 Four decades ago, J.R. Lucas (1964) argued that Gödel’s first incompleteness theorem entails that no machine can ever reach human-level intelligence. His argument has not proved to be compelling, but Lucas initiated a debate that has produced more formidable arguments. One of Lucas’ indefatigable defenders is the physicist Roger Penrose, whose first attempt to vindicate Lucas was a Gödelian attack on “Strong” AI articulated in his The Emperor’s New Mind (1989). This first attempt fell short, and Penrose published a more elaborate and more fastidious Gödelian case, expressed in Chapters 2 and 3 of his Shadows of the Mind (1994). 