My book states the following theorem with no proof. Can anyone give an outline of the proof, or an explanation of how the formal statement is to be constructed?

Theorem:

Suppose that $M(x_1,...,x_n)$ is a decidable predicate. Then it is possible to construct a statement $σ(x_1,...,x_n)$ of $L$ [formal language of arithmetic] that is a formal counterpart of $M(x_1,...,x_n)$ in the following sense: for any $a_1,...,a_n ∈ ℕ$,

$M(a_1,...,a_n)$ holds iff $σ(a_1,...,a_n)$ is true.