Jensen's covering theorem
Introduction 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  cannot be covered by a constructible set of cardinality less than .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.
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