Jensen's covering theorem

In <a href="/facts/Set_theory/1ptfsvUP">set theory</a>, Jensen's covering theorem states that if <a href="/facts/Zero_sharp/u0xrkn6l">0#</a> does not exist then every <a href="/facts/Uncountable/zl2WqyJQ">uncountable</a> set of <a href="/facts/Ordinal_number/GelFLjJt">ordinals</a> is contained in a <a href="/facts/Constructible_universe/pxU0cmL3">constructible set</a> of the same <a href="/facts/Cardinality/m6VPojwT">cardinality</a>. Informally this conclusion says that the <a href="/facts/Constructible_universe/pxU0cmL3">constructible universe</a> is close to the universe of all sets. The first proof appeared in (Devlin & Jensen 1975). <a href="/facts/Jack_Silver/wzrk5XVR">Silver</a> later gave a fine-structure-free proof using his <a href="/facts/Silver_machine/E81Tb5ed">machines</a> and finally <a href="/facts/Menachem_Magidor/jL79brqd">Magidor</a> (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 }}
 
 cannot be covered by a constructible set of cardinality less than 
 
 
 
 
 ℵ
 
 ω
 
 
 
 
 {\displaystyle \aleph _{\omega }}
 
.
In his book Proper Forcing, <a href="/facts/Saharon_Shelah/oI5WSK8p">Shelah</a> proved a strong form of Jensen's covering lemma.
<a href="/facts/Hugh_Woodin/gvqYIXWt">Hugh Woodin</a> 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.

<ul><li><a href="/facts/Keith_Devlin/9YsdjXjk">Devlin, Keith I.</a>; <a href="/facts/Ronald_Bj%C3%B6rn_Jensen/Q5PxW1Mr">Jensen, R. Björn</a> (1975), <a href="https://books.google.com/books?id=9UHU_bq-wc8C&q=Marginalia+to+a+theorem+of+Silver">"Marginalia to a theorem of Silver"</a>, ISILC Logic Conference (Proc. Internat. Summer Inst. and Logic Colloq., Kiel, 1974), Lecture Notes in Mathematics, vol. 499, Berlin, New York: <a href="/facts/Springer-Verlag/nAesf6nT">Springer-Verlag</a>, pp. 115–142, <a href="/facts/Doi_(identifier)/muM9Etpq">doi</a>:<a href="https://doi.org/10.1007%2FBFb0079419">10.1007/BFb0079419</a>, <a href="/facts/ISBN_(identifier)/15AdSPa9">ISBN</a> 978-3-540-07534-9, <a href="/facts/MR_(identifier)/uP137L11">MR</a> <a href="https://mathscinet.ams.org/mathscinet-getitem?mr=0480036">0480036</a></li>
<li>Magidor, Menachem (1990), "Representing sets of ordinals as countable unions of sets in the core model", <a href="/facts/Transactions_of_the_American_Mathematical_Society/JmKtyCwK">Transactions of the American Mathematical Society</a>, 317 (1): 91–126, <a href="/facts/Doi_(identifier)/muM9Etpq">doi</a>:<a href="https://doi.org/10.2307%2F2001455">10.2307/2001455</a>, <a href="/facts/ISSN_(identifier)/DPAflDvU">ISSN</a> <a href="https://search.worldcat.org/issn/0002-9947">0002-9947</a>, <a href="/facts/JSTOR_(identifier)/YTeVmaJ7">JSTOR</a> <a href="https://www.jstor.org/stable/2001455">2001455</a>, <a href="/facts/MR_(identifier)/uP137L11">MR</a> <a href="https://mathscinet.ams.org/mathscinet-getitem?mr=0939805">0939805</a></li>
<li>Mitchell, William (2010), "The covering lemma", Handbook of Set Theory, Springer, pp. 1497–1594, <a href="/facts/Doi_(identifier)/muM9Etpq">doi</a>:<a href="https://doi.org/10.1007%2F978-1-4020-5764-9_19">10.1007/978-1-4020-5764-9_19</a>, <a href="/facts/ISBN_(identifier)/15AdSPa9">ISBN</a> 978-1-4020-4843-2</li>
<li><a href="/facts/Saharon_Shelah/oI5WSK8p">Shelah, Saharon</a> (1982), Proper Forcing, Lecture Notes in Mathematics, vol. 940, Berlin, New York: <a href="/facts/Springer-Verlag/nAesf6nT">Springer-Verlag</a>, <a href="/facts/Doi_(identifier)/muM9Etpq">doi</a>:<a href="https://doi.org/10.1007%2FBFb0096536">10.1007/BFb0096536</a>, <a href="/facts/Hdl_(identifier)/rdebSxmC">hdl</a>:<a href="https://hdl.handle.net/10338.dmlcz%2F143570">10338.dmlcz/143570</a>, <a href="/facts/ISBN_(identifier)/15AdSPa9">ISBN</a> 978-3-540-11593-9, <a href="/facts/MR_(identifier)/uP137L11">MR</a> <a href="https://mathscinet.ams.org/mathscinet-getitem?mr=0675955">0675955</a></li></ul>

Jensen's covering theorem open-in-new

Jensen's covering theorem