Taken together, the two incompleteness theorems say that it is not possible to find a set of axioms for arithmetic which are totally adequate. The first theorem says no set of axioms for arithmetic can be consistent and complete. So, any formal system that proves certain basic arithmetic truths must contain an arithmetical statement that is true but which cannot be proved from the axioms. The second theorem is really just a tightening‐up of a particular aspect of the first incompleteness theorem. It states that if a set of axioms A is consistent then the consistency of A cannot be proved by A.

The key implication of these incompleteness theorems is that you might be able to prove all true statements about numbers (or, equivalently, about any other branch of mathematics) within a system by going outside the system to define new rules or axioms, but if you do so then you only create a larger system which will have its own unprovable statements.