A short note on the significance of the Penrose-halting theorem

In The Emperor’s New Mind [Penrose, 1989], Roger Penrose proves a variant of the halting problem, and uses it to argue that humans have cognitive capacities beyond the computable. In this short note I explicate his argument, and show how it fails, via a corollary of his result. My response to Penrose is in fact of a kind with a number of prior responses: he assumes human powers, that (as the corollary shows) no computer could have. However, as far as I am aware, no one has previously addressed this specific form of the argument, in the direct way that I will.