Gödel's Incompleteness Theorems/First

Theorem
Let $T$ be the set of theorems of some recursive set of sentences in the language of arithmetic such that $T$ contains minimal arithmetic.

$T$ cannot be both consistent and complete.

Corollary
If $T$ is both consistent and complete, it does not contain minimal arithmetic.

Proof
Aiming for a contradiction, suppose that such a $T$ is consistent and complete.

By the Undecidability Theorem, since $T$ is consistent and contains $Q$, it is not recursive.

But, by Complete Recursively Axiomatized Theories are Recursive, since $T$ is complete and is the set of theorems of a recursive set, it is recursive.

Thus, we have a contradiction.

Proof of corollary
This is simply the contrapositive of the main theorem.