Jensen's covering theorem explained

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[1] 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

\aleph\omega

cannot be covered by a constructible set of cardinality less than

\aleph\omega

.

In his book Proper Forcing, Shelah proved a strong form of Jensen's covering lemma.

Hugh Woodin states it as:[2]

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.

Notes and References

  1. W. Mitchell, Inner models for large cardinals (2012, p.16). Accessed 2022-12-08.
  2. "In search of Ultimate-L" Version: January 30, 2017