Two theorems widely interpreted to mean that it is impossible to find a complete and consistent set of axioms on which to base all mathematics. n.b. It is nevertheless possible to formalize a basis for the vast majority of mathematics people actually do - see Zermelo–Fraenkel Set Theory.
First incompleteness theorem: For any consistent formal system F within which a certain amount of elementary arithmetic can be carried out, there are statements in the language of F which can neither be proved nor disproved in F. That is, F is "incomplete".
Second incompleteness theorem: For any consistent system F within which a certain amount of elementary arithmetic can be carried out, the consistency of F cannot be proved in F itself.