Undecidability Theorem

Theorem
Let $T$ be the set of theorems of some consistent theory in the language of arithmetic which contains minimal arithmetic.

$T$ is not recursive.

Proof
By Set of Gödel Numbers of Arithmetic Theorems Not Definable in Arithmetic, since $T$ is a consistent extension of $Q$, the set $\Theta$ of Gödel numbers of the theorems in $T$ is not a definable set in $T$.

Since Recursive Sets are Definable in Arithmetic, this means that $\Theta$ is not recursive.

Suppose $T$ were recursive.

Then $\Theta$ would be recursive since by Gödel Numbering is Recursive we could recursively go from $\Theta$ to $T$, use the recursivity of $T$, then recursively go back to $\Theta$.

This would be a contradiction.

Thus, $T$ is not recursive.