We will discuss new decidability results for mixed characteristic henselian fields, whose proof goes via reduction to positive characteristic. In order to achieve the reduction, we will use extensively the theory of perfectoid fields/p-adic Hodge theory and also the earlier Krasner-Kazhdan-Deligne principle. Our main results are the following: (1) A relative decidability result for perfectoid fields. This says that, under a certain natural assumption, a perfectoid field K is decidable relative to its tilt K^?. As an application, we obtain several decidability results for tame fields of mixed characteristic, transposing a recent result of Lisinski (building on earlier work of Kuhlmann). We also obtain a different application by transposing work of Anscombe-Fehm to mixed characteristic. (2) An undecidability result for the asymptotic theory of p-adic fields (fixed p). This says that the set of sentences in the language of valued fields with cross-section which are true in all but finitely many finite extensions of Qp is undecidable. This should be contrasted with the Ax-Kochen/Ershov Theorem, saying that each individual p-adic field is decidable in said language.