Classical Metalogic An Introduction to Classical Model Theory & Computability Notes by R.J. Buehler January 5, 2015 2 Preface What follows are my personal notes created in preparation for the UC-Berkeley Group in Logic preliminary exam. I am not a computability theorist, nor a model theorist; I am a graduate student with some knowledge who is–alas–quite fallible. Accordingly, this text is made available as a convenient reference, set of notes, and summary, but without even the slight hint of a guarantee that everything contained within is factual and correct (indeed, some areas are entirely unchanged from the moment I copied them off the blackboard). This said, if you find a serious error, I would greatly appreciate it if you would let me know so that it can be corrected. The material for these notes derives from a wide variety of sources: Lectures by Wes Holliday Lectures by Antonio Montalban Lectures by John Steel Kevin Kelly’s computability theory notes David Marker’s “Model Theory: An Introduction” Wilfrid Hodge’s “A Shorter Model Theory” Robert Soare’s “Recursively Enumerable Sets and Degrees" Richard Kaye’s “Models of Peano Arithmetic” Chang and Keisler’s “Model Theory" While I certainly hope my notes are beneficial, if you’re attempting to learn the contained material for the first time, I would highly suggest picking up (at least) a copy of Marker and Soare’s texts in addition. To those Group in Logic students who may be using these notes to help themselves prepare for their preliminary exam, chapters 1-5, 7, and 9-18 contain relevant material, as well as chapter 8, section 3. i ii PREFACE Contents Preface i 1 First-Order Logic 1 1.1 Signatures.................................................1 1.2 Languages.................................................1 1.2.1 Terms...............................................1 1.2.2 Atomic Formulas and Sentences.................................2 1.2.3 Formal Languages........................................2 1.2.4 Standard Abbreviations......................................4 1.3 Normal Forms...............................................4 I Classical Model Theory5 2 The Basics of Models 7 2.1 Models..................................................7 2.2 Models and Formal Languages......................................8 2.2.1 Model-Theoretic Logical Consequence.............................9 2.3 Theories.................................................. 10 2.3.1 Theories and Diagrams of Models................................ 10 2.3.2 Canonical Models........................................ 11 3 Relations Between Models 13 3.1 From Homomorphism to Isomorphism.................................. 13 3.2 Submodels................................................. 14 3.3 Preserving Formulas........................................... 15 3.3.1 8 and 9 Preservation....................................... 16 3.3.2 Elementary Preservation..................................... 18 3.4 Chains of Models............................................. 18 3.4.1 Elementary Chains of Models.................................. 19 3.5 Reduction and Expansion......................................... 19 4 Compactness and Strong Completeness 21 4.1 Hintikka Sets............................................... 21 4.2 Compactness for First-Order Logic.................................... 22 4.3 Strong Soundness and Completeness for First-Order Logic....................... 23 5 Definability 25 5.1 Definability of Relations......................................... 25 5.2 Definable Classes of Models....................................... 26 5.3 The Skolem Theorems.......................................... 26 5.3.1 Skolemization.......................................... 27 5.3.2 The Downward Löwenheim-Skolem Theorem.......................... 27 iii iv CONTENTS 5.3.3 The Upward Löwenheim-Skolem Theorem........................... 28 6 Amalgamation 31 6.1 Elementary Amalgamation Property................................... 31 6.2 Strong Elementary Amalgamation Property............................... 32 6.3 The Strong Amalgam Property and Algebraicity............................. 32 6.4 Interpolation and Definability....................................... 34 7 Types, Saturation, and Categoricity 37 7.1 Types................................................... 37 7.1.1 Tuples with the Same Type.................................... 37 7.1.2 Principal and Supported Types.................................. 38 7.1.3 Towards Saturation........................................ 38 7.2 Omitting Types.............................................. 40 7.3 Atomic and Prime Models........................................ 41 7.4 Saturation................................................. 43 7.4.1 Homogeneity........................................... 44 7.4.2 Universality............................................ 45 7.5 Categoricity................................................ 46 8 Other Model-Theoretic Constructions 49 8.1 Fraïssé’s Construction........................................... 49 8.1.1 Constructing the Fraïssé Limit.................................. 51 8.1.2 Countable Categoricity and Fraïssé Limits............................ 52 8.2 Ultraproducts............................................... 55 8.2.1 Ultrafilters............................................ 55 8.3 Ehrenfeucht-Mostowski Models..................................... 57 8.3.1 Indiscernible Sequences..................................... 57 8.3.2 E-M Models........................................... 58 II Computability Theory 61 9 Of Algorithms 63 9.1 Turing Machines............................................. 63 9.1.1 The Church-Turing Thesis.................................... 64 9.2 Functions and Recursion......................................... 64 10 The Primitive and µ-Recursive Functions 65 10.1 The Primitive Recursive Functions.................................... 65 10.1.1 Primitive Recursive Relations.................................. 65 10.2 Primitive Recursive Operations...................................... 66 10.2.1 Piecewise Definition....................................... 67 10.2.2 Bounded Search......................................... 67 10.2.3 Coding Sequences........................................ 67 10.2.4 Simultaneous and Course of Values Recursion......................... 68 10.3 The Computable Functions?....................................... 69 10.4 Unbounded Search & µ-Recursion.................................... 70 10.5 The Computable Functions........................................ 71 10.5.1 Computable Relations...................................... 71 CONTENTS v 11 The Fundamental Theorems of Computability Theory 73 11.1 The Padding Lemma........................................... 73 11.2 The Normal Form Theorem........................................ 73 11.3 The Enumeration Theorem........................................ 74 m 11.4 The sn Theorem.............................................. 75 11.5 The Halting Problem........................................... 76 11.6 Kleene’s Fixed Point Theorem...................................... 76 12 Non-Computable Relations 81 12.1 A Three-Fold Distinction......................................... 82 12.1.1 Closure Properties and Connections............................... 84 12.2 Computably Enumerable Relations.................................... 85 12.3 Reduction and Computable Separability................................. 87 12.4 Many-One Reducibility.......................................... 88 12.4.1 The m-Degrees.......................................... 89 0 0 12.4.2 Characterizing the Σ1/Π1-Complete Sets............................. 89 13 Relative Computation and Turing Reducibility 95 13.1 Higher Complexity Relations....................................... 95 13.1.1 Closure Properties & Basic Results............................... 96 13.1.2 Canonical Problems....................................... 97 13.1.3 Trees and Tree Arguments.................................... 98 13.2 Relative Computability.......................................... 99 13.2.1 The Relativized Theorems.................................... 100 13.2.2 The Relative Arithmetical Hierarchy............................... 101 13.3 Turing Reducibility............................................ 102 13.3.1 The Turing Jump Operator.................................... 103 14 Post’s Problem & Oracle Constructions 107 14.0.2 Simple Relations......................................... 107 14.1 Incomparable Turing Degrees....................................... 108 14.2 Friedburg Completeness......................................... 110 14.2.1 Minimal Degrees......................................... 111 14.2.2 Constructing c.e. sets....................................... 112 III Computability and First-Order Logic 115 15 Languages, Theories, and Computability 117 15.1 Languages, Theories, and Axioms.................................... 117 15.2 Decidability and Undecidability..................................... 118 15.3 Proving Decidability........................................... 120 15.3.1 Quantifier Elimination...................................... 120 15.3.2 Model Theory........................................... 121 15.3.3 Interpretation........................................... 122 15.3.4 The Decidability of Validity................................... 123 15.4 Proving Undecidability.......................................... 123 16 Arithmetic & Incompleteness 125 16.1 True Arithmetic.............................................