Jensen's covering theorem
In
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[1] 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 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:[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.
References
- MR 0480036
- Magidor, Menachem (1990), "Representing sets of ordinals as countable unions of sets in the core model", MR 0939805
- Mitchell, William (2010), "The covering lemma", Handbook of Set Theory, Springer, pp. 1497–1594, ISBN 978-1-4020-4843-2
- MR 0675955
Notes
- ^ W. Mitchell, Inner models for large cardinals (2012, p.16). Accessed 2022-12-08.
- ^ "In search of Ultimate-L" Version: January 30, 2017
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