Soare in a yet unpublished textbook (I happened to be in a class taught by one of his former graduate students where we were field-testing a rough draft of his new textbook) Computability Theory and Applications: The Art of Classical Computability (coming out in 2014 if I remember correctly) describes an acceptable numbering using the following conditions (slightly paraphrased below):
- Define $\mathcal{P}$ to be the class of partial computable (p.c.) functions of one variable.
- A numbering of the p.c. functions is a map $\pi$ from $\omega$ onto $\mathcal{P}$.
- The standard numbering of Turing programs is given by coding its 5-tuples (i.e. the standard Godel numbering).
- Let $\hat{\pi}$ be another numbering and let $\psi _e$ denote $\hat{\pi}(e)$. Then $\hat{\pi}$ is an acceptable numbering if there are computable functions $f$ and $g$ such that $\varphi_{f(x)} = \psi _x$ and $\psi _{g(x)} = \varphi _x$ (i.e. if we can move computably between the standard numbering and $\hat{\pi}$).
My question is why does Soare specify $\mathcal{P}$ to be the class of p.c. functions of one variable?
Is there any meaningful difference between a Turing program of one variable and a Turing program of multiple variables (since variables are just separated by blanks and therefore presumably any Turing program associated with a p.c. function is both "one variable" and "multi-variable")?
In the interest of keeping this post to a reasonable length, I won't provide examples unless asked, but it doesn't seem like the subsequent theorems about acceptable numbering as well as subsequent exercises rely at all on the "one variable" part.