Arithmetic Analogues of P versus BPP
There are several notions of randomness in computability theory (/the arithmetic hierarchy; lookup "Martin-Lof randomness", "Kurtz random", "Schnorr random", ...), but I think the ones that are analogous to $\mathsf{BPP}$ become trivial in the setting of the arithmetic hierarchy. The reason is essentially that a randomized Turing machine with bounded error can be simulated by a deterministic one: the deterministic one simulates the random one with all settings of the randomness and then takes the majority vote. If the original machine took time $t(n)$ then the new machine takes time $2^{t(n)}$, which is why this trick tends not to work in the resource-bounded case. 

(Though of course it works for any deterministic time-bounded class where if time bound $t(n)$ is allowed in the class then so is $2^{t(n)}$. In particular, this trick works in $\mathsf{DTIME}(\mathcal{E}^4)$, where $\mathcal{E}^4$ is the fourth level of the Grzegorczyk hierarchy of primitive recursive functions. But that's still a pretty big class.)