As I said, the "zeroth successor" of a number is itself. That is, zero is the result of applying the successor function to itself zero times. You have to apply a function at least once in order to have applied the function (and thus obtained a result of applying the function, e.g., calculated a successor).

If you don't like the term, you can think of it this way:

• P0(x): x = 0

• P1(x): x = S0

• P2(x): x = SS0

and so forth.