Jensen's Covering Theorem
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set theory Set theory is the branch of mathematical logic that studies Set (mathematics), sets, which can be informally described as collections of objects. Although objects of any kind can be collected into a set, set theory – as a branch of mathema ...
, Jensen's covering theorem states that if 0# does not exist then every
uncountable In mathematics, an uncountable set, informally, is an infinite set that contains too many elements to be countable. The uncountability of a set is closely related to its cardinal number: a set is uncountable if its cardinal number is larger tha ...
set of ordinals is contained in a constructible set of the same
cardinality The thumb is the first digit of the hand, next to the index finger. When a person is standing in the medical anatomical position (where the palm is facing to the front), the thumb is the outermost digit. The Medical Latin English noun for thum ...
. Informally this conclusion says that the
constructible universe In mathematics, in set theory, the constructible universe (or Gödel's constructible universe), denoted by L, is a particular Class (set theory), class of Set (mathematics), sets that can be described entirely in terms of simpler sets. L is the un ...
is close to the universe of all sets. The first proof appeared in .
Silver Silver is a chemical element; it has Symbol (chemistry), symbol Ag () and atomic number 47. A soft, whitish-gray, lustrous transition metal, it exhibits the highest electrical conductivity, thermal conductivity, and reflectivity of any metal. ...
later gave a fine-structure-free proof using his
machines A machine is a physical system that uses power to apply forces and control movement to perform an action. The term is commonly applied to artificial devices, such as those employing engines or motors, but also to natural biological macromolec ...
and finally 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 Shelah may refer to: * Shelah (son of Judah), a son of Judah according to the Bible * Shelah (name), a Hebrew personal name * Shlach, the 37th weekly Torah portion (parshah) in the annual Jewish cycle of Torah reading * Salih, a prophet described i ...
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.


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Notes

Set theory {{mathlogic-stub