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Effective model theory: the number of models and their complexity
 MODELS AND COMPUTABILITY
, 1999
"... Effective model theory studies model theoretic notions with an eye towards issues of computability and effectiveness. We consider two possible starting points. If the basic objects are taken to be theories, then the appropriate effective version investigates decidable theories (the set of theorems i ..."
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Cited by 19 (6 self)
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Effective model theory studies model theoretic notions with an eye towards issues of computability and effectiveness. We consider two possible starting points. If the basic objects are taken to be theories, then the appropriate effective version investigates decidable theories (the set of theorems is computable) and decidable structures (ones with decidable theories). If the objects of initial interest are typical mathematical structures, then the starting point is computable structures. We present an introduction to both of these aspects of effective model theory organized roughly around the themes of the number and types of models of theories with particular attention to categoricity (as either a hypothesis or a conclusion) and the analysis of various computability issues in families of models.
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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Cited by 18 (4 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 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.
Degree spectra of prime models
 J. Symbolic Logic
, 2004
"... 2.1 Notation from model theory................... 4 2.2 F ..."
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Cited by 10 (2 self)
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2.1 Notation from model theory................... 4 2.2 F
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.
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 ..."
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Cited by 3 (1 self)
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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.
Open Problems in the Theory of Constructive Algebraic Systems
, 2000
"... In this paper we concentrate on open problems in two directions in the development of the theory of constructive algebraic systems. The first direction deals with universal algebras whose positive open diagrams can be computably enumerated. These algebras are called positive algebras. Here we em ..."
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Cited by 1 (0 self)
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In this paper we concentrate on open problems in two directions in the development of the theory of constructive algebraic systems. The first direction deals with universal algebras whose positive open diagrams can be computably enumerated. These algebras are called positive algebras. Here we emphasize the interplay between universal algebra and computability theory. We propose a systematic study of positive algebras as a new direction in the development of the theory of constructive algebraic systems. The second direction concerns the traditional topics in constructive model theory. First we propose the study of constructive models of theories with few models such as countably categorical theories, uncountably categorical theories, and Ehrenfeucht theories. Next, we propose the study of computable isomorphisms and computable dimensions of such models. We also discuss issues related to the computabilitytheoretic complexity of relations in constructive algebraic systems.
Recursive Models of Theories with Few Models
"... We begin by presenting some basic definitions from effective model theory. A recursive structure is one with a recursive domain and uniformly recursive atomic relations. Without lost of generality, we can always suppose that the domain of every recursive structure is the set of all naturall numbers ..."
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We begin by presenting some basic definitions from effective model theory. A recursive structure is one with a recursive domain and uniformly recursive atomic relations. Without lost of generality, we can always suppose that the domain of every recursive structure is the set of all naturall numbers ω and
Computable Models Spectra of Ehrenfeucht Theories
"... This study is devoted to the class of Ehrenfeucht theories, this class of theories has been well studied and has attracted considerable attention. ..."
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This study is devoted to the class of Ehrenfeucht theories, this class of theories has been well studied and has attracted considerable attention.