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Degree spectra of prime models
 J. Symbolic Logic
, 2004
"... 2.1 Notation from model theory................... 4 2.2 F ..."
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2.1 Notation from model theory................... 4 2.2 F
Computable trees, prime models, and relative decidability
 PROC. AMER. MATH. SOC
, 2005
"... We show that for every computable tree T with no dead ends and all paths computable, and every D>T ∅, there is a Dcomputable listing of the isolated paths of T. It follows that for every complete decidable theory T such that all the types of T are computable and every D>T ∅, there is a Ddecidable ..."
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We show that for every computable tree T with no dead ends and all paths computable, and every D>T ∅, there is a Dcomputable listing of the isolated paths of T. It follows that for every complete decidable theory T such that all the types of T are computable and every D>T ∅, there is a Ddecidable prime model of T. This result extends a theorem of Csima and yields a stronger version of the theorem, due independently to Slaman and Wehner, that there is a structure with presentations of every nonzero degree but no computable presentation.
The atomic model theorem and type omitting
 Trans. Amer. Math. Soc
"... We investigate the complexity of several classical model theoretic theorems about prime and atomic models and omitting types. Some are provable in RCA0, others are equivalent to ACA0. One, that every atomic theory has an atomic model, is not provable in RCA0 but is incomparable with WKL0, more than ..."
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We investigate the complexity of several classical model theoretic theorems about prime and atomic models and omitting types. Some are provable in RCA0, others are equivalent to ACA0. One, that every atomic theory has an atomic model, is not provable in RCA0 but is incomparable with WKL0, more than Π1 1 conservative over RCA0 and strictly weaker than all the combinatorial principles of Hirschfeldt and Shore [2007] that are not Π1 1 conservative over RCA0. A priority argument with Shore blocking shows that it is also Π1 1conservative over BΣ2. We also provide a theorem provable by a finite injury priority argument that is conservative over IΣ1 but implies IΣ2 over BΣ2, and a type omitting theorem that is equivalent to the principle that for every X there is a set that is hyperimmune relative to X. Finally, we give a version of the atomic model theorem that is equivalent to the principle that for every X there is a set that is not recursive in X, and is thus in a sense the weakest possible natural principle not true in the ωmodel consisting of the recursive sets.
The atomic model theorem
"... We investigate the complexity of several classical model theoretic theorems about prime and atomic models and omitting types. Some are provable in RCA0, others are equivalent to ACA0. One, that every atomic theory has an atomic model, is not provable in RCA0 but is incomparable with WKL0, more than ..."
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Cited by 3 (1 self)
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We investigate the complexity of several classical model theoretic theorems about prime and atomic models and omitting types. Some are provable in RCA0, others are equivalent to ACA0. One, that every atomic theory has an atomic model, is not provable in RCA0 but is incomparable with WKL0, more than Π1 1 conservative over RCA0 and strictly weaker than all the combinatorial principles of Hirschfeldt and Shore [2007] that are not Π1 1 conservative over RCA0. A priority argument with Shore blocking shows that it is also Π 1 1conservative over BΣ2. We also provide a theorem provable by a finite injury priority argument that is conservative over IΣ1 but implies IΣ2 over BΣ2, and a type omitting theorem that is equivalent to the principle that for every X there is a set that is hyperimmune relative to X. Finally, we give a version of the atomic model theorem that is equivalent to the principle that for every X there is a set that is not recursive in X, and is thus in a sense the weakest possible natural principle not true in the ωmodel consisting of the recursive sets.
Soare, Bounding homogeneous models
"... A Turing degree d is homogeneous bounding if every complete decidable (CD) theory has a ddecidable 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 Arithm ..."
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A Turing degree d is homogeneous bounding if every complete decidable (CD) theory has a ddecidable 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 degree is homogeneous bounding if and only if it is a PA degree. We do this by showing that there is a single CD theory T such that every homogeneous model of T has a PA degree. 1
EFFECTIVELY CATEGORICAL ABELIAN GROUPS
"... We study effective categoricity of computable abelian groups of the form ⊕ i∈ω H, where H is a subgroup of (Q, +). Such groups are called homogeneous completely decomposable. It is wellknown that a homogeneous completely decomposable group is computably categorical if and only if its rank is finit ..."
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We study effective categoricity of computable abelian groups of the form ⊕ i∈ω H, where H is a subgroup of (Q, +). Such groups are called homogeneous completely decomposable. It is wellknown that a homogeneous completely decomposable group is computably categorical if and only if its rank is finite. We study ∆0 ncategoricity in this class of groups, for n> 1. We introduce a new algebraic concept of Sindependence which is a generalization of the wellknown notion of pindependence. We develop the theory of Pindependent sets. We apply these techniques to show that every homogeneous completely decomposable group is ∆0 3categorical. We prove that a homogeneous completely decomposable group of infinite rank is ∆0 2categorical if and only if it is isomorphic to the free module over the localization of Z by a computably enumerable set of primes P with the semilow complement (within the set of all primes). Finally, we apply these results and techniques to study the complexity of generating bases of computable free modules over localizations of integers, including the free abelian group.
ηREPRESENTATION OF SETS AND DEGREES
"... Let A = {a0 < a1 < a2 <...} which does not include 0 or 1 and η the order type of the rationals. When A can be represented by a computable linear order of the ordertype ..."
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Let A = {a0 < a1 < a2 <...} which does not include 0 or 1 and η the order type of the rationals. When A can be represented by a computable linear order of the ordertype
THESIS SUMMARY
"... The main goal of my thesis is the application of logical and computabilitytheoretic techniques to better understand the foundational nature of structures and theorems from different branches of mathematics. To achieve this goal, I examined the effective (i.e. computable) content of structures and t ..."
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The main goal of my thesis is the application of logical and computabilitytheoretic techniques to better understand the foundational nature of structures and theorems from different branches of mathematics. To achieve this goal, I examined the effective (i.e. computable) content of structures and theorems from several different branches of math, including computability theory and model theory (Chapter 1), fractal dimension