The idea of axioms in mathematics is not quite the same as it is with physics. Physicists don't need the type of mathematical rigor such as the set theory axio

The idea of axioms in mathematics is not quite the same as it is with physics. Physicists don't need the type of mathematical rigor such as the set theory axioms to build up all of mathematics. In fact, by keeping in mind the multitudes of paths from which one part of physics could be related or derived from another part, you are exploring the possibly undiscovered laws in nature. It will only be relevant to axiomatically define physics (from a set of basic axioms to reach all laws) after we confirmed to have discovered all of physics. That day hasn't come yet - there's no grand unified theory so far, and there's still stuff that we dont know surely.