I have been studying primitive recursive functions for a while now using Kleene's "Introduction to Metamathematics". He says in his book that "We say that a function $\phi(x_1, ..., x_n)$ is the representing function of a predicate $P(x_1, ..., x_n)$ if $\phi$ takes only values $0$ and $1$, and is $0$ when predicate is true and $1$ when predicate is false."
I have trouble understanding this definition. We want the theory of primitive recursive functions to be finitary just as metamathematics because we will apply this theory of primitive recursive functions to prove Godel's theorem (according to Kleene).
I have trouble understanding what does finitary means here and how it is the case that this definition is finitary. So, if I understand correctly, then predicate is some rule which for each of its variables gives values true or false depending on to each variable being a specific natural number. Now, if I want to show that predicate has representing function then for all possible values I should check whether when $P$ evaluates to true $\phi$ evaluates to $0$, and when $P$ evalutes to false, $\phi$ evaluates to $1$. But because the domain is not finite, that would require me infinite steps. Isn't that a deviation from the plan of this theory being finitary?
I would appreciate your help and suggestions. Probably I am not quite understanding the whole goal of primitive recursive functions and I am being confused by definitions.