Given this answer by user34445 to the question: Final theory in Physics: a mathematical existence proof? https://physics.stackexchange.com/questions/76058/final-theory-in-physics-a-mathematical-existence-proof
How would anyone know if they saw the equation of everything?
Physics presupposes Mathematics. So any "Final Theory of Physics" would also need to presuppose Mathematics for it to have efficacy. Therefore, because this 'Final Theory must presuppose Mathematics, it cannot prove (or disprove) the validity of Mathematics. Likewise, Mathematics presuppose the validity of Logic, so Mathematics cannot therefore prove (or disprove) the validity of Logic. Therefore Physics cannot prove (or disprove) the validity of Logic or Math. Since Physics cannot prove (or disprove) all metaphysical laws (such as Logic or Mathematics), therefore Physics is incomplete.
Now you might say, bah, that is merely math and logic - metaphysics not physics, which is true, except that anything physical is necessarily also metaphysical since anything that actually exists, can be thought to exist. Physics is a subset of metaphysics. We can imagine many things, even impossible things, that do not actually exist, but nothing that actually exists cannot be 'not imagined', whether or not we experience it directly. The domain of all things physical must be a subset of the domain of all things metaphysical. So if physics is incomplete metaphysically, it must also be incomplete physically, since physics is a subset of metaphysics. Therefore, we have good reason to believe that a 'Physics Final Theory' must be incomplete.
Notwithstanding these very good reasons for believing 'The Final Theory in Physics' must be incomplete, lets assume, for the sake of argument, that such a theory did exist, and was both consistent and complete, contradicting both of Gödel's Incompleteness Theorems; such a theorem could then prove its own consistency, the question would become 'How complex would proofs be involving this theorem?' Would proving the consistency or completeness of this 'The Final Theory of Physics' be an NP-complete problem? What about proving both consistency and completeness?
The following are possible:
Option 1. A 'Final Theory of Physics' exists that is both consistent and complete.
Although we've already shown that it could not be complete, we've assumed otherwise. We know that there are physical problems in Quantum Mechanics that are NP-Complete hard, so if it were a Final Theory and complete, proving this would also be an NP-Complete hard problem since proving some of its parts are. Similar reasoning shows that proving it's consistency would also be NP-Complete, so proving this theory to be both complete and consistent will take beyond the age of the universe.
Option 2. A 'Final Theory of Physics' exists that is consistent, but is not complete - with the following possibilities:
2.1. Proving the consistency of this theory is an NP-complete problem, so effectively not provable (at least not in the age of the universe)
2.2. Proving the consistency of this theory is not an NP-complete problem and is provable in some finite amount of time.
Looking at 2.2, we've already shown that because NP-Complete problems exist in Quantum Mechanics, which would be parts of the Final Theory, proving the consistency of the theory is an NP-complete problem, which contradicts our presupposition in 2.2. So Option 2.2 is impossible.
Option 3. A 'Final Theory of Physics' exists, that is complete, but not consistent.
Because the theory in this option is not consistent, we could both prove and disprove that it is complete, a contradiction. Any theory that contradicts itself is useless as a theory. This option's 'Final Theory' is useless.
Option 4. A complete 'Final Theory of Physics' does not exist, but better and better aggregate approximations exist which are consistent, but not complete, which give us better understandings of the laws of physics over time (I am assuming the laws of physics are universal since the non-universality of physical laws is another topic entirely).
What can we conclude from this? We can conclude that even if such a Final Theory exists that is consistent, it is going to be impossible to prove that it is the 'Final Theory', whether or not it is complete. So there is really no way to know once we've discovered the 'Final Theory'.
so once again, does anyone think they would know if they saw the equation of everything and how would they know?