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). 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 [math]\displaystyle{ \aleph_\omega }[/math] cannot be covered by a constructible set of cardinality less than [math]\displaystyle{ \aleph_\omega }[/math].

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, 499, Berlin, New York: Springer-Verlag, pp. 115–142, doi:10.1007/BFb0079419, ISBN 978-3-540-07534-9, https://books.google.com/books?id=9UHU_bq-wc8C&q=Marginalia+to+a+theorem+of+Silver 
• 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 
• 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, 940, Berlin, New York: Springer-Verlag, doi:10.1007/BFb0096536, ISBN 978-3-540-11593-9