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What's Q? They lost me there. A is all computable functions in... Something?

Also, if Turing completeness indicates that a language can simulate any mechanical process, and any Turing complete language must be Gödel incomplete, then that means there must be some physical facts that are simply inexpressible?

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Not inexpressible. Unprovable. There could be a set of correct programs that could never be proven correct by a type checker, for example.

Q is the set of all functions that take an integer as an input and return a 0 or a 1. A is the set of all function in Q that are computable.

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