In an (otherwise very enlightening) answer to another question of mine the question came up
What functions are allowed as building blocks for computable functions?
I was astonished that there is a matter of discussion: for me a computable function $f: \mathbb{N} \rightarrow \mathbb{N}$ has been just any $\mu$-recursive function. And there has not been a discussion about fancy "allowed building blocks" like $x\rightarrow \sqrt x$.
How do I have to deal with this objection?
If I want to remain in the realm of natural numbers, (why) do I have to consider "building blocks" like $x\rightarrow \sqrt x$, which may yield intermediate results not finitely expressible by natural numbers (without introducing new symbols)?