If you start from a sufficient number of complex axioms/assumptions, then the system will either be complete or consistent. If it's suffiently complex and consistent it can't be complete. If it is not consistent it is fundamentally flawed. So some statements which are true can not be reached by the axioms if the axioms are consistent. But the statement is true.
Science isn't the same thing as an axiomatic system of logic like Peano arithmetic; but the proofs show that there are things a computer can not solve, and there are other similar proofs that show there are limitations to what a computer can do. But there is no proof that human mind = computer.
Hume is much better at kicking Science's ass.
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It's funny that actually, the more we try to "go deep" and brew some kind of TOE, the less these theories become predictive in scope.
Take the standard model for example. It's only good at computing scattering cross sections between subatomic particles, and requires enormous amount of processing power and human sweating to get anything out of it.
Trying to make a model and predict a small atomic nucleus starting only from the standard model ? Good luck, you would need a computer the size of a small nation and an army of graduate students.
A model of every human interactions ? You would need a Laplace's demon as big as the solar system, and even there. the l Every time we try to predict something we have to embed, to virtualize a small part of the world into a bigger part of the world, for example a computer or a human brain. The only perfect simulation, the simulation which requires no more no less than what the movement one simulates, is the world itself.
A theory of everything won't be any good at predicting anything, only at telling you what is within the bounds of possibilities. These possibilities are spoken in terms of symmetries.
So as of today we are pretty confident that anything is possible within the bounds of a local SU(3) X SU(2) X U(1) symmetry and a global translation X SO(3,1) symmetry, nothing more, nothing less.
Anyway, it is in that sense that those who speculate about theories of everything following creative impulses brought about by reading New Scientist and their sensationalistic covers telling of strings and loops, those are doing so without understanding what these TOE really entails.
I'm all for silencing these speculations, but these speculation are not what's going on in research. Let the quantum mystical magical guru have wet dreams of Laplace's demons, but also let the scientist do what he likes.
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