I don't agree with your claim that a theory of consciousness is fundamental to any theory of physics. In mathematics we have to deal with the possibility that the theorems we're trying to prove are "independent" of the system of proofs we're working in (such statements are known to exist and are neither provable nor disprovable, for example, the existence of certain numbers). As a consequence, any physicist working in a mathematical framework must consider the possibility that a theory of consciousness is independent of any theory of physics consistent with what we observe. In other words, it may be the case that physics, just like mathematics, cannot be both complete and consistent.
That premise is definitely true. It is a result of Gödel's Incompleteness Theorems. 'Physics' certainly satisfies the constraint: 'of sufficient complexity to encode the natural numbers'.
I wonder how you come to that conclusion, because physics is not proven to be of "infinite complexity" (loose statement).
Some things that we are used to from mathematics might not be true in physics. Take for example the fact that in mathematics the real numbers are uncountable. Now in reality (physics), the whole set of real numbers may not exist. It is only an abstract concept from mathematics. And while it may be possible to reproduce any real number in physics as some quantity, you are reproducing them as you go, making the "real" (physical) real numbers countable.
Pretty much any system of logic worth looking at (including any which the sciences may be based on and the one running inside each human brain) is going to at least be as complex as this set of axioms.
Also, you really should read the JR Lucas material-- it explains this. And, I'd suggest Nagel & Newman's _Godel's Proof_ for a great introductory explanation of the Incompleteness Theorems.
We don't know that the physical world has infinite precision. In particular, time doesn't even seem to be continuous as far as we can tell (cf. Planck constant). And we think there's a finite amount of mass in the universe, so how could we encode arbitrarily large natural numbers (as is required to model Peano arithmetic)?
We don't have to encode arbitrarily large natural numbers. Rather we have to encode the rules that allow us to construct them (which is quite simple actually). And, I think 'digital physics' is more compatible with Incompleteness implications than the alternatives; not less.
I don't agree with your claim that a theory of consciousness is fundamental to any theory of physics. In mathematics we have to deal with the possibility that the theorems we're trying to prove are "independent" of the system of proofs we're working in (such statements are known to exist and are neither provable nor disprovable, for example, the existence of certain numbers). As a consequence, any physicist working in a mathematical framework must consider the possibility that a theory of consciousness is independent of any theory of physics consistent with what we observe. In other words, it may be the case that physics, just like mathematics, cannot be both complete and consistent.