> given any formal system F that we might want to take as a foundation for mathematics (for example, Peano Arithmetic or Zermelo-Fraenkel set theory), Gödel tells us that there are Turing machines that run forever, but that can’t be proved to run forever in F.

Wow. I've never understood Godel's theorem before. I've never seen it put that way. Thank you! Is Godel's incompleteness theorem effectively the same thing as the halting problem then? Or rather, a result of it?