TITLE: Why are $\Delta_1$ sentences of arithmetic called recursive?
QUESTION [14 upvotes]: The arithmetic hierarchy defines the $\Pi_1$ formulae of arithmetic to be formulae that are provably equivalent to a formula in prenex normal form that only has universal quantifiers, and $\Sigma_1$ if it is provably equivalent to a prenex normal form with only existential quantifiers.
A formula is $\Delta_1$ if it is both $\Pi_1$ and $\Sigma_1.$  These formulae are often called recursive: why?

REPLY [4 votes]: The term recursive in computability theory is the same as the term computable, which may or may not give a better intuition of the concept of $\Delta_1^0$. 
Anyway, you can understand a set $A$ (or relation $\Theta$) to be computable if there is an algorithmic process such that for any $n$, in finite time will return the answer of whether $n \in A$ or $n \notin A$. 
A relation $\Theta$ is $\Sigma_1^0$ if it is of the form $(\exists k)\varphi(x,k)$ where $\varphi$ is something known to be computable. So if $\Theta(n)$ holds, then by searching, one will find a witness $k$ in finite amount of time. However, in finite time you can not ascertain if $\neg\Theta(x)$ holds since it requires checking that every $k$ is not a witness. 
A relation is $\Pi_1^0$ if its complement is $\Sigma_1^0$. 
Now so understand why $\Delta_1^0$ relations are consider recursive or computable. So $\Theta$ being $\Delta_1^0$ means that it an its complement is $\Sigma_1^0$. Suppose that $\Theta(x) = (\exists k)(\varphi(x))$ and $\neg\Theta(x) = (\exists k)(\psi(x))$ where both $\varphi$ and $\psi$ are computable. The claim is that determining $\Theta$ is computable. To do this just search through all $k$ such and ask if $\varphi(x,k)$ or $\psi(x,k)$. If the first is found then $\Theta(x)$ if the second is found then $\neg\Theta(x)$. Such a $k$ will definitely be found for one or the other since for all $x$, either $\Theta(x)$ or $\neg\Theta(x)$. So $\Delta_1^0$ relations satisfies the intuition that computable relations are those which one can determine satisfiability or nonsatisfiability in finite amount of time.