Gödel's incompleteness theorem shows that no consistent, recursively enumerable theory (that is, one whose theorems form a recursively enumerable set) in which the concept of natural numbers can be expressed, can include all true statements about them. As a result, some domains of knowledge cannot be formalized, accurately and completely, as mathematical theories. (Here, formalizing accurately and completely means that all true propositions—and only true propositions—are derivable within the mathematical system.) This limitation, however, in no way precludes the construction of mathematical theories that formalize large bodies of scientific knowledge.
gödel's incompleteness theorem shows that no consistent recursively enumerable theory (that is one whose theorems form a recursively enumerable set) in which the concept of natural numbers can be expressed can include all true statements about them as a result some domains of knowledge cannot be formalized accurately and completely as mathematical theories (here formalizing accurately and completely means that all true propositions—and only true propositions—are derivable within the mathematical system) this limitation however in no way precludes the construction of mathematical theories that formalize large bodies of scientific knowledge