that

Γ

=

1

⋅

l

+

1

,

2

⋅

l

+

1

,

…

,

n

⋅

l

+

1

,

(

n

+

1

)

⋅

l

+

1

.

3.

If

l

≥

max

⁢

(

a

o

,

…

,

a

n

)

.

4.

Then

a

k

<

(

k

+

1

)

⋅

l

+

1

.

5.

But by the previous

proposition

the numbers

(

k

+

1

)

⁢

…

⁢

l

+

1

are pairwise relatively prime.

6.

This implies that the simultaneous congruences

x

≡

a

o

[

(

mod

1

⋅

l

+

1

)

]

⋮

x

≡

a

n

[

(

mod

(

n

+

1

)

⋅

l

+

1

)

]

have a common solution

m

, by the

chinese remainder theorem

.

7.

Therefore

m

≡

a

k

[

(

mod

(

k

+

1

)

⋅

l

+

1

)

]

.

8.

This means that

a

k

=

R

⁢

[

m

,

(

k

+

1

)

⋅

l

+

1

]

.

9.

But that is

a

k

=

β

⁢

(

m

,

l

,

k

)

.

It is easily seen that the

β

function is

primitive recursive

.

For that we only have to redenominated

m

,

l

and

k

as:

m

=

x

1

l

=

x

2

k

=

x

3

.

Then

β

⁢

(

x

1

,

x

2

,

x

3

)

=

R

⁢

[

x

1

,

(

x

3

+

1

)

⋅

x

2

+

1

]

.

But the functions "+", "

⋅

" e "

R

" are primitive recursive.

Therefore

β

is primitive recursive.

References

1

Bernays, P., Hilbert, D.,

Grundlagen der Mathematik

, 2.Auflage, Berlin, 1968.

2

Gödel, K.,

Collected works

, ed. S. Feferman, Oxford, 1987-2003.

3

Kleene, S.,

Introduction

to

metamathematics

, North-Holland, Amsterdam, 1964.

Title

Gödel’s

beta function

Canonical name

GodelsBetaFunction

Date of creation

2013-03-22 18:34:57

Last modified on

2013-03-22 18:34:57

Owner

gribskoff (21395)

Last modified by

gribskoff (21395)

Numerical id

18

Author

gribskoff (21395)

Entry type

Definition

Classification

msc 03A05

Classification

msc 03-01

Classification

msc 11A07

Synonym

representation of sequences in formal systems

Related topic

ChineseRemainderTheorem

Related topic

BeyondFormalism

Related topic

FromHilbertsTenthProblemToGodelsTrichotomy

Related topic

LinearCongruence

Defines

gamma sequence

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