SneerClub

This is outside my own department, but I think there's a problem with Aaronson's treatment of Gödel's incompleteness theorems. He says that Gödel's first incompleteness theorem follows directly from Turing's proof that the halting problem is undecidable. This doesn't quite work, as I understand it. The result conventionally known as Gödel's theorem is stronger than what you can get from the undecidability of the halting problem. In other words, the result that the Turing machines get you depends upon a more demanding precondition than "consistency", and so it is somewhat less impressive than what was desired. My best stab at a semi-intuitive explanation would be in the vein of, "When you're discussing the consistency of mathematics itself, you have to be double-special-careful that ideas like the number of steps a Turing machine takes really do make sense."

The historical problem is that Turing himself did not prove the undecidability of the halting problem. He wasn't even focused on halting. His main concern was computing real numbers, where naturally a successful description of a number could be a machine that doesn't stop. The "halting state" as we know and love it today was due to Emil Post.

Moreover, this is one of the passages where Aaronson seems to be offering the one and only true Nerd Opinion. He is dismissive of any way to understand Gödel's theorems apart from the story he offers, to the extent that a person who had only read Aaronson would be befuddled by anyone who used Gödel numbering after 1936.