set theory Set theory is the branch of mathematical logic that studies 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 mathematics, is mostly concer ...

, 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 In mathematics, in set theory, the constructible universe (or Gödel's constructible universe), denoted by , is a particular class of sets that can be described entirely in terms of simpler sets. is the union of the constructible hierarchy . It ...

is close to the universe of all sets. The first proof appeared in .

Silver Silver is a chemical element with the Symbol (chemistry), symbol Ag (from the Latin ', derived from the Proto-Indo-European wikt:Reconstruction:Proto-Indo-European/h₂erǵ-, ''h₂erǵ'': "shiny" or "white") and atomic number 47. A soft, whi ...

later gave a fine-structure-free proof using his

machines A machine is a physical system using 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 macromolecul ...

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 ℵ_ω cannot be covered by a constructible set of cardinality less than ℵ_ω. 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 ...

proved a strong form of Jensen's covering lemma.

Hugh Woodin William Hugh Woodin (born April 23, 1955) is an American mathematician and set theorist at Harvard University. He has made many notable contributions to the theory of inner models and determinacy. A type of large cardinals, the Woodin cardinals, ...

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

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Notes

Set theory {{mathlogic-stub