I remember reading somewhere (although I can't find a source) that for any computable function $f: \mathbb{N} \to \mathbb{N}$ (that is, a function such that we can write an algorithm that computes $f(n)$ given $n$), the formula $f(x) = y$ with two free variables is expressible in the language of Peano Arithmetic.
However, when reading the Wikipedia article Gödel β function it states:
The β function is used, in particular, in showing that the class of arithmetically definable functions is closed under primitive recursion, and therefore includes all primitive recursive functions.
My understanding is that the primitive recursive functions are those functions which can be computed by an algorithm which would use for loops, but no while loops.
Doesn't the class of arithmetically definable functions include all computable functions (primitive recursive or not)? If yes, are Gödel β function useful in showing that or do we need some other tools?