There are a couple well known proofs of incompleteness based on properties of PA degrees, which have been studied extensively in recursion theory.

A PA degree is a Turing degree that can compute a complete extension of PA. Obviously, to prove the incompleteness theorem, it's enough to show that no PA degree can be recursive. (If we had a consistent r.e. theory T extending PA that was not incomplete, then its completion would be recursive -- to decide whether $T \models \varphi$ or $T \models \neg \varphi$, simply look for a proof of $\varphi$ or $\neg \varphi$ from $T$, and because $T$ is assumed to be complete, this process always terminates and is hence recursive).

One way to see that there are no recursive PA degrees is to observe that any PA degree can compute a nonstandard model of PA via compactness and a Henkin construction. Now apply Tennenbaum's theorem that there are no recursive nonstandard models of PA.

Another way to see that there are no recursive PA degrees is with $\Pi^0_1$ classes. Every PA degree can compute a path through each $\Pi^0_1$ class (this is one direction of the Scott basis theorem). To finish, note that one can construct $\Pi^0_1$ classes that do not contain any recursive elements. For instance, there are $\Pi^0_1$ classes that contain only diagonally nonrecursive elements.

There are a couple well known proofs of incompleteness based on properties of PA degrees. PA degrees have been studied extensively in recursion theory.

A PA degree is a Turing degree that can compute a complete extension of PA. Obviously, to prove the incompleteness theorem, it's enough to show that no PA degree can be recursive. (If we had a consistent r.e. theory T extending PA that was not incomplete, then its completion would be recursive -- to decide whether $T \models \varphi$ or $T \models \neg \varphi$, simply look for a proof of $\varphi$ or $\neg \varphi$ from $T$, and because $T$ is assumed to be complete, this process always terminates and is hence recursive).

One way to see that there are no recursive PA degrees is to observe that any PA degree can compute a nonstandard model of PA via compactness and a Henkin construction. Now apply Tennenbaum's theorem that there are no recursive nonstandard models of PA.

Another way to see that there are no recursive PA degrees is with $\Pi^0_1$ classes. Every PA degree can compute a path through each $\Pi^0_1$ class (this is one direction of the Scott basis theorem). To finish, note that one can construct $\Pi^0_1$ classes that do not contain any recursive elements. For instance, there are $\Pi^0_1$ classes that contain only diagonally nonrecursive elements.