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The theory of the metarecursively enumerable degrees
"... Abstract. Sacks [Sa1966a] asks if the metarecursivley enumerable degrees are elementarily equivalent to the r.e. degrees. In unpublished work, Slaman and Shore proved that they are not. This paper provides a simpler proof of that result and characterizes the degree of the theory as O (ω) or, equival ..."
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Abstract. Sacks [Sa1966a] asks if the metarecursivley enumerable degrees are elementarily equivalent to the r.e. degrees. In unpublished work, Slaman and Shore proved that they are not. This paper provides a simpler proof of that result and characterizes the degree of the theory as O (ω) or, equivalently, that of the truth set of L ω CK
The Role of True Finiteness in the Admissible Recursively Enumerable Degrees
 Memoirs of the American Mathematical Society
"... Abstract. When attempting to generalize recursion theory to admissible ordinals, it may seem as if all classical priority constructions can be lifted to any admissible ordinal satisfying a sufficiently strong fragment of the replacement scheme. We show, however, that this is not always the case. In ..."
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Abstract. When attempting to generalize recursion theory to admissible ordinals, it may seem as if all classical priority constructions can be lifted to any admissible ordinal satisfying a sufficiently strong fragment of the replacement scheme. We show, however, that this is not always the case. In fact, there are some constructions which make an essential use of the notion of finiteness which cannot be replaced by the generalized notion of αfiniteness. As examples we discuss both codings of models of arithmetic into the recursively enumerable degrees, and nondistributive lattice embeddings into these degrees. We show that if an admissible ordinal α is effectively close to ω (where this closeness can be measured by size or by cofinality) then such constructions may be performed in the αr.e. degrees, but otherwise they fail. The results of these constructions can be expressed in the firstorder language of partially ordered sets, and so these results also show that there are natural elementary differences between the structures of αr.e. degrees for various classes of admissible ordinals α. Together with coding work which shows that for some α, the theory of the αr.e. degrees is complicated, we get that for every admissible ordinal
Slicing the Truth: On the Computability Theoretic and Reverse Mathematical Analysis of
 Institute for Mathematical Sciences, National University of Singapore, World Scientific
"... In this expository article, we discuss two closely related approaches to studying the relative strength of mathematical principles: computable mathematics and reverse mathematics. Drawing our examples from combinatorics and model theory, we explore a variety of phenomena and techniques in these ar ..."
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In this expository article, we discuss two closely related approaches to studying the relative strength of mathematical principles: computable mathematics and reverse mathematics. Drawing our examples from combinatorics and model theory, we explore a variety of phenomena and techniques in these areas. We begin with variations on König’s Lemma, and give an introduction to reverse mathematics and related parts of computability theory. We then focus on Ramsey’s Theorem as a case study in the computability theoretic and reverse mathematical analysis of combinatorial principles. We study Ramsey’s Theorem for Pairs (RT22) in detail, focusing on fundamental tools such as stability, cohesiveness, and Mathias forcing; and on combinatorial and model theoretic consequences of RT22. We also discuss the important theme of conservativity results. In the final section, we explore several topics that reveal various aspects of computable mathematics and reverse mathematics. An appendix contains a proof of Liu’s recent result that RT22 does not imply Weak König’s Lemma. There are exercises and open questions throughout the article. ∗Please send any corrections to
INDUCTION, BOUNDING, WEAK COMBINATORIAL PRINCIPLES, AND THE HOMOGENEOUS MODEL THEOREM
, 2014
"... Goncharov and Peretyat’kin independently gave necessary and sufficient conditions for when a set of types of a complete theory T is the type spectrum of some homogeneous model of T. Their result can be stated as a principle of second order arithmetic, which we call the Homogeneous Model Theorem (HM ..."
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Goncharov and Peretyat’kin independently gave necessary and sufficient conditions for when a set of types of a complete theory T is the type spectrum of some homogeneous model of T. Their result can be stated as a principle of second order arithmetic, which we call the Homogeneous Model Theorem (HMT), and analyzed from the points of view of computability theory and reverse mathematics. Previous computability theoretic results by Lange suggested a close connection between HMT and the Atomic Model Theorem (AMT), which states that every complete atomic theory has an atomic model. We show that HMT and AMT are indeed equivalent in the sense of reverse mathematics, as well as in a strong computability theoretic sense. We do the same for an analogous result of Peretyat’kin giving necessary and sufficient conditions for when a set of types is the type spectrum of some model.
Nonstandard methods in Ramsey’s theorem for pairs, Computational prospects of infinity
 World Sci. Publ
, 2008
"... Abstract. We discuss the use of nonstandard methods in the study of Ramsey type problems, and illustrate this with an example concerning the existence of definable solutions in models of BΣ02 for the combinatorial principles of Ramsey’s Theorem for pairs and cohesiveness. 1. ..."
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Abstract. We discuss the use of nonstandard methods in the study of Ramsey type problems, and illustrate this with an example concerning the existence of definable solutions in models of BΣ02 for the combinatorial principles of Ramsey’s Theorem for pairs and cohesiveness. 1.
Transfinite Machine Models
, 2011
"... In recent years there has emerged the study of discrete computational models which are allowed to act transfinitely. By ‘discrete ’ we mean that the machine models considered are not analogue machines, but compute by means of distinct stages or in units of time. The paradigm of such models is, of co ..."
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In recent years there has emerged the study of discrete computational models which are allowed to act transfinitely. By ‘discrete ’ we mean that the machine models considered are not analogue machines, but compute by means of distinct stages or in units of time. The paradigm of such models is, of course, Turing’s original
Discrete Transfinite Computation
"... We describe various computational models based initially, but not exclusively, on that of the Turing machine, that are generalized to allow for transfinitely many computational steps. Variants of such machines are considered that have longer tapes than the standard model, or that work on ordinals r ..."
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We describe various computational models based initially, but not exclusively, on that of the Turing machine, that are generalized to allow for transfinitely many computational steps. Variants of such machines are considered that have longer tapes than the standard model, or that work on ordinals rather than numbers. We outline the connections between such models and the older theories of recursion in higher types, generalized recursion theory, and recursion on ordinals such as αrecursion. We conclude that, in particular, polynomial time computation on ωstrings is well modelled by several convergent conceptions. 1