Sy-David Friedman

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There have been numerous results showing that a measurable cardinal κ can carry exactly α normal measures in a model of GCH, where α is a cardinal at most κ++. Starting with just one measurable cardinal, we have [9] (for α = 1), [10] (for α = κ++, the maximum possible) and [1] (for α = κ+, after collapsing κ++). In addition, under stronger large cardinal(More)
In this paper we introduce some fusion properties of forcing notions which guarantee that an iteration with supports of size ≤ κ not only does not collapse κ but also preserves the strength of κ (after a suitable preparatory forcing). This provides a general theory covering the known cases of tree iterations which preserve large cardinals (cf. [4, 5, 6, 8,(More)
By forcing over amodel of ZFC+ GCH (aboveא0) with a class-sized partial order preserving this theory we produce a model in which there is a locally defined well-order of the universe; that is, one whose restriction to all levels H(κ) (κ ≥ ω2 a regular cardinal) is a well-order of H(κ) definable over the structure 〈H(κ),∈〉 by a parameter-free formula.(More)
The study of Borel equivalence relations under Borel reducibility has developed into an important area of descriptive set theory. The dichotomies of Silver ([19]) and Harrington-Kechris-Louveau ([5]) show that with respect to Borel reducibility, any Borel equivalence relation strictly above equality on ω is above equality on P(ω), the power set of ω, and(More)
We study the structure of Σ1 equivalence relations on hyperarithmetical subsets of ω under reducibilities given by hyperarithmetical or computable functions, called h-reducibility and FF-reducibility, respectively. We show that the structure is rich even when one fixes the number of properly Σ1 (i.e. Σ 1 1 but not ∆ 1 1) equivalence classes. We also show(More)
The continuum function F on regular cardinals is known to have great freedom; if α, β are regular cardinals, then F needs only obey the following two restrictions: (1) cf(F (α)) > α, (2) α < β → F (α) ≤ F (β). However, if we wish to preserve measurable cardinals in the generic extension, new restrictions must be put on F . We say that κ is F(More)
We study the complexity of the isomorphism relation on classes of computable structures. We use the notion of FF -reducibility introduced in [9] to show completeness of the isomorphism relation on many familiar classes in the context of all Σ1 equivalence relations on hyperarithmetical subsets of ω.