Tarski's Undefinability Theorem

Theorem
Let $\mathcal{Z}$ be the standard structure $(\Z, +, \cdot, s, <, 0)$ for the language of arithmetic.

Let $\mathrm{Th}_\mathcal{Z}$ be the sentences which are true in $\mathcal{Z}$.

Let $\Theta$ be the set of Gödel numbers of those sentences in $\mathrm{Th}_\mathcal{Z}$.

$\Theta$ is not definable in $\mathrm{Th}_\mathcal{Z}$.

Note
Informally, the theorem says that the set of true statements about arithmetic can't be defined arithmetically.

Proof
$\mathrm{Th}_\mathcal{Z}$ is easily seen to be a consistent extension of minimal arithmetic. (In fact, the axioms in minimal arithmetic were selected based on the behavior of standard arithmetic.)

Thus, the theorem is a special case of Set of Gödel Numbers of Arithmetic Theorems Not Definable in Arithmetic (and can be seen to follow immediately).