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Model Theory Pascal Vanier Saturday 26th March, 2016 09:22 2 Sources The basis of this course are the model theory courses of Université Claude Bernard Lyon 1 and in particular the lecture notes of Emmanuel Jeandel for the course of Frank Wagner and Bruno Poizat of which this document is mainly an english translation. We also used the lecture notes of the course of Tuna Altinel and Julien Melleray and the proof of the compactness theorem of the course A crèche course on model theory of Domenico Zambella (Università di Torino). Contents I Model theory 5 1 Languages and structures 7 1.1 Formulae . .7 1.2 Structures . .8 1.3 Theories and models . 10 1.4 Diagrams . 11 1.5 Exercises . 12 2 Equivalence between models 13 2.1 Elementary equivalence . 13 2.2 Substructures . 14 2.3 Morphisms . 15 2.4 Diagrams . 17 2.5 Tarski’s test . 17 2.6 Skolemisation . 18 2.7 Definability . 20 2.8 Exercises . 20 3 Compactness theorem 21 3.1 Compactness of quantifier free sets . 21 3.2 Upward Löwenheim-Skolem . 23 3.3 Axiomatisation . 24 3.4 κ-categoricity . 25 3.5 Exercises . 25 4 Back and forth 27 4.1 1-isomorphisms . 27 4.2 Isomorphic structures . 29 4.3 Elementary equivalence . 30 4.4 !-saturated models . 30 4.5 Applications to completeness . 33 4.5.1 Dense orders without extremities . 33 4.6 Discrete orders without extremities . 34 4.7 Exercises . 35 3 4 CONTENTS Part I Model theory 5 Chapter 1 Languages and structures 1.1 Formulae Definition 1.1.1 Language A language L is a set of : • constant symbols (ci), • function symbols (fi), • relation symbols (Ri). We will always assume that equality is among the relation symbols. Example 1.1.2 −1 • The language of groups Lgroups = f1; ; ·} in the multiplicative notation and Lgroups = f0; +; −}. −1 • The language of rings Lfield = f0; 1; +; −; ·; g. At this stage, a language does not give any meaning to the symbols, it should only be considered as an alphabet with which sentences may be written. These sentences will be called terms: Definition 1.1.3 Term Let L be a language, the set of L-terms is the smallest set T such that: • any ci is in T , • T contains some set of symbols V disjoint from the (ci) called variables. • T contains all fi(a1; : : : ; ani ) with ai 2 T . A term is just the composition of function symbols with constants and some variables of the set V . With this we are now able to define what a formula is. Formulae on L correspond to first order logic on the language L: 7 8 CHAPTER 1. LANGUAGES AND STRUCTURES Definition 1.1.4 Formula Let L be a language and T be the set of L-terms. The set of L-formulae F is the smallest set such that: • F contains all atomic formulae: the Ri(t1; : : : ; tn) with tj 2 T and Ri a relation in L. • F is closed by boolean combinations: if φ, are formulae of F , then so are φ _ , φ ^ and :φ. • F is closed by existential and universal quantifiers : for any vari- able v 2 V , and formula φ 2 F , 8vφ and 9vφ are formulae in F . As the attentive reader may have noted, formulae may contain variable of V not "caught" by any quantifier, such variables are said to be free variables, and this notion may be defined more formally: Definition 1.1.5 Free variable A free variable for a formula is defined inductively in the following way: • The free variables of an atomic formula are the variables v 2 V appearing in this formula. • If φ = _ ρ or φ = ^ ρ, the free variables of φ are the variables that are free in either or ρ. • If φ = : , the free variables of φ are the free variables of . • If φ = 9v or φ = 8v the free variables of phi are the free variables of minus v. In the rest of the course, we will note φ(x) 2 F for a formula whose free variables are x = (x1; : : : ; xn).A sentence is a formula without free variables. 1.2 Structures For the moment, we only defined languages and formulae without giving them any meaning or semantics. The aim of model theory is to study the structures in which some set of sentences are true : 1.2. STRUCTURES 9 Definition 1.2.1 Structure Let L be a language, an L-structure M is formed of a base set M, also called the universe or domain of M and of: M • a set of constants (ci ) of M associated to each of the symbols ci of L, M ni • a set of functions (fi : M ! M) associated to each of the symbols fi of L, M mi • a set of relations (Ri ⊆ M ) associated to each of the symbols Ri of L. This set of relations always associates the equality symbol to the relation f(x; x) j x 2 Mg. Now that we have associated a meaning to each of the symbols in our lan- guage, we can define the meaning of a term in a structure very naturally: Definition 1.2.2 Let L be a language, M be an L-structure and m elements of M, the universe of M. The interpretation of a term t(x) with parameter m may be defined in an inductive way: • The interpretation of a constant c is cM. • The interpretation of x 2 V is the corresponding element in m. M M M M • The interpretation of f(t1; : : : ; tn) is f (t1 ; : : : ; tn ) with ti the interpretation of ti in M. This in turn allows us to define what the satifiability of a formula is: Definition 1.2.3 Satisfiability of a formula Let L be a language and M be an L-structure. If m is a vector of elements of M, the satisfaction of a formula φ(x) with parameter m is defined inductively by : • φ(m) is an atomic formula R(t1; : : : ; tn), φ(m) is satisfied if and M M M only if (t1 ; : : : ; tn ) 2 R where ti is the interpretation of ti with parameter m. • φ(m) = (m) ^ ρ(m) is satisfied iff (m) and ρ(m) are both satisfied. • φ(m) = (m) _ ρ(m) is satisfied iff at least one of (m) or ρ(m) is satisfied. • φ = :ρ is satisfied iff ρ is not satisfied. • φ(m) = 9y (m; y) is satisfied with parameter m iff there exists y 2 M such that is satisfied with the parameter m; y. • φ(m) = 8y (m; y) is satisfied iff for all y 2 M, (m; y) is satis- fied. When φ(x) is satisfied in M with parameter m, we use the notation M j= φ(m), which means "the structure M satisfies φ(m). 10 CHAPTER 1. LANGUAGES AND STRUCTURES Definition 1.2.4 Equivalent formulae Two formulae φ(x) and (x) are equivalent iff for any L-structure M and parameter m 2 M, M j= φ(m) iff M j= (m). Exercise 1.2.1. Prove that the following formulae are equivalent: • φ and ::φ • :(φ ^ ) and (:φ _: ) • :(φ _ ) and (:φ ^ : ) • :9xφ and 8x:φ • :8xφ and 9x:φ The usual notation φ ) stands for :φ _ and φ , stands for (φ ) ) ^ ( ) φ). 1.3 Theories and models We may now get to the core of this course: what is a theory and a model for this theory: Definition 1.3.1 Theory Let L be a language, T a set of sentences on L. T is a theory if and only if there exists an L-structure M such that 8φ 2 T; M j= φ M is then called a model of theory T . In order for a theory to have a model, it needs to be coherent, in the sense that there exists a structure satisfying all of its sentences. For instance a theory containing :φ and φ at the same time cannot be coherent, as they cannot be simultaneously satisfied by definition. Definition 1.3.2 Consistent theory A theory T is consistent if it has a model. 1.4. DIAGRAMS 11 Example 1.3.3 We may get back to example 1.1.2, where we defined the language of groups, we may now give a meaning to this language with the theory corresponding to groups: • 8x; x · 1 = x • 8x; x · x−1 = 1 • 8x; x · x−1 · x = 1 • 8x; y; z; (x · y) · z = x · (y · z) A theory that will be used sometimes as an example in this course is the theory of orders: Example 1.3.4 The language of the theory of orders contains only one relation and is the following: • 8x; x < y • 8x; y; (x < y) ):(y < x) • 8x; y; z; (x < y) ^ (y < z) ) (x < z) If we want a theory of total orders, we can add the sentence 8x; y; (x < y) _ (y < x) _ (x = y). We will note T j= φ if and only if for any model M of T , M j= φ, that is to say if the truth of φ depends only on the theory and not on the model. Property 1.3.1. If φ 2 T , then T j= φ. Warning! T 6j= φ is not the same as T j= :φ: in the first case, there exists a model of T that does not satisfy φ (i.e.it satisfies :φ), and in the second case every model of T satisfies :φ.
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