Learn More
We give the first systematic study of strong isomorphism reductions , a notion of reduction more appropriate than polynomial time reduction when, for example, comparing the computational complexity of the isomor-phim problem for different classes of structures. We show that the partial ordering of its degrees is quite rich. We analyze its relationship to a(More)
There are two standard ways to establish consistency in set theory. One is to prove consistency using inner models, in the way that Gödel proved the consistency of GCH using the inner model L. The other is to prove consistency using outer models, in the way that Cohen proved the consistency of the negation of CH by enlarging L to a forcing extension L[G].(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 structure of Σ 1 1 equivalence relations on hyperarith-metical 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 Σ 1 1 (i.e. Σ 1 1 but not ∆ 1 1) equivalence classes. We also(More)
Shelah-Woodin [10] investigate the possibility of violating instances of GCH through the addition of a single real. In particular they show that it is possible to obtain a failure of CH by adding a single real to a model of GCH, preserving cofinalities. In this article we strengthen their result by showing that it is possible to violate GCH at all infinite(More)
If L is a finite relational language then all computable L-structures can be effectively enumerated in a sequence {An}n∈ω in such a way that for every computable L-structure B an index n of its iso-morphic copy An can be found effectively and uniformly. Having such a universal computable numbering, we can identify computable structures with their indices in(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)