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The Polynomial Hierarchy does not collapse
The arithmetical hierarchy (AH) is similar to the polynomial hierarchy (PH). Unlike the PH, the AH does not collapse relative to any oracle. A $Σ_{k+1}$-complete language is computably enumerable (c.e.) relative to the k-th level of the AH. So, given an oracle in the k-th level of the AH, we require
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