I could interpret you to make the claim that mathematical results are exactly pairs consisting of a formal system and a derivation in it . In the context of a discussion of the incompleteness theorem , I would suggest that that is exactly the correct notion to be considering . While this certainly makes things simple , it has the problem that it is far removed from the praxis of most mathematicians . The incompleteness theorem is equally far removed from the praxis of most mathematicians . In fact , I could argue that it is even further removed since most mathematicians are at least implicitly assuming that their proofs could , in principle if not in practice , be converted to formal proofs in ZFC but the vast majority are utterly unconcerned with the potential of ZFC being inconsistent and uninterested in it being incomplete . Technically speaking , the article says it ' s not so simple , not that the theorems become invalid . The article didn ' t seem to be saying anything all that technical to me . But my point was really just about the opening lines and implications of them , as I said the rest of it was okay . it does not seem simple if you consider mathematical results to have meaning independent of formal derivations , to be mental constructions or to be facts about the real or potential world whether the physical world or a mathematical platonic worldThe moment you bring reality into the picture , regardless of whether you mean physical reality or some sort of platonic notion , you have basically declared a priori that there exists a model of some form of logical theory and presumably one capable of encoding arithmetic since we are discussing math , at which point the incompleteness theorem essentially becomes nothing but a curiosity of logic as we have taken a priori the assumption of a model , hence consistency completeness theorem .