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So I was recently reading the proof of the incompleteness theorem (again), and there is one thing thats bugging me. I understood everything until the point that you create a self referential statement; "The formula with Godel number $G$ cannot be proven". This statement has the Godel number $G$. Now, one can conclude that this statement cannot be false, and hence must be true and unprovable so math is incomplete. However, doesn't this just prove that this particular statement cannot be proven? How does this show that there must be an actual mathematical theorem or conjecture that cannot be proven using this system. Also if you substitute Godel number $G = \text{The statement with Godel number $G$ cannot be proven}$, it just becomes some absurd infinite nested statements which is also a bit confusing.

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We can write some actual arithmetic (with quantifiers and arithmetic operations, without English words) formula $\Box(n)$ which says "$n$ is Godel number of provable formula".

Then we can use fixed point theorem to show that there is a formula $\phi$ s.t. it's provable in arithmetic that $\phi \leftrightarrow \neg\Box(\ulcorner \phi \urcorner)$, thus avoiding infinite nesting (IMO the existence of fixed point is the most important part of the proof).

Now, we have some arithmetic formula $\phi$ which is unprovable in arithmetic.

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