We consider a de’Liguoro-Piperno-style extension of the pure lambda calculus with a non-deterministic choice operator as well as a non-deterministic iterator construct, with the aim of studying its normalization properties. We provide a simple characterization of non-strongly normalizable terms by means of the so called “zoom-in” perpetual reduction strategy. We then show that this characterization implies the strong normalization of the simply typed version of the calculus. As straightforward corollary of these results we obtain a new proof of strong normalization of Goedel’s System T by a simple translation of this latter system into the former.