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.

Proof
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.

The result follows by Proof by Contradiction.

Also see

 * Gödel's Second Incompleteness Theorem