Proof of Godel's incompleteness Theorem

 This proof borrows from the book 'Godel's proof' by Nagel and Newman.

First Incompleteness theorem:

The first incompleteness theorem states that 'Any consistent formal system F within which a certain amount of elementary arithmetic can be carried out is incomplete; i.e., there are statements of the language of F which can neither be proved nor disproved in F.'

1. Uniquely number every theorem in a system of mathematics. We can do this by assigning 10 symbols like $0$,$1$,$\subset$ (short for ‘if . . . then . . .’) etc a number between 1 to 10. Also, we can assign numerical variables, sentential variables and predicate variables to first, second and third powers of prime numbers greater or equal to 11, respectively. Note that the use of prime number ensures that all these variables get assigned a unique number, because otherwise $11^2$ is nothing but 121. Another way of doing it could have been assigning 11, 12 and 13 with variable of each type sequentially, and so on.
2. Now that each letter of the language has a unique numerical representation, we assign a number to a formula by raising the nth prime to the power of the numerical value of the nth symbol, and taking product of all such numbers. Strings of formulas separated by comma is also a formula.
3. The fact that a formula with Godel number x is the proof of a Godel number z hints towards a mathematical property that these two numbers share in the general case. This mathematical property depends upon the syntax as well as the rule of inference of the proof system. Let us denote the fact that the numbers x and z share this specific property by the formula $ Dem(x,z)$. This formula might not be true for a given value of x and z. Note that here, $Dem$ is a predicate variable.
4. Next, let $sub(m,13,m)$ denote the Godel number of the formula that you get when you replace in the formula with Godel number m, each occurrence of the variable with Godel number 13 (here y) the number m.
5. Now, note the meaning of $\sim \exists x Dem(x,sub(n,13,n))$ where n is Godel number of the statement $\sim \exists x Dem(x,sub(y,13,y))$. Note that $sub(n,13,n)$ is just the Godel number of the first formula. Hence, the first formula states that it cannot be proven within the system. (Although seen at mathematical level, the first formula just claims that a number does not have a given mathematical property. It is only when interpreted at metamathematical level that we infer a second layer of meaning to this statement- that it is claiming that it cannot be proven within the system). Now, if this statement is false, then it can be proven. A system of mathematics that can prove even false results is inconsistent. Hence, it must be true, and hence cannot be proven.
6. Note that $\sim \exists x Dem(x,sub(n,13,n))$ expresses a property that no number possesses. Hence, it represents a new axiom of mathematics.

Notes:-

1. If there are finitely many rules of inference, then Godel's theorem holds. If there are infinitely many of them, then we get another Godel like situation, where instead of infinite axioms, there infinite rules of inference. But the question is, is there an equivalent formulation of logic where there are only axioms but no rule of inference? I don't think so. Rules of inference are necessary to get from one axiom to another. They can be encompassed by a property that Godel numbers possess, as given by $Dem(x,z)$. Hence the rules of inference link numbers based on some property. Godel statement of a system of axioms is a statement that numbers don't possess a certain property.
2. How does the choice of syntax affect the numerical relations as in $Dem(x,z)$? There should be some invariance regarding some general properties of these relations, independent of the choice of syntax.

Second Incompleteness theorem

The second incompleteness theorem states that 'Assume F is a consistent formalized system which contains elementary arithmetic. Then the consistency of F is not provable within F.'

The proof of the second incompleteness theorem follows from the first one. Even though the Godel statement G of a system F states that it is not provable, the reason why we cannot prove it within the system is that we are assuming that our system is consistent, and hence cannot prove a statement which says that it is not provable. Hence the consistency of F implies its incompleteness. The last sentence itself can be formalized as a theorem in F, and can also be proven within F. The proof follows because if we know that our system is consistent, then there is no more barrier in proving that G must be true. It is the lack of proof of consistency that prevents a proof a G. But if G is proven, then we will get an absurd result. Hence if F is consistent, its consistency itself is necessarily not provable to establish the non-provability of G.