deductions are Δ_1

deductions are

Δ

1

Using the example of Gödel numbering, we can show that

Proves

⁡

(

a

,

x

)

(the statement that

a

is a proof of

x

, which will be formally defined below) is

Δ

1

.

First,

Term

⁡

(

x

)

should be true if and only if

x

is the Gödel number of a term. Thanks to

primitive recursion

, we can define it by:

Term

⁡

(

x

)

↔

∃

i

<

x

[

x

=

⟨

0

,

i

⟩

]

∨

x

=

⟨

5

⟩

∨

∃

y

<

x

[

x

=

⟨

6

,

y

⟩

∧

Term

(

y

)

]

∨

∃

y

,

z

<

x

[

x

=

⟨

8

,

y

,

z

⟩

∧

Term

(

y

)

∧

Term

(

z

)

]

∨

∃

y

,

z

<

x

[

x

=

⟨

9

,

y

,

z

⟩

∧

Term

(

y

)

∧

Term

(

z

)

]

Then

AtForm

⁡

(

x

)

, which is true when

x

is the Gödel number of an

atomic formula

, is defined by:

AtForm

⁡

(

x

)

↔

∃

y

,

z

<

x

[

x

=

⟨

1

,

y

,

z

⟩

∧

Term

(

y

)

∧

Term

(

z

)

]

∨

∃

y

,

z

<

x

[

x

=

⟨

7

,

y

,

z

⟩

∧

Term

(

y

)

∧

Term

(

z

)

]

Next,

Form

⁡

(

x

)

, which is true only if

x

is the Gödel number of a

formula

, is defined recursively by:

Form

⁡

(

x

)

↔

AtForm

⁡

(

x

)

∨

∃

i

,

y

<

x

[

x

=

⟨

2

,

i

,

y

⟩

∧

Form

(

y

)

]

∨

∃

y

<

x

[

x

=

⟨

3

,

y

⟩

∧

Form

(

y

)

]

∨

∃

y

,

z

<

x

[

x

=

⟨

4

,

y

,

z

⟩

∧

Form

(

y

)

∧

Form

(

z

)

]

The definition of

QFForm

⁡

(

x

)

, which is true when

x

is the Gödel number of a

quantifier free formula

, is defined the same way except without the second clause.

Next we want to show that the set of logical

tautologies

is

Δ

1

. This will be done by formalizing the

concept

of

truth tables

, which will require some development. First we show that

AtForms

⁡

(

a

)

, which is a

sequence

containing the (unique) atomic formulas of

a

is

Δ

1

. Define it by:

AtForms

⁡

(

a

,

t

)

↔

(

¬

Form

(

a

)

∧

t

=

0

)

∨

Form

(

a

)

∧

(

∃

x

,

y

<

a

[

a

=

⟨

1

,

x

,

y

⟩

t

=

a

]

∨

∃

x

,

y

<

a

[

a

=

⟨

7

,

x

,

y

⟩

∧

t

=

a

]

∨

∃

i

,

x

<

a

[

a

=

⟨

2

,

i

,

x

⟩

∧

t

=

AtForms

(

x

)

]

∨

∃

x

<

a

[

a

=

⟨

3

,

x

⟩

∧

t

=

AtForms

(

x

)

]

∨

∃

x

,

y

<

a

[

a

=

⟨

4

,

x

,

y

⟩

∧

t

=

AtForms

(

x

)

*

u

AtForms

(

y

)

]

)

We say

v

is a

truth assignment

if it is a sequence of pairs with the first member of each pair being a atomic formula and the second being either

1

or

0

:

T

A

(

v

)

↔

∀

i

<

len

(

v

)

∃

x

,

y

<

(

v

)

i

[

(

v

)

i

=

⟨

x

,

y

⟩

∧

AtForm

(

x

)

∧

(

y

=

1

∨

y

=

0

)

]

Then

v

is a truth assignment for

a

if

v

is a truth assignment,

a

is quantifier free, and every atomic formula in

a

is the first member of one of the pairs in

v

. That is:

T

A

f

(

v

,

a

)

↔

T

A

(

v

)

∧

QFForm

(

a

)

∧

∀

i

<

len

(

AtForms

(

a

)

)

∃

j

<

len

(

v

)

[

(

(

v

)

j

)

0

=

(

AtForms

(

a

)

)

i

]

Then we can define when

v

makes

a

true by:

True

⁡

(

v

,

a

)

↔

T

⁢

A

⁢

f

⁢

(

v

,

a

)

∧

AtForm

(

a

)

∧

∃

i

<

len

(

v

)

[

(

(

v

)

i

)

0

=

a

∧

(

(

v

)

i

)

1

=

1

]

∨

∃

y

<

x

[

x

=

⟨

3

,

y

⟩

∧

True

(

v

,

y

)

]

∨

∃

y

,

z

<

x

[

x

=

⟨

4

,

y

,

z

⟩

∧

True

(

v

,

y

)

→

True

(

v

,

z

)

]

Then

a

is a tautology if every truth assignment makes it true:

Taut

(

a

)

∀

v

<

2

2

AtForms

⁡

(

a

)

[

T

A

f

(

v

,

a

)

→

True

(

v

,

a

)

]

We say that a number

a

is a

deduction

of

ϕ

if it encodes a proof of

ϕ

from a set of axioms

A

⁢

x

. This means that

a

is a sequence where for each

(

a

)

i

either:

•

(

a

)

i

is the Gödel number of an axiom

•

(

a

)

i

is a logical tautology

or

•

there are some

j

,

k

<

i

such that

(

a

)

j

=

⟨

4

,

(

a

)

k

,

(

a

)

i

⟩

(that is,

(

a

)

i

is a

conclusion

under

modus ponens

from

(

a

)

j

and

(

a

)

k

).

and the last element of

a

is

⌜

⁢

ϕ

⁢

⌝

.

If

A

⁢

x

is

Δ

1

(almost every system of axioms, including

P

⁢

A

, is

Δ

1

) then

Proves

⁡

(

a

,

x

)

, which is true if

a

is a deduction whose last value is

x

, is also

Δ

1

. This is fairly simple to see from the above results (let

Ax

⁡

(

x

)

be the

relation

specifying that

x

is the Gödel number of an axiom):

Proves

(

a

,

x

)

↔

∀

i

<

len

(

a

)

[

Ax

(

(

a

)

i

)

∨

∃

j

,

k

<

i

[

(

a

)

j

=

⟨

4

,

(

a

)

k

,

(

a

)

i

⟩

]

∨

Taut

(

(

a

)

i

)

]

Title

deductions are

Δ

1

Canonical name

DeductionsAreDelta1

Date of creation

2013-03-22 12:58:36

Last modified on

2013-03-22 12:58:36

Owner

mathcam (2727)

Last modified by

mathcam (2727)

Numerical id

9

Author

mathcam (2727)

Entry type

Theorem

Classification

msc 03B10

Synonym

deductions are delta 1

Defines

truth assignment

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