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 [ 1] 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
ℵ
ω
{\displaystyle \aleph _{\omega }}
ℵ
ω
{\displaystyle \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.
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
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