Results 1 
4 of
4
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 Ddec ..."
Abstract

Cited by 7 (4 self)
 Add to MetaCart
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.
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 area ..."
Abstract

Cited by 3 (1 self)
 Add to MetaCart
(Show Context)
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.
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 ..."
Abstract

Cited by 2 (2 self)
 Add to MetaCart
(Show Context)
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.