Post by bitScript

Godel's ACTUAL Proof ☺

1. (∃x)Dem(x, z) means there is a proof with Gödel Number x which demonstrates z or more simply z is demonstrable
2. ~ (∃x) Dem(x, z) means z is not demonstrable
3. A specific case of this expression is obtained by swapping z by Sub(y, 17, y)
4. We get ~ (∃x) Dem(x, Sub(y, 17, y)) - lets call this formula B
5. B means: the formula with Gödel number Sub(y, 17, y) is not demonstrable
6. Lets call the Gödel number of B: n
7. Lets replace all instances of the variable y with n to get ~ (∃x) Dem(x, Sub(n, 17, n)) – lets call this formula G with a
   Gödel number of g
8. G means: the formula with Gödel number Sub(n, 17, n) is not demonstrable
9. But Sub(n, 17, n) is what we did on line 7 therefore Sub(n, 17, n) has a Gödel number of g
10. Therefore, G means: the formula with Gödel number g is not demonstrable
11. Gödel proved that G is demonstrable if and only if ~G is demonstrable
12. But if the calculus is to be consistent, then G can never be demonstrable otherwise there would be inconsistencies as
    its formal negation must also be demonstrable
13. Since G and its negation ~G are tautological (they are the only 2 options so one must be true) either G is true or ~G is
    true.
14. It is easy to see that G is true as it states that ‘G is not demonstrable’ which we have just proved to be the case if the
    calculus is to be consistent
15. Therefore G is true and can not be demonstrated. Therefore the calculus is incomplete