I'm reading this book, logic and complexity by Richard Lassaigne, where there is a recursive function $\phi(x,y)$ which enumerates all the recursive functions with one parameter (for the sake of this question that last part isn't really important).
For the proof that the Halting problem is undecidable, the author takes the function $g(x)=\phi(x,x)$ and proves that its domain is not a recursive set and that leads to the desired result.
I'm having a hard time understanding why $g(x)$ is still a recursive function. Intuitively I can say "sure..", but i haven't been able to use the definition of recursive function to prove it.
In the book, a recursive function is defined as a function obtained by the applying the $\mu$ operator, recursion and compostition to the simple functions (constant function, projetion and succesor). My problem comes from the arity, since every time they define a recursive function they specify the number of parameters the function must have and in no definition is this number reduced from 2 to 1.
Thanks in advance
The precise definitions may change from book to book, so I'll go with the wikipedia one: using the composition rule ($4$), take $k=2$, $m=1$ and $g_1,g_2$ both be the identity on $\mathbb{N}$.