In this, the third and final proof of the Incompleteness Theorem, we will show that the existence of a self-referential sentence implies that the language of arithmetic itself is incomplete.
Assume that there is a sentence φ in the language of first-order arithmetic which expresses the sentence "This sentence is not provable."
By the diagonal lemma, there exists a Gödel number g such that the sentence "φ(g) is not true" is the same as the sentence φ.
Now, if φ is true, then φ(g) is true, so "φ(g) is not true" must be false, which means φ(g) is true, which means φ is true, which means φ(g) is true, which means... (and so on)
This shows that φ(g) is true, which means that φ is true, which means that the theory of arithmetic is complete.
But wait, that can't be. We wanted to show that the theory of arithmetic is incomplete.
So, let us instead assume that φ is false. Then φ(g) is false, which means that "φ(g) is not true" is true, which means φ(g) is true, which means... (and so on).
This shows that φ is false, which means that there is no sentence expressing the sentence "This sentence is not provable."
Which means that the theory of arithmetic is incomplete.
QED.