Based on the Tarski's Indefinability Theorem (TA in standard model is not arithmetic (no FOL formula can represent TA,
a formula represent a predicate relation definition: under Tarski's first order language structure, for every element of the Universe (0,1, 2,....) n there exists a FOL formula P_n such that standard model satisfies P_n if and only if n is in the relation set)
and so not recursive enumerable), is it reasonable to pose the following argument:
Assuming real world fits into some formal system (use Phi to refer any possible things into the system, which can be categorized into some groups by some syntactical formalism) then since the real world contains the standard model, which means TA is also in the True group and so True group will not be arithmetic, not recursive enumerable, and also not axiomatizable at least in the sense of modern day of Turing Machine.
On the other hand, if real world does not fit into any formal system, then at least in the sense of modern day of Turing Machine, given there is no syntactical formalism, TM cannot even encode or compute it. (I know this direction's argument is sloppy, I really have little experience on the computation procedure if something cannot fit into formal system)
So the conclusion is, for the axiomatization of real world, if it wants to explain everything, then it must entail the computation more than the level of Turing Machine, and the concept of axiomatization needs to be adjusted, which given todays computation it's quite impossible.
And this basically implies the imperfection of physics in a theoretical sense where as long as our axiomatization of physics is within the power of Turing Machine, then it cannot explain everything in reality.