Model Theory

Total Page:16

File Type:pdf, Size:1020Kb

Model Theory 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.
Recommended publications
  • ⊥N AS an ABSTRACT ELEMENTARY CLASS We Show
    ⊥N AS AN ABSTRACT ELEMENTARY CLASS JOHN T. BALDWIN, PAUL C. EKLOF, AND JAN TRLIFAJ We show that the concept of an Abstract Elementary Class (AEC) provides a unifying notion for several properties of classes of modules and discuss the stability class of these AEC. An abstract elementary class consists of a class of models K and a strengthening of the notion of submodel ≺K such that (K, ≺K) satisfies ⊥ the properties described below. Here we deal with various classes ( N, ≺N ); the precise definition of the class of modules ⊥N is given below. A key innovation is ⊥ that A≺N B means A ⊆ B and B/A ∈ N. We define in the text the main notions used here; important background defini- tions and proofs from the theory of modules can be found in [EM02] and [GT06]; concepts of AEC are due to Shelah (e.g. [She87]) but collected in [Bal]. The surprising fact is that some of the basic model theoretic properties of the ⊥ ⊥ class ( N, ≺N ) translate directly to algebraic properties of the class N and the module N over the ring R that have previously been studied in quite a different context (approximation theory of modules, infinite dimensional tilting theory etc.). Our main results, stated with increasingly strong conditions on the ring R, are: ⊥ Theorem 0.1. (1) Let R be any ring and N an R–module. If ( N, ≺N ) is an AEC then N is a cotorsion module. Conversely, if N is pure–injective, ⊥ then the class ( N, ≺N ) is an AEC. (2) Let R be a right noetherian ring and N be an R–module of injective di- ⊥ ⊥ mension ≤ 1.
    [Show full text]
  • Scheme Representation for First-Logic
    Scheme representation for first-order logic Spencer Breiner Carnegie Mellon University Advisor: Steve Awodey Committee: Jeremy Avigad Hans Halvorson Pieter Hofstra February 12, 2014 arXiv:1402.2600v1 [math.LO] 11 Feb 2014 Contents 1 Logical spectra 11 1.1 The spectrum M0 .......................... 11 1.2 Sheaves on M0 ............................ 20 1.3 Thegenericmodel .......................... 24 1.4 The spectral groupoid M = Spec(T)................ 27 1.5 Stability, compactness and definability . 31 1.6 Classicalfirst-orderlogic. 37 2 Pretopos Logic 43 2.1 Coherentlogicandpretoposes. 43 2.2 Factorization in Ptop ........................ 51 2.3 Semantics, slices and localization . 57 2.4 Themethodofdiagrams. .. .. .. .. .. .. .. .. .. 61 2.5 ClassicaltheoriesandBooleanpretoposes . 70 3 Logical Schemes 74 3.1 StacksandSheaves.......................... 74 3.2 Affineschemes ............................ 80 3.3 Thecategoryoflogicalschemes . 87 3.4 Opensubschemesandgluing . 93 3.5 Limitsofschemes........................... 98 1 4 Applications 110 4.1 OT asasite..............................111 4.2 Structuresheafasuniverse . 119 4.3 Isotropy ................................125 4.4 ConceptualCompleteness . 134 2 Introduction Although contemporary model theory has been called “algebraic geometry mi- nus fields” [15], the formal methods of the two fields are radically different. This dissertation aims to shrink that gap by presenting a theory of “logical schemes,” geometric entities which relate to first-order logical theories in much the same way that
    [Show full text]
  • Continuous First-Order Logic
    Continuous First-Order Logic Wesley Calvert Calcutta Logic Circle 4 September 2011 Wesley Calvert (SIU / IMSc) Continuous First-Order Logic 4 September 2011 1 / 45 Definition A first-order theory T is said to be stable iff there are less than the maximum possible number of types over T , up to equivalence. Theorem (Shelah's Main Gap Theorem) If T is a first-order theory and is stable and . , then the class of models looks like . Otherwise, there's no hope. Model-Theoretic Beginnings Problem Given a theory T , describe the structure of models of T . Wesley Calvert (SIU / IMSc) Continuous First-Order Logic 4 September 2011 2 / 45 Theorem (Shelah's Main Gap Theorem) If T is a first-order theory and is stable and . , then the class of models looks like . Otherwise, there's no hope. Model-Theoretic Beginnings Problem Given a theory T , describe the structure of models of T . Definition A first-order theory T is said to be stable iff there are less than the maximum possible number of types over T , up to equivalence. Wesley Calvert (SIU / IMSc) Continuous First-Order Logic 4 September 2011 2 / 45 Model-Theoretic Beginnings Problem Given a theory T , describe the structure of models of T . Definition A first-order theory T is said to be stable iff there are less than the maximum possible number of types over T , up to equivalence. Theorem (Shelah's Main Gap Theorem) If T is a first-order theory and is stable and . , then the class of models looks like . Otherwise, there's no hope.
    [Show full text]
  • Proof Diagrams for Multiplicative Linear Logic
    Proof Diagrams for Multiplicative Linear Logic Matteo Acclavio I2M Marseille, France Aix-Marseile Universit´e [email protected] The original idea of proof nets can be formulated by means of interaction nets syntax. Additional machinery as switching, jumps and graph connectivity is needed in order to ensure correspondence between a proof structure and a correct proof in sequent calculus. In this paper we give an interpretationof proof nets in the syntax of string diagrams. Even though we lose standard proof equivalence, our construction allows to define a framework where soundness and well-typeness of a diagram can be verified in linear time. Introduction Proof nets are a geometrical representation of linear logic proofs introduced by J-Y.Girard [5]. The build- ing blocks of proof nets are called proof structures that have been generalized by Y. Lafont [11] in the so-called interaction nets. To recognize if a proof structure is a proof net one needs to verify its sequen- tializability property, that is, whether it corresponds to a linear logic proof derivation. Following Girard’s original correction criterion, others methods have been introduced, notably by Danos-Regnier [4], that ensures graph acyclicity by a notion of switchings on ⊗ cells, and by Guerrini [7], that reformulates correction by means of graph contractability. Proof structures allow to recover the semantic equivalence of derivation under commutation and permutation of some inference rules. Unfortunately this property makes ineffective the aforementioned criteria in presence of the multiplicative unit ⊥. In order to recover a sequentialization condition for the multiplicative fragment with units, Girard has introduced the notion of jumps [6].
    [Show full text]
  • Ultraproducts and Elementary Classes *
    MATHEMATICS ULTRAPRODUCTS AND ELEMENTARY CLASSES * BY H. JEROME KEISLER (Communicated by Prof. A. HEYTING at the meeting of May 27, 1961 INTRODUCTION In the theory of models of the first order predicate logic with identity we study the relationship between notions from logic and purely mathe­ matical objects, called relational systems, or (first order) structures, m, consisting of a set A and a sequence of relations on A corresponding to the predicate symbols of the logic. In such studies some of the most useful methods are those which permit us to form structures which have certain semantical properties. For example, the completeness theorem permits us to form a structure which satisfies a given consistent set of sentences. A more recent tool is the reduced product operation, which is particularly simple and direct from the mathematical point of view. Speaking very roughly, a reduced product Iliei ill:i/D of the structures mi, i El, is formed by first taking their direct product and then forming the quotient system determined in a certain way by the filter D on the index set 1. If D is an ultrafilter we have an ultraproduct (or prime reduced product), and if each m:i coincides with a fixed system ill: we have a reduced power, or ultrapower, ill: I j D. Ultraproducts have recently been applied to obtain new, comparatively direct mathematical proofs and at the same time stronger forms of a number of theorems in model theory. An outstanding example is the proof of the compactness theorem outlined in [33]. They have also been used to obtain several completely new results in the theory of models (for example, see [35]).
    [Show full text]
  • Notes on Mathematical Logic David W. Kueker
    Notes on Mathematical Logic David W. Kueker University of Maryland, College Park E-mail address: [email protected] URL: http://www-users.math.umd.edu/~dwk/ Contents Chapter 0. Introduction: What Is Logic? 1 Part 1. Elementary Logic 5 Chapter 1. Sentential Logic 7 0. Introduction 7 1. Sentences of Sentential Logic 8 2. Truth Assignments 11 3. Logical Consequence 13 4. Compactness 17 5. Formal Deductions 19 6. Exercises 20 20 Chapter 2. First-Order Logic 23 0. Introduction 23 1. Formulas of First Order Logic 24 2. Structures for First Order Logic 28 3. Logical Consequence and Validity 33 4. Formal Deductions 37 5. Theories and Their Models 42 6. Exercises 46 46 Chapter 3. The Completeness Theorem 49 0. Introduction 49 1. Henkin Sets and Their Models 49 2. Constructing Henkin Sets 52 3. Consequences of the Completeness Theorem 54 4. Completeness Categoricity, Quantifier Elimination 57 5. Exercises 58 58 Part 2. Model Theory 59 Chapter 4. Some Methods in Model Theory 61 0. Introduction 61 1. Realizing and Omitting Types 61 2. Elementary Extensions and Chains 66 3. The Back-and-Forth Method 69 i ii CONTENTS 4. Exercises 71 71 Chapter 5. Countable Models of Complete Theories 73 0. Introduction 73 1. Prime Models 73 2. Universal and Saturated Models 75 3. Theories with Just Finitely Many Countable Models 77 4. Exercises 79 79 Chapter 6. Further Topics in Model Theory 81 0. Introduction 81 1. Interpolation and Definability 81 2. Saturated Models 84 3. Skolem Functions and Indescernables 87 4. Some Applications 91 5.
    [Show full text]
  • Elementary Model Theory Notes for MATH 762
    George F. McNulty Elementary Model Theory Notes for MATH 762 drawings by the author University of South Carolina Fall 2011 Model Theory MATH 762 Fall 2011 TTh 12:30p.m.–1:45 p.m. in LeConte 303B Office Hours: 2:00pm to 3:30 pm Monday through Thursday Instructor: George F. McNulty Recommended Text: Introduction to Model Theory By Philipp Rothmaler What We Will Cover After a couple of weeks to introduce the fundamental concepts and set the context (material chosen from the first three chapters of the text), the course will proceed with the development of first-order model theory. In the text this is the material covered beginning in Chapter 4. Our aim is cover most of the material in the text (although not all the examples) as well as some material that extends beyond the topics covered in the text (notably a proof of Morley’s Categoricity Theorem). The Work Once the introductory phase of the course is completed, there will be a series of problem sets to entertain and challenge each student. Mastering the problem sets should give each student a detailed familiarity with the main concepts and theorems of model theory and how these concepts and theorems might be applied. So working through the problems sets is really the heart of the course. Most of the problems require some reflection and can usually not be resolved in just one sitting. Grades The grades in this course will be based on each student’s work on the problem sets. Roughly speaking, an A will be assigned to students whose problems sets eventually reveal a mastery of the central concepts and theorems of model theory; a B will be assigned to students whose work reveals a grasp of the basic concepts and a reasonable competence, short of mastery, in putting this grasp into play to solve problems.
    [Show full text]
  • Elementary Equivalence Versus Isomorphism
    Elementary equivalence versus Isomorphism by Florian Pop at Bonn Lecture 1: Introduction and motivation / The Question It is one of the basic questions of Algebra to \classify" algebraic objects, like for instance fields, up to isomorphism. There are several aspects of this question, the main one being a formalized definition of \classifying". Rather than riding on such themes, we will start by giving two typical examples: 1) The isomorphy type of a finite field K is given by its cardinality jKj, i.e., if K and L are such fields, then K =∼ L iff jKj = jLj. 1) The isomorphy type of an algebraically closed field K is determined by two invariants: (i) Absolute transcendence degree td(K), (ii) The characteristic p = char(K) ≥ 0. In other words, if K and L are algebraically closed fields, then K =∼ L iff td(K) = td(L) and char(K) = char(L). Nevertheless, if we want to classify fields K up to isomorphism even in an \only a little bit" more general context, then we run into very serious difficulties... A typical example here is the attempt to give the isomorphy types of real closed fields (it seems first tried by Artin and Schreier). A real closed field Kpis \as close as possible" to its algebraic closure K, as [K : K] = 2, and K = K[ −1]. These fields were introduced by Artin in his famous proof of Hilbert's 17th Problem. Roughly speaking, the real closed fields are the fields having exactly the same algebraic properties as the reals R. One knows quite a lot about real closed fields, see e.g.
    [Show full text]
  • Elementary Equivalence, Partial Isomorphisms, and Scott-Karp Analysis
    Elementary Equivalence, Partial Isomorphisms, and Scott-Karp analysis 1 These are self-study notes I prepared when I was trying to understand the subject. 1 Elementary equivalence and Finite back and forth We begin by analyzing the power of finitary analysis of countable structures. In doing so, we introduce finitary versions of all the tools we will see later. This helps to clarify the need for the more powerful infinitary tools. Definition 1.1 Let A and B be two s-structures. A partial embedding between two structures A and A is a map p whose domain is a finite subset of A and range is a finite subset of B that preserves constants and relations. Namely, p has the following properties: (i) It is injective, (ii) For every constant c 2 s, cA 2 dom(p) and p(cA ) = cB , and (iii) For every n-ary relation R 2 s, and all a1;:::;an 2 dom(p), A B R (a1;:::;an) , R (p(a1);:::; p(an)): The above definition is purely structural. We could define a partial embedding using language as follows: A partial embedding between two s-structures A and B is a finite relation A B f(a1;b1);··· ;(an;bn)g [ f(c ;c ): c a constant in sg ⊂ A × B such that the simple extensions (A ;a1;:::;an) and (B ;b1;:::;bn) satisfy the same atomic sen- tences. Equivalently, for all j atomic, A j= j(a1;:::;an) , B j= j(b1;:::;bn). This definition allows us to define an extension of an embedding p as an embedding q whose graph contains the graph of p.
    [Show full text]
  • Model Theory on Finite Structures∗
    Model Theory on Finite Structures∗ Anuj Dawar Department of Computer Science University of Wales Swansea Swansea, SA2 8PP, U.K. e-mail: [email protected] 1 Introduction In mathematical logic, the notions of mathematical structure, language and proof themselves become the subject of mathematical investigation, and are treated as first class mathematical objects in their own right. Model theory is the branch of mathematical logic that is particularly concerned with the relationship between structure and language. It seeks to establish the limits of the expressive power of our formal language (usually the first order predicate calculus) by investigating what can or cannot be expressed in the language. The kinds of questions that are asked are: What properties can or cannot be formulated in first order logic? What structures or relations can or cannot be defined in first order logic? How does the expressive power of different logical languages compare? Model Theory arose in the context of classical logic, which was concerned with resolving the paradoxes of infinity and elucidating the nature of the infinite. The main constructions of model theory yield infinite structures and the methods and results assume that structures are, in general, infinite. As we shall see later, many, if not most, of these methods fail when we confine ourselves to finite structures Many questions that arise in computer science can be seen as being model theoretic in nature, in that they investigate the relationship between a formal language and a structure. However, most structures of interest in computer science are finite. The interest in finite model theory grew out of questions in theoretical computer science (particularly, database theory and complexity theory).
    [Show full text]
  • The Tarski Theorems and Elementary Equivalence of Group Rings
    Advances in Pure Mathematics, 2017, 7, 199-212 http://www.scirp.org/journal/apm ISSN Online: 2160-0384 ISSN Print: 2160-0368 The Tarski Theorems and Elementary Equivalence of Group Rings Benjamin Fine1, Anthony Gaglione2, Gerhard Rosenberger3, Dennis Spellman4 1Department of Mathematics, Fairfield University, Fairfield, CT, USA 2Department of Mathematics, United States Naval Academy, Annapolis, MD, USA 3Fachbereich Mathematik, University of Hamburg, Hamburg, Germany 4Department of Statistics, Temple University, Philadelphia, PA, USA How to cite this paper: Fine, B., Gaglione, Abstract A., Rosenberger, G. and Spellman, D. (2017) The Tarski Theorems and Elementary Equi- The Tarski theorems, proved by Myasnikov and Kharlampovich and inde- valence of Group Rings. Advances in Pure pendently by Sela say that all nonabelian free groups satisfy the same first- Mathematics, 7, 199-212. order or elementary theory. Kharlampovich and Myasnikov also prove that https://doi.org/10.4236/apm.2017.72011 the elementary theory of free groups is decidable. For a group ring they have Received: November 23, 2016 proved that the first-order theory (in the language of ring theory) is not de- Accepted: February 11, 2017 cidable and have studied equations over group rings, especially for torsion- Published: February 14, 2017 free hyperbolic groups. In this note we examine and survey extensions of Copyright © 2017 by authors and Tarksi-like results to the collection of group rings and examine relationships Scientific Research Publishing Inc. between the universal and elementary theories of the corresponding groups This work is licensed under the Creative and rings and the corresponding universal theory of the formed group ring.
    [Show full text]
  • Elementary Equivalences and Accessible Functors
    ELEMENTARY EQUIVALENCES AND ACCESSIBLE FUNCTORS T. BEKE AND J. ROSICKY´ ∗ ABSTRACT. We introduce the notion of λ-equivalence and λ-embeddings of objects in suitable cat- egories. This notion specializes to L1λ-equivalence and L1λ-elementary embedding for categories of structures in a language of arity less than λ, and interacts well with functors and λ-directed col- imits. We recover and extend results of Feferman and Eklof on “local functors” without fixing a language in advance. This is convenient for formalizing Lefschetz’s principle in algebraic geometry, which was one of the main applications of the work of Eklof. INTRODUCTION Elementary equivalence seems to be an intrinsically syntactic notion, and thus an unlikely sub- ject for categorical model theory. Categorical model theory, after all, focuses on the properties of models describable via a corresponding category of models, independent of the underlying lan- guage. Karp’s theorem, however, equates L1λ-equivalence of structures with the existence of a set of partial isomorphisms satisfying certain extension properties. This paper will introduce a calculus of equivalence of objects in suitable categories that specializes to L1λ-equivalence for categories of structures. Our interest was drawn to this problem by papers of Feferman [12], Eklof [9] and Hodges [13] from the 1970’s, focused on operations on structures that preserve L1λ-equivalence. A beautiful application was Eklof’s formalization in [10] of the Lefschetz–Weil principle, namely, that there is “only one” algebraic geometry over any universal domain, that is, algebraically closed field of infinite transcendence degree in a given characteristic. Since any two universal domains of the same characteristic are L1!-equivalent, it is natural to take the meaning of “only one” to be “same, up to L1!-equivalence” and to characterize families of operations preserving this equivalence.
    [Show full text]