The incompleteness theorem does not only apply to mathematical systems. It applies to all formal systems in the realm of logic. If we assume that all of reality

The incompleteness theorem does not only apply to mathematical systems. It applies to all formal systems in the realm of logic. If we assume that all of reality can be reduced to a formal logical system then the incompleteness theorems apply. Thus even non-axiomatic concepts may be unexplainable. However, you bring up a good point. Explanations rely on evidence and evidence must be observed. Additionally, the tools which we employ to observe evidence are imperfect. What does this mean? It means in the world of logic only some things can't be proven, but in the world of science (aka reality as we know it): Nothing can be proven. Nothing can be fully explained because additional evidence can always be observed in a later point in time that disproves an explanation. As Einstein once said: "The scientific theorist is not to be envied. For Nature, or more precisely experiment, is an inexorable and not very friendly judge of his work. It never says "Yes" to a theory. In the most favorable cases it says "Maybe," and in the great majority of cases simply "No." If an experiment agrees with a theory it means for the latter "Maybe," and if it does not agree it means "No." Probably every theory will someday experience its "No"—most theories, soon after conception." or Karl Popper who put the concept simply: "...no matter how many instances of white swans we may have observed, this does not explain the conclusion that all swans are white." In short, no amount of evidence or explanation can justify any statement. Interestingly though, it takes only one observed black swan to disprove a statement. Hence the role of falsifiability in the scientific method.