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 {{harv|Devlin|Jensen|1975}}. Silver later gave a fine-structure-free proof using his machinesW. Mitchell, [https://www.semanticscholar.org/paper/Inner-Models-for-Large-Cardinals-Mitchell/ecf7380a4468e233a23282157b318e20156e3a1a Inner models for large cardinals] (2012, p.16). Accessed 2022-12-08. and finally {{harvs|txt|authorlink=Menachem Magidor|last=Magidor|year=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:"In search of Ultimate-L" Version: January 30, 2017
: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
- {{Citation | author1-link=Keith Devlin | last1=Devlin | first1=Keith I. | author2-link=Ronald Björn Jensen | last2=Jensen | first2=R. Björn | title=ISILC Logic Conference (Proc. Internat. Summer Inst. and Logic Colloq., Kiel, 1974) | publisher=Springer-Verlag | location=Berlin, New York | series=Lecture Notes in Mathematics | isbn=978-3-540-07534-9 | doi=10.1007/BFb0079419 | mr=0480036 | year=1975 | volume=499 | chapter=Marginalia to a theorem of Silver | chapter-url=https://books.google.com/books?id=9UHU_bq-wc8C&q=Marginalia+to+a+theorem+of+Silver | pages=115–142}}
- {{Citation | last1=Magidor | first1=Menachem | title=Representing sets of ordinals as countable unions of sets in the core model | mr=939805 | year=1990 | journal=Transactions of the American Mathematical Society | issn=0002-9947 | volume=317 | issue=1 | pages=91–126 | doi=10.2307/2001455| jstor=2001455 | doi-access=free }}
- {{citation | last1=Mitchell|first1=William|chapter=The covering lemma|title=Handbook of Set Theory|publisher=Springer|year=2010|doi=10.1007/978-1-4020-5764-9_19|pages= 1497–1594| isbn=978-1-4020-4843-2 }}
- {{Citation | last1=Shelah | first1=Saharon | author1-link=Saharon Shelah | title=Proper Forcing | publisher=Springer-Verlag | location=Berlin, New York | series=Lecture Notes in Mathematics | isbn=978-3-540-11593-9 | doi=10.1007/BFb0096536 | mr=675955 | year=1982 | volume=940| hdl=10338.dmlcz/143570 | hdl-access=free }}