 The First Incompleteness Theorem provides a counterexample to completeness by exhibiting an arithmetic statement which is neither provable nor refutable in Peano arithmetic, though true in the standard model. The Second Incompleteness Theorem shows that the consistency of arithmetic cannot be proved in arithmetic itself. Thus Gödel’s theorems demonstrated the infeasibility of the Hilbert program, if it is to be characterized by those particular desiderata, consistency and completeness.