> Godel explored this exhaustively and concluded (though did not prove) that this is not possible
While I agree that Aaronson's take on philosophy is not very original or profound, I don't think that Gödel concluded anything of the sort, and if he said something in that vein (assuming you're referring to the discussion on mechanism in the link you're provided), it is more an expression of his faith that any problem can be solved (and since not every problem can be solved by a machine, the mind is more powerful than any machine) -- hardly a conclusion.
I think it's just sloppy discussion.
Any hard-determinist will say that every interaction is pre-determined based on a previous interaction. The compatibilist approach tries to reconcile hard-determinism with the illusion of free will in a reference frame of the individual - but I think falls apart. I think Aaronson is trying to convey this particular philosophy.
"if you knew the complete state of the universe, you could use it to calculate [either deterministically or probabilistically] everything I’d do in the future."
Fortunately for us, Godel explored this exhaustively and concluded (though did not prove) that this is not possible. See also the Munchhausen Trilemma of proofs for problems proving any epistemological inquiry.
[1]https://en.wikipedia.org/wiki/Hard_determinism
[2]https://en.wikipedia.org/wiki/Compatibilism
[3]http://plato.stanford.edu/entries/goedel-incompleteness/
[4]https://en.wikipedia.org/wiki/M%C3%BCnchhausen_trilemma