The arithmetic hierarchy has a few different equivalent definitions but let's use one based on computability. We define inductively there hierarchies, Σi0, Πi0 and Δi0. Σ00=Π00=Δ00 are the computable sets and
- Δi+10 are the sets computable with a Σi0 oracle.
- Σi+10 are the sets computably enumerable with a Σi0 oracle.
- Πi0 = co-Σi0.
We completely know the structure of the arithmetic hierarchy, for i > 0, Σi0 ≠ Πi0 and for i ≥ 0, Δi0 = Σi+10 ∩ Πi+10.
The arithmetic hierarchy inspired the polynomial-time hierarchy in complexity theory. Unlike the arithmetic hierarchy, separations in the polynomial-time hierarchy remain open and any separation implies P ≠ NP. While we have relativized worlds which do quite a few different separations and collapses in the polynomial-time hierarchy the following remains open: Does there exist a relativized world where the polynomial-time hierarchy looks like the arithmetic hierarchy, i.e., for i > 0, Σip ≠ Πip and for i ≥ 0, Δip = Σi+1p ∩ Πi+1p?