Porpora offers an a priori argument for the conclusion that there are infinitely many thoughts that it is physically possible for us to think. That there should be such an a priori argument is astonishing enough. That the argument should be simple enough to teach to a first-year undergraduate class in about 20 min, as Porpora’s is, is more astonishing still. Porpora’s main target is Max Tegmark’s recent argument for the claim that if current physics is right, then there are (...) mental duplicates of us in far flung regions of the Universe. His argument is directed against Tegmark’s assumption that mental facts supervene upon physical facts. So, if Porpora’s argument is sound then not only is Tegmark’s argument unsound, but physicalism is also false. So, Porpora’s argument is powerful indeed. Who would have thought that a simple a priori argument, together with the physical facts, could solve the issue of whether physicalism is true?. Not I. In this paper I take a closer look at Porpora’s argument and show that it is fallacious. I also consider the other reasons Porpora gives for thinking there are infinitely many thinkable thoughts and find them similarly lacking. (shrink)