  Once again Gödel provided the first steps in the search for new axioms.  In 1946 he proposed as new axioms large cardinal   axioms—axioms of infinity that assert that there are very large levels of the hierarchy of types—and he went so far as to entertain a generalized completeness theorem for such axioms, according to which all statements of set theory could be settled by such axioms (Gödel 1946, 151).