Jensen's covering theorem

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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 (Devlin & Jensen 1975). Silver later gave a fine structure free proof using his machines and finally Magidor (1990) 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.