Jensen's covering theorem
Encyclopedia
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 . Silver
later gave a fine structure free proof using his machines
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 ℵω cannot be covered by a constructible set of cardinality less than ℵω.
In his book Proper Forcing, Shelah
proved a strong form of Jensen's covering lemma.
Set theory
Set theory is the branch of mathematics that studies sets, which are collections of objects. Although any type of object can be collected into a set, set theory is applied most often to objects that are relevant to mathematics...
, Jensen's covering theorem states that if 0#
Zero sharp
In the mathematical discipline of set theory, 0# is the set of true formulas about indiscernibles in the Gödel constructible universe. It is often encoded as a subset of the integers , or as a subset of the hereditarily finite sets, or as a real number...
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
Constructible universe
In mathematics, the constructible universe , denoted L, is a particular class of sets which can be described entirely in terms of simpler sets. It was introduced by Kurt Gödel in his 1938 paper "The Consistency of the Axiom of Choice and of the Generalized Continuum-Hypothesis"...
is close to the universe of all sets. The first proof appeared in . Silver
Jack Silver
Jack Howard Silver is a set theorist and logician at the University of California, Berkeley. He has made several deep contributions to set theory...
later gave a fine structure free proof using his machines
Silver machine
In set theory, Silver machines are devices used for bypassing the use of fine structure in proofs of statements holding in L. They were invented by set theorist Jack Silver as a means of proving global square holds in the constructible universe....
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 ℵω cannot be covered by a constructible set of cardinality less than ℵω.
In his book Proper Forcing, Shelah
Saharon Shelah
Saharon Shelah is an Israeli mathematician. He is a professor of mathematics at the Hebrew University of Jerusalem and Rutgers University in New Jersey.-Biography:...
proved a strong form of Jensen's covering lemma.