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The number of normal measures
TLDR
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.
Fine Structure and Class Forcing
Forcing with Finite Conditions
We present a generalisation to ω 2 of Baumgartner’s forcing for adding a CUB subset of ω 1 with finite conditions.
Generalized Descriptive Set Theory and Classification Theory
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.
Isomorphism relations on computable structures
TLDR
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.
THE EIGHTFOLD WAY
TLDR
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 ${\aleph _\omega }$.
Perfect trees and elementary embeddings
TLDR
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#.
Large cardinals and $L$-like universes
TLDR
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.
Homogeneous iteration and measure one covering relative to HOD
TLDR
This paper proves that there is a superstrong cardinal and for every regular cardinal κ, κ+ is greater than κ- of HOD.
Measurable cardinals and the cofinality of the symmetric group
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
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