Oracle Relations Between Complexity Classes I'm trying to get a better handle on oracle separations between complexity classes but I keep running up against some (seemingly) silly issues that make me think that I'm fundamentally misunderstanding something (or more than one thing).

For example, I understand that for an oracle separation to hold in the unrelativized case it must hold for all possible oracles (which is why we cannot answer P=NP? this way). However, is it not true that for any two complexity classes we can always construct an oracle such that they are equivalent in relation to that oracle? For instance, by choosing a higher level of the arithmetic hierarchy to use as the oracle?

i.e. given two complexity classes $A$ and $B$ such that $A,B \subseteq \Sigma_{n}^{0}$, is it not always the case that $A^{\Sigma_{n+1}^{0}} = B^{\Sigma_{n+1}^{0}}$, given that the arithmetic hierarchy doesn't collapse? If not, why?
You seem to understand the concept even though you described it wrong. ​ ​ ​ Unless exactly one of {A,B} is uniform, an oracle separation of A from B either means ​ [showing that there is an oracle O such that ​ AO ≠ BO ] , ​ or means [exhibitings an oracle O such that ​ AO ≠ BO ].

Yes, it is "not true that for any two complexity classes we can always construct an oracle such that they are equivalent in relation to that oracle". ​ In particular, most hierarchy theorems relativize. ​ However, it is the case "that for any two complexity classes we can always construct" a class-oracle "such that they are equivalent in relation to that" class-oracle. ​ (For example, this one always does that.) ​ The triangle levels do not have complete sets with respect to syntactic subclasses of R, and each other non-computable level is not low for itself.

Given "two complexity classes $A$ and $B$ such that $A,B \subseteq \Sigma_{n}^{0}$," it is indeed "not always the case that $A^{\Sigma_{n+1}^{0}} = B^{\Sigma_{n+1}^{0}}$," even though "the arithmetic hierarchy doesn't collapse". ​ See two sentences before this sentence.