Barbara F. Csima

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We consider the Turing degrees of prime models of complete decidable theories. In particular we show that every complete decidable atomic theory has a prime model whose elementary diagram is low. We combine the construction used in the proof with other constructions to show that complete decidable atomic theories have low prime models with added properties.(More)
A set X is prime bounding if for every complete atomic decidable (CAD) theory T there is a prime model A of T decidable in X. It is easy to see that X = 0′ is prime bounding. Denisov claimed that every X <T 0′ is not prime bounding, but we discovered this to be incorrect. Here we give the correct characterization that the prime bounding sets X ≤T 0′ are(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 RCA0 of reverse mathematics. We answer the question of the converse implication(More)
A computable structure A is x-computably categorical for some Turing degree x, if for every computable structure B ∼= A there is an isomorphism f : B → A with f ≤T x. A degree x is a degree of categoricity if there is a computable structure A such that A is x-computably categorical, and for all y, if A is y-computably categorical then x ≤T y. We construct a(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(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 degree(More)
Let N be a countable algebraic structure in a finite language Σ (where the equality symbol = belongs to Σ) whose universe is a subset of ω. We denote via D(N ) the set of all atomic sentences and their negations which are true inN (here we use some computable numbering of the sentences in the language Σ∗ = Σ ∪ ω to identify D(N ) with a subset of ω). The(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 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 is a degree of categoricity.(More)
Fix a finite set L and an infinite list of variables v0, v1, v2, . . . , vn, . . .. For m,n ≤ ω, W (L,m, n) is the set of sequences w of elements of L ∪ {vi | i < m} of length n with the property that vi occurs in w for each i < m and the first occurrence of vi is before the first occurrence of vj whenever i < j < m. When m ∈ ω, W (L,m) is ⋃ n∈ωW (L,m, n).(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)
Fräıssé 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 Fräıssé(More)