Computability function - how to express it in set theory/arithmetic hierarchy

Question

Let's say that $f$ is computable function such that for particular inputs $x$ and $y$, $f(x) = 0$ and $f(y) = 0$.

If we want to express this in logical form (arithmetic hierarchy formula), what would it be like? Or is this impossible to be represented in first-order logic?

$x$ and $y$ are specific (fixed) numbers.

Answer 1

For a given function $f$, the statement that the computation of $f(x)$ halts and returns a number $z$ is a $\Sigma^0_1$ formula. The conjunction of two $\Sigma^0_1$ formulas is again $\Sigma^0_1$.