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As Andres Caicedo points out in his comment (to the Question), the modest fragment $\sf{PRA}$ (Primitive Recursive Arithmetic) of $\sf{PA}$ (Peano arithmetic) is already is able to verify the incompleteness theorems.

Indeed, the proof of the Gödel–Rosser incompleteness proof is entirely syntactic and can be readily implemented in a fragment of $\sf{PRA}$ known as $I\Delta_0 + exp$, where $I\Delta_0$ is the weakening of $PA$ in which the induction scheme is only available for $\Delta_0$-formulas, and $exp$ asserts the totality of the exponential function $2^x$ (it is well-known that $I\Delta_0$ is unable to prove the totality of the exponential function).

It is worth noting that in the above $I\Delta_0 + exp$ can be even reduced to $I\Delta_0 + \Omega_1$, where $\Omega_1$ is the axiom asserting the totality of the function $2^{\left| x\right|^2 }$, where $\left| x\right|$ denotes the length of the binary expansion of $x$. The theory $I\Delta_0 + \Omega_1$ is commonly viewed as the weakest fragment of $\sf{PA}$ in which one can develop a workable "theory of syntax".

PS. As pointed out by Jeřábek, the incompleteness theorems can be implemented in even weaker systems.

As Andres Caicedo points out in his comment (to the Question), the modest fragment $\sf{PRA}$ (Primitive Recursive Arithmetic) of $\sf{PA}$ (Peano arithmetic) is already able to verify the incompleteness theorems.

Indeed, the proof of the Gödel–Rosser incompleteness proof is entirely syntactic and can be readily implemented in a fragment of $\sf{PRA}$ known as $I\Delta_0 + exp$, where $I\Delta_0$ is the weakening of $PA$ in which the induction scheme is only available for $\Delta_0$-formulas, and $exp$ asserts the totality of the exponential function $2^x$ (it is well-known that $I\Delta_0$ is unable to prove the totality of the exponential function).

It is worth noting that in the above $I\Delta_0 + exp$ can be even reduced to $I\Delta_0 + \Omega_1$, where $\Omega_1$ is the axiom asserting the totality of the function $2^{\left| x\right|^2 }$, where $\left| x\right|$ denotes the length of the binary expansion of $x$. The theory $I\Delta_0 + \Omega_1$ is commonly viewed as the weakest fragment of $\sf{PA}$ in which one can develop a workable "theory of syntax".

PS. As pointed out by Jeřábek, the incompleteness theorems can be implemented in even weaker systems.

As Andres Caicedo points out in his comment (to the Question), the modest fragment $\sf{PRA}$ (Primitive Recursive Arithmetic) of $\sf{PA}$ (Peano arithmetic) is already is able to verify the incompleteness theorems.

Indeed, the proof of the Gödel-Rosser incompleteness proof is entirely syntactic and can be readily implemented in a fragment of $\sf{PRA}$ known as $I\Delta_0 + exp$, where $I\Delta_0$ is the weakening of $PA$ in which the induction scheme is only available for $\Delta_0$-formulas, and $exp$ asserts the totality of the exponential function $2^x$ (it is well-known that $I\Delta_0$ is unable to prove the totality of the exponential function).

It is worth noting that in the above $I\Delta_0 + exp$ can be even reduced to $I\Delta_0 + \Omega_1$, where $\Omega_1$ is the axiom asserting the totality of the function $2^{\left| x\right|^2 }$, where $\left| x\right|$ denotes the length of the binary expansion of $x$. The theory $I\Delta_0 + \Omega_1$ is commonly viewed as the weakest fragment of $\sf{PA}$ in which one can develop a workable "theory of syntax".

PS. As pointed out by Jeřábek, the incompleteness theorems can be implemented in even weaker systems.

As Andres Caicedo points out in his comment (to the Question), the modest fragment $\sf{PRA}$ (Primitive Recursive Arithmetic) of $\sf{PA}$ (Peano arithmetic) is already is able to verify the incompleteness theorems.

Indeed, the proof of the Gödel–Rosser incompleteness proof is entirely syntactic and can be readily implemented in a fragment of $\sf{PRA}$ known as $I\Delta_0 + exp$, where $I\Delta_0$ is the weakening of $PA$ in which the induction scheme is only available for $\Delta_0$-formulas, and $exp$ asserts the totality of the exponential function $2^x$ (it is well-known that $I\Delta_0$ is unable to prove the totality of the exponential function).

It is worth noting that in the above $I\Delta_0 + exp$ can be even reduced to $I\Delta_0 + \Omega_1$, where $\Omega_1$ is the axiom asserting the totality of the function $2^{\left| x\right|^2 }$, where $\left| x\right|$ denotes the length of the binary expansion of $x$. The theory $I\Delta_0 + \Omega_1$ is commonly viewed as the weakest fragment of $\sf{PA}$ in which one can develop a workable "theory of syntax".

PS. As pointed out by Jeřábek, the incompleteness theorems can be implemented in even weaker systems.

As Andres Caicedo points out in his comment (to the Question), the modest fragment $\sf{PRA}$ (Primitive Recursive Arithmetic) of $\sf{PA}$ (Peano arithmetic) is already is able to verify the incompleteness theorems.

Indeed, the proof of the Gödel-Rosser incompleteness proof is entirely syntactic and can be readily implemented in a fragment of $\sf{PRA}$ known as $I\Delta_0 + exp$, where $I\Delta_0$ is the weakening of $PA$ in which the induction scheme is only available for $\Delta_0$-formulas, and $exp$ asserts the totality of the exponential function $2^x$ (it is well-known that $I\Delta_0$ is unable to prove the totality of the exponential function).

It is worth noting that in the above $I\Delta_0 + exp$ can be even reduced to $I\Delta_0 + \Omega_1$, where $\Omega_1$ is the axiom asserting the totality of the function $2^{\left| x\right|^2 }$, where $\left| x\right|$ denotes the length of the binary expansion of $x$. The theory $I\Delta_0 + \Omega_1$ is commonly viewed as the weakest fragment of $\sf{PA}$ in which one can develop a workable "theory of syntax".

PS. There are (a) other equivalent formulations of $\Omega_1$; and (b) weaker systems than $I\Delta_0 + \Omega_1$ within which the incompleteness theorems can be implemented. See the comments by Goldstern and Jeřábek below.

As Andres Caicedo points out in his comment (to the Question), the modest fragment $\sf{PRA}$ (Primitive Recursive Arithmetic) of $\sf{PA}$ (Peano arithmetic) is already is able to verify the incompleteness theorems.

Indeed, the proof of the Gödel-Rosser incompleteness proof is entirely syntactic and can be readily implemented in a fragment of $\sf{PRA}$ known as $I\Delta_0 + exp$, where $I\Delta_0$ is the weakening of $PA$ in which the induction scheme is only available for $\Delta_0$-formulas, and $exp$ asserts the totality of the exponential function $2^x$ (it is well-known that $I\Delta_0$ is unable to prove the totality of the exponential function).

It is worth noting that in the above $I\Delta_0 + exp$ can be even reduced to $I\Delta_0 + \Omega_1$, where $\Omega_1$ is the axiom asserting the totality of the function $2^{\left| x\right|^2 }$, where $\left| x\right|$ denotes the length of the binary expansion of $x$. The theory $I\Delta_0 + \Omega_1$ is commonly viewed as the weakest fragment of $\sf{PA}$ in which one can develop a workable "theory of syntax".

PS. As pointed out by Jeřábek, the incompleteness theorems can be implemented in even weaker systems.