Handbook of Set Theory

pp 1497-1594


The Covering Lemma

  • William J. MitchellAffiliated withDepartment of Mathematics, University of Florida Email author 

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Our understanding of inner models was transformed in 1974 by as set of handwritten notes of Ronald Jensen with the modest title “Marginalia on a Theorem of Silver”. Before the covering lemma, Gödel’s class L of constructible sets was a model of set theory of which much was know, but which had little connection with the universe; with the covering lemma it is, in the absence of large cardinals, the supporting skeleton of the universe.

The first section of this chapter discusses the forms of the covering lemma, beginning with Jensen’s original lemma for L, going to extensions taking into account arbitrary sequences of measurable cardinals, and then describing variants of the statement: the weak covering lemma, which is basic to inner models beyond measurable cardinals, the strong covering lemma, and variants such as that of Magidor which avoid the need for second order closure. Later sections discuss applications, give proofs of the statements, and discuss what is known up to and beyond a Woodin cardinal.