Godel2

How to svmbolize a logical formula with numbers:

Linę I:    (3x)(x-sy)

Linę 2:	(	3	X	)	(	X	=5	s	y	)
	4	4	4	4	4	4	4	4	4	4
Linę 3:	8	4	11	9	8	11	5	7	13	9
		4	4	i	4	4	4	4	4	4
	8	4	11	9	8	u	5	7	13	9
Linę 4:	2	^x 3	X 5	X 7	X 11	X 13	X 17	X 19	x 23	x 29

Every number says something in logie or mathematics.

Two definitions: Dem (x,y)

in for put

Sub(x, 13, y)

<    (kucek** tflU y

means:    *

Formula x is a logical proof of formula y.

means:    v    x    ^    3

This is the Godeł number of the formula that we obtain from formula x when we substitute for the symbol with Godeł number 13 a formula with Godeł number y.

Gddel Theorem:    <    *    #

U (n): ^ (x) ~ Dcm(x,y) which means:

For every x, it is not true that x is the proof of y.

(G):    (x) ~ Dem(x, sub(n, 13,n))    which means:

^    For every x, it is not truć that x is the proof of sub(n,13,n), or

For every x, it is not truć that x is the proof of formula n, if for y in n you put formula n.

In other words: In (x) - Dem(x,y) we put (n) in the place of y:

______t_

I (x) - Dem(x,y),

and obtain    (x) ~ Dem(x,(x) - Dem(x,y));

For every x it is not true that x is the proof of the formula that says that

for every x, x is not the proof of any formula y; or

There is no proof for a formula that says there is no proof for it.

But (x) ~ Dem(x,(x) - Dem(x,y)) is equivalent to (G): (x) - Dem(x, sub(n, 13,n)).

So (G) says "I do not have a proof', and it is true.

(G) is true but cannot be proved.