In set theory, Jensen's covering theorem states that if 0# does not exist then every uncountable set of ordinals is contained in a constructible set of the same cardinality. Informally this conclusion says that the constructible universe is close to the universe of all sets. The first proof appeared in . Silver later gave a fine-structure-free proof using his machines and finally gave an even simpler proof. The converse of Jensen's covering theorem is also true: if 0# exists then the countable set of all cardinals less than ℵω cannot be covered by a constructible set of cardinality less than ℵω. In his book Proper Forcing, Shelah proved a strong form of Jensen's covering lemma. Hugh Woodin states it as: Theorem 3.33 (Jensen). One of the following holds. (1) Suppose λ is a singular cardinal. Then λ is singular in L and its successor cardinal is its successor cardinal in L. (2) Every uncountable cardinal is inaccessible in L. References Notes Set theory