Not All Natural Number Functions are Primitive Recursive

Theorem
Not all functions $$f: \N \to \N$$ are primitive recursive.

Proof
All primitive recursive functions are URM computable.

The set of $$\mathbf{U}$$ of URM programs is countably infinite.

The set of $$\mathbb{F}$$ of natural number functions is uncountably infinite.

Hence there is no surjection from $$\mathbf{U} \to \mathbb{F}$$.

Hence $$\mathbf{U} \subsetneq \mathbb{F}$$.

Hence $$\exists f \in \mathbb{F}: f \notin \mathbf{U}$$.