We discuss new decidability and undecidability results for mixed characteristic henselian fields, whose proof goes via reduction to positive characteristic. The reduction uses extensively the theory of perfectoid fields and also the earlier Krasner-Kazhdan-Deligne principle. Our main results will be:

(1) A relative decidability theorem for perfectoid fields. Using this, we obtain decidability of certain tame fields of mixed characteristic.

(2) An undecidability result for the asymptotic theory of all finite extensions of Qp (fixed p) with cross-section. We will also discuss the arithmetic of models of the asymptotic theory and present a model-theoretic way of seeing the Fontaine-Wintenberger theorem.