Jensen's covering theorem

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 $\aleph _{\omega }$ 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:

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

• Devlin, Keith I.; Jensen, R. Björn (1975), "Marginalia to a theorem of Silver", ISILC Logic Conference (Proc. Internat. Summer Inst. and Logic Colloq., Kiel, 1974), Lecture Notes in Mathematics, vol. 499, Berlin, New York: Springer-Verlag, pp. 115–142, doi:10.1007/BFb0079419, ISBN 978-3-540-07534-9, MR 0480036
• Magidor, Menachem (1990), "Representing sets of ordinals as countable unions of sets in the core model", Transactions of the American Mathematical Society, 317 (1): 91–126, doi:10.2307/2001455, ISSN 0002-9947, JSTOR 2001455, MR 0939805
• Mitchell, William (2010), "The covering lemma", Handbook of Set Theory, Springer, pp. 1497–1594, doi:10.1007/978-1-4020-5764-9_19, ISBN 978-1-4020-4843-2
• Shelah, Saharon (1982), Proper Forcing, Lecture Notes in Mathematics, vol. 940, Berlin, New York: Springer-Verlag, doi:10.1007/BFb0096536, hdl:10338.dmlcz/143570, ISBN 978-3-540-11593-9, MR 0675955