This article treats all cases by a uniform argument, starting with only one measurable cardinal and applying a cofinality-preserving forcing, and explores the possibilities for the number of normal measures on a cardinal at which the GCH fails.Expand

Descriptive set theory is mainly concerned with studying subsets of the space of all countable binary sequences. In this paper we study the generalization where countable is replaced by uncountable.… Expand

The notion of FF-reducibility introduced in [9] is used to show completeness of the isomorphism relation on many familiar classes in the context of all equivalence relations on hyperarithmetical subsets of ω.Expand

The proofs combine the techniques of adding and then destroying a nonreflecting stationary set or a ${\kappa ^{ + + }}$-Souslin tree, variants of Mitchell’s forcing to obtain the tree property, together with the Prikry-collapse poset for turning a large cardinal into ${\aleph _\omega }$.Expand

The technique is applied to provide a new proof of Woodin's theorem as well as the new result that global domination at inaccessibles (the statement that d(κ) is less than 2κ for inaccessible κ) is internally consistent, given the existence of 0#.Expand

The possibilities for this assertion of V = L, asserting that every set is constructible, are explored, for various notions of " L-like " and for various types of large cardinals.Expand

Assuming the existence of a hypermeasurable cardinal, we construct a model of Set Theory with a measurable cardinalsuch that 2 • = • ++ and the group Sym(•) of all permutations ofcannot be written as… Expand