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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)

Degrees that are not Degrees of Categoricity

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)

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)

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)

We show that Sacks' and Shoenfield's analogs of jump inversion fail for both tt-and wtt-reducibilities in a strong way. In particular we show that there is a ∆ 0 2 set B >tt ∅ such that there is no c.e. set A with A ≡wtt B. We also show that there is a Σ 0 2 set C >tt ∅ such that there is no ∆ 0 2 set D with D ≡wtt C.