Which is—to use the latest philosophy lingo—dumb. To be fair to Penrose, the “Gödel’s theory about formal systems proves that souls exist” is an extremely common take; anyone following LLM discussions has likely seen it rediscovered at least once or twice.

To pull from the relevant part of Hofstadter’s incredible I am a Strange Loop (a book also happens to more rigorously invoke Gödel for cognitive science):

  And this is our central quandary. Either we believe in a nonmaterial soul that lives outside the laws of physics, which amounts to a nonscientific belief in magic, or we reject that idea, in which case the eternally beckoning question "What could ever make a mere physical pattern be me?”
  After all, a phrase like "physical system" or "physical substrate" brings to mind for most people… an intricate structure consisting of vast numbers of interlocked wheels, gears, rods, tubes, balls, pendula, and so forth, even if they are tiny, invisible, perfectly silent, and possibly even probabilistic. Such an array of interacting inanimate stuff seems to most people as unconscious and devoid of inner light as a flush toilet, an automobile transmission, a fancy Swiss watch (mechanical or electronic), a cog railway, an ocean liner, or an oil refinery. Such a system is not just probably unconscious, it is *necessarily* so, as they see it. This is the kind of single-level intuition so skillfully exploited by John Searle in his attempts to convince people that computers could never be conscious, no matter what abstract patterns might reside in them, and could never mean anything at all by whatever long chains of lexical items they might string together.

Highly recommend it for anyone who liked Gödel, Escher, Bach, but wants more explicit scientific theses! He basically wrote it to clarify the more artsy/rhetorical points made in the former book.