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Copyright line will be provided by the publisher We analyze the degree spectra of structures in which different types of immunity conditions are encoded. In particular , we give an example of a structure whose degree spectrum coincides with the hyperimmune degrees. As a corollary, this shows the existence of an almost computable structure the complement of… (More)

We construct a minimal pair of K-degrees. We do this by showing the existence of an unbounded nondecreasing function f which forces K-triviality in the sense that γ ∈ 2 ω is K-trivial if and only if for all n, K(γ n) ≤ K(n) + f (n) + O(1).

Degrees that are not Degrees of Categoricity

A Turing degree d is homogeneous bounding if every complete decidable (CD) theory has a d-decidable homogeneous model A, i.e., the elementary diagram D e (A) has degree d. It follows from results of Macintyre and Marker that every PA degree (i.e., every degree of a complete extension of Peano Arithmetic) is homogeneous bounding. We prove that in fact a… (More)

Topologists Nabutovsky and Weinberger discovered how to embed computably enumerable (c.e.) sets into the geometry of Riemannian metrics modulo diffeomorphisms. They used the complexity of the settling times of the c.e. sets to exhibit a much greater complexity of the depth and density of local minima for the diameter function than previously imagined. Their… (More)

Tarski defined a way of assigning to each boolean algebra, B, an invariant inv(B) ∈ In, where In is a set of triples from N, such that two boolean algebras have the same invariant if and only if they are elementarily equivalent. Moreover, given the invariant of a boolean algebra, there is a computable procedure that decides its elementary theory. If we… (More)

The Rainbow Ramsey Theorem is essentially an " anti-Ramsey " theorem which states that certain types of colorings must be injective on a large subset (rather than constant on a large subset). Surprisingly, this version follows easily from Ramsey's Theorem, even in the weak system RCA 0 of reverse mathematics. We answer the question of the converse… (More)

We study arithmetic and hyperarithmetic degrees of categoricity. We extend a result of Fokina, Kalimullin, and R. Miller to show that for every computable ordinal α, 0 (α) is the degree of categoricity of some computable structure A. We show additionally that for α a computable successor ordinal, every degree 2-c.e. in and above 0 (α) is a degree of… (More)