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)
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 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.
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)
Fra¨ıssé studied countable structures S through analysis of the age of S, i.e., the set of all finitely generated substructures of S. We investigate the effectiveness of his analysis, considering effectively presented lists of finitely generated structures and asking when such a list is the age of a computable structure. We focus particularly on the… (More)