I'm a bit confused with accurately defining partial functions. I'm trying to do the following exercise.
Define the partial function $f$ from $\mathbb{N} \to \mathbb{N} $ where if n is the Godel number of a statement with proof in formal arithmetic, then $f(n) = 1$; otherwise $f(n) = 0$.
This is what I have and I'm not sure if what I'm writing has anything to do with partial functions as I'm getting partial functions confused with partial recursiveness and being Turing computable:
$$f(n) = \begin{cases} 1, &\text{if n is the Godel number of a statement with proof in formal arithmetic} \\ 0, & \text{otherwise} \end{cases}$$