TITLE: Is $\Delta_0=\Delta_1$ in arithmetical hierarchy?
QUESTION [7 upvotes]: I have seen a definition (e.g. http://www.math.ubc.ca/~bwallace/ArithmeticalHierarchy.pdf) of an arithmetical hierarchy in computability starting with: "let $\Delta_0=\Sigma_0=\Pi_0$ be the set of all computable subsets of natural numbers...", then of course $\Delta_0=\Delta_1$.
However, I have seen another definition where "$\Delta_0=\Sigma_0=\Pi_0$ is the set of all subsets of natural numbers definable by a formula of first-order arithmetic with no quantifiers...".
For the later definition, I do believe that $\Delta_0 \subset \Delta_1$, but cannot prove the proper inclusion. Clearly, $\mathtt{Even}$, the set of even numbers, is in $\Delta_1$, but could I prove that $\mathtt{Even} \not\in \Delta_0$? Or is it perhaps true that $\Delta_0=\Delta_1$?

REPLY [8 votes]: There are several conventions about this, depending on which area you work in. They become equivalent at the level $\Delta^0_1$ and higher, but vary on $\Delta^0_0$, $\Sigma^0_0$, and $\Pi^0_0$. I will drop the superscripts for the rest of this answer.
In computability theory, the most common "starting point" is the computable sets. So it is relatively common, as in Soare's book Computably Enumerable Sets and Degrees, to define a set to be $\Delta_0$, $\Sigma_0$, and/or $\Pi_0$ if it is computable. I suspect this relates to the viewpoint that interprets $\Sigma_{n+1}$ sets as projections of $\Pi_n$ sets: this viewpoint doesn't need any notion of "formula" to define the arithmetical hierarchy. So a $\Sigma_1$ subset of $\mathbb{N}$ is exactly a projection of a computable subset of $\mathbb{N}\times\mathbb{N}$, etc. 
On the other hand, in formalized arithmetic, the most common "starting point" is the set of bounded-quantifier formulas. So it is relatively common, as in Kaye's book Models of Peano Arithmetic, to define a formula to be $\Sigma_0$, $\Pi_0$ and/or $\Delta_0$ if it has only bounded quantifiers.  
Of course, Post's theorem shows there is a tight link between the formula viewpoint and the computability/projection viewpoint. But the exact statement can be different at the lowest levels of the arithmetical hierarchy depending on how the hierarchy is defined.
There may be some other variations, e.g. I think I might have once seen someone define a set to be $\Delta_0$ if the set is primitive recursive.