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Topics in the Theory of Recursive Functions SETS, MODELS, AND PROOFS: TOPICS IN THE THEORY OF RECURSIVE FUNCTIONS A Dissertation Presented to the Faculty of the Graduate School of Cornell University in Partial Fulfillment of the Requirements for the Degree of Doctor of Philosophy by David R. Belanger August 2015 This document is in the public domain. SETS, MODELS, AND PROOFS: TOPICS IN THE THEORY OF RECURSIVE FUNCTIONS David R. Belanger, Ph.D. Cornell University We prove results in various areas of recursion theory. First, in joint work with Richard Shore, we prove a new jump-inversion result for ideals of recursively enu- merable (r.e.) degrees; this defeats what had seemed to be a promising tack on the automorphism problem for the semilattice R of r.e. degrees. Second, in work spanning two chapters, we calibrate the reverse-mathematical strength of a number of theorems of basic model theory, such as the Ryll-Nardzewski atomic-model theorem, Vaught's no-two-model theorem, Ehrenfeucht's three-model theorem, and the existence theorems for homogeneous and saturated models. Whereas most of these are equivalent over RCA0 to one of RCA0, WKL0, ACA0, as usual, we also uncover model-theoretic statements with exotic complexities such as :WKL0 _ 0 ACA0 and WKL0 _ IΣ2. Third, we examine the possible weak truth table (wtt) degree spectra of count- able first-order structures. We find several points at which the wtt- and Turing- degree cases differ, notably that the most direct wtt analogue of Knight's di- chotomy theorem does not hold. Yet we find weaker analogies between the two, including a new trichotomy theorem for wtt degree spectra in the spirit of Knight's. BIOGRAPHICAL SKETCH David Belanger was born and raised in Belleville, Ontario, Canada. He attended the University of Waterloo, where he earned Bachelor's and Master's degrees in mathematics. During his Master's studies he concentrated on the area of math- ematical logic, and in 2009 he moved to Cornell University for a Ph.D. in the same. iii ACKNOWLEDGEMENTS First on the list is my adviser, Richard Shore, who has shown great patience and provided sound guidance, for what must add up to hundreds of hours, in the preparation in this thesis and of its constituent articles|and who actually wrote a sizable part of Chapter 1. I am grateful to Bob Constable, who has shared plenty of his time and knowledge with me, to Barbara Csima, who first introduced me to logic at a research level, and to all the other professors and experts who have done similar favors for me and for my fellow students. I am equally thankful to those students, my colleagues, and to those writers of letters, postcards, and electronic communications who have helped me to live and work through the snowed-in, numb-fingered Ithaca winters. Last and most important are family, near and far, but never estranged; spanning provinces, countries, and now continents; among whom I am most at home, and without whom I could never get very far. iv TABLE OF CONTENTS 0 Global Introduction 1 0.1 Recursive sets and Turing degrees . .2 0.2 The structures of D and R .......................4 0.3 Recursive model theory . .5 0.4 Recursive sets and the arithmetical hierarchy . .7 0.5 Reverse mathematics . .8 1 Recursively enumerable sets and the question of rigidity 11 1.1 Introduction . 11 1.2 Proving part (1) of the theorem . 17 1.2.1 Requirements . 18 1.2.2 Notation and bookkeeping . 20 1.2.3 The basic strategies and outcomes; defining δs ........ 22 1.2.4 Verification . 24 1.3 Proving part (2) of the Theorem . 30 1.4 From the Conjecture to Rigidity . 31 1.5 Questions and Observations . 33 2 Reverse mathematics of model theory 38 2.1 Introduction . 38 2.1.1 Notation for reverse mathematics . 40 2.1.2 Background and notation for model theory . 42 2.2 Main results . 46 2.2.1 Reverse mathematics and @0-categorical theories . 46 2.2.2 Reverse mathematics and theories with finitely many models 50 2.3 Coding an extendable binary tree as a theory . 53 2.3.1 Construction . 54 2.3.2 Verification . 54 2.3.3 Applications . 57 2.4 A theory with infinitely many 1-types, whose every nonprincipal type computes KZ ........................... 59 2.4.1 Construction . 59 2.4.2 Verification . 60 2.4.3 Applications . 64 2.5 Models from a tree of Henkin constructions . 66 2.5.1 Construction . 67 2.5.2 Verification . 68 2.5.3 Applications . 69 2.6 Theories with only finitely many n-types for every n ......... 71 2.6.1 Construction . 72 2.6.2 Verification . 73 v 2.6.3 Applications . 80 2.7 Theories with finitely many models . 83 2.7.1 Construction . 85 2.7.2 Verification . 86 2.7.3 Application . 88 3 Further results in the reverse mathematics of model theory 89 3.1 Introduction . 89 3.1.1 Conventions and organization . 90 3.1.2 Formalizing model theory . 91 0 0 3.1.3 The basics of WKL0, ACA0, IΣ2, and BΣ2 ........... 93 3.2 Main Results . 98 3.2.1 Existence theorems for homogeneous models . 98 3.2.2 Existence theorems for saturated models . 100 3.2.3 Type amalgamation, WKL0, and induction . 105 3.2.4 Elementary embeddings and universal models . 107 3.2.5 Indiscernibles . 109 3.3 Models and embeddings from a tree of Henkin constructions . 110 3.3.1 Applications . 112 3.4 A Controlled failure of compactness . 120 3.4.1 Construction . 121 3.4.2 Verification . 122 3.4.3 Applications . 126 3.5 1-Homogeneity vs strong 1-homogeneity . 127 3.5.1 Construction . 128 3.5.2 Verification . 129 3.5.3 Application . 133 3.6 A theory with the finite free amalgamation property, but without the 1-point full amalgamation property . 133 3.6.1 Construction . 134 3.6.2 Verification . 135 3.6.3 Applications . 136 0 3.7 The case with neither WKL0 nor Σ2 induction . 136 3.7.1 Construction . 137 3.7.2 Verification . 138 3.7.3 The first application . 142 3.7.4 The second application . 144 4 Weak truth table degrees of models 147 4.1 Introduction . 147 4.1.1 Background and overview . 148 4.2 Knight's dichotomy for Turing degree spectra . 153 4.3 A Dichotomy for the wtt degree spectrum . 157 4.4 Proof of Theorem 4.3.6 . 161 vi 4.5 Structures with finite signature . 170 4.6 Some specific examples . 175 Bibliography 183 vii LIST OF TABLES 2.1 Implications for Theorems 2.2.2 through 2.2.4. 47 3.1 Implications for Theorem 3.2.24. 107 viii CHAPTER 0 GLOBAL INTRODUCTION In the course of developing a mathematical theory, it is natural to consider the complexity of the examples and counterexamples that are supposed to be produced. The precise definition of simplicity and complexity may vary. Some well-known notions of a simple object are: an equation that can be solved by radicals; a length that can be constructed using a compass and straightedge; a subset of R that is Borel; an optimization problem that can be solved by a computer program in polynomial time; and a subset of N that can be computed by some effective procedure. These are inequivalent in general, and each is a reasonable definition in its proper context. Recursion theory is the area of mathematical logic that studies the last of these|the effective computability of sets|and related questions of effective enumerability and relative computability. Following the publication of Turing's 1936 article [70], the community has gen- erally come to agree that an effective procedure is one which is computable by one of his abstract Turing machines. The idea of an effective procedure, however, far predates any such consensus. In 1900, for instance, Hilbert asked as the tenth of his famous Problems for a procedure to determine whether a given multivariate polynomial has integer roots. Such a procedure, if found, would constitute a pos- itive solution with no need for a formal definition. The problem was eventually solved in the negative, however, when it was shown through the combined efforts of Matiyasevich, Robinson, Davis, and Putnam that no suitable Turing machine exists. With Turing's definition as a starting point, mathematicians have made rapid progress on relatively concrete decision problems|obtaining negative solutions along the lines of Hilbert's tenth problem to the word problem for groups, to the 1 problem of logical entailment, and to many more besides|but also on computabil- ity in the abstract, such as the theorem of Kleene and Post that there exists a pair of subsets of N neither of which can be used to compute the other. Recursion theory today focuses more on the latter type of question, but the strength of this preference varies among the numerous subfields. In what follows we will consider the semilattice of recursively enumerable degrees (a topic of pure recursion the- ory), recursive model theory (where recursion theory meets classical model theory), and reverse mathematics (where recursion theory meets proof theory, and where classical decision problems frequently reappear). This dissertation collects all of the mathematical research that the author sub- mitted for publication during the course of his doctoral studies. Not counting the present introductory Chapter 0, there are four chapters: each comprising, more or less unchanged, one journal article. The corresponding citations are provided at the beginning of each chapter. With the exception of Chapter 1, which was written in collaboration with Richard Shore, all of what follows is my work alone.
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