談邏輯與數學界線之淡化（修正版）
    在近代數學研究中，邏輯與數學界線之淡化。這種現象值得關注。
    在“模型論入門”博文中，我們指出，進入二十一世紀，世界數學形式化、公理化高潮興起。“知識共享”無窮小微積分就是在這種發展浪潮中崛起的。
    反觀我們國內，數學教育界的學術認識水平似乎仍然停留在世界數學形式化、公理化發展階段之前。
    無論如何，我們的數學教育落伍了。
    說明：本本附件，是關於模型論的長篇論文，資料詳實，值得存檔備查。
袁萌  陳啟清   12月17日
附件：模型論（Model Theory）
Class Notes for Mathematics 571 Spring 2010
Model Theory
written by C. Ward Henson
Mathematics Department University of Illinois 1409 West Green Street Urbana, Illinois 61801 email:

Introduction
The purpose of Math 571 is to give a thorough introduction to the methods of model theory for first order logic. Model theory is the branch of logic that deals with mathematical structures and the formal languages they interpret. First order logic is the most important formal language and its model theory is a rich and interesting subject with significant applications to the main body of mathematics. Model theory began as a serious subject in the 1950s with the work of Abraham Robinson and Alfred Tarski, and since then it has been an active and successful area of research. Beyond the core techniques and results of model theory, Math 571 places a lot of emphasis on examples and applications, in order to show clearly the variety of ways in which model theory can be useful in mathematics. For example, we give a thorough treatment of the model theory of the field of real numbers (real closed fields) and show how this can be used to obtain the characterization of positive semi-definite rational functions that gives a solution to Hilbert’s 17th Problem. A highlight of Math 571 is a proof of Morley’s Theorem: if T is a complete theory in a countable language, and T is κ-categorical for some uncountable κ, then T is categorical for all uncountable κ. The machinery needed for this proof includes the concepts of Morley rank and degree for formulas in ω-stable theories. The methods needed for this proof illustrate ideas that have become central to modern research in model theory. To succeed in Math 571, it is necessary to have exposure to the syntax and semantics of first order logic, and experience with expressing mathematical properties via first order formulas. A good undergraduate course in logic will usually provide the necessary background. The canonical prerequisite course at UIUC is Math 570, but this covers many things that are not needed as background for Math 571. In the lecture notes for Math 570 (written by Prof. van den Dries) the material necessary for Math 571 is presented in sections 2.3 through 2.6 (pages 24–37 in the 2009 version). These lecture notes are available at http://www.math.uiuc.edu/ vddries/410notes/main.dvi. A standard undergraduate text in logic is A Mathematical Introduction to Logic by Herbert B. Enderton (Academic Press; second edition, 2001). Here the material needed for Math 571 is covered in sections 2.0 through 2.2 (pages 67–104). This material is also discussed in Model Theory by David Marker (see sections 1.1 and 1.2, and the first half of 1.3, as well as many of the exercises at the end of chapter 1) and in many other textbooks in model theory. For Math 571 it is not necessary to have any exposure to a proof system for first order logic, nor to G¨odel’s completeness theorem. Math 571 begins with a proof of the compactness theorem for first order languages, and this is all one needs for model theory.
We close this introduction by discussing a number of books of possible interest to anyone studying model theory. The first two books listed are now the standard graduate texts in model theory; they can be used as background references for most of what is done in Math 571. David Marker, Model Theory: an Introduction. Bruno Poizat, A Course in Model Theory.
The next book listed was the standard graduate text in model theory from its first publication in the 1960s until recently. It is somewhat out of date and incomplete from a modern viewpoint, but for much of the content of Math 571 it is a suitable reference. C. C. Chang and H. J. Keisler, Model Theory.
Another recent monograph on model theory is Model Theory by Wilfrid Hodges. This book contains many results and examples that are otherwise only available in journal articles, and gives a very comprehensive treatment of basic model theory. However it is very long and it is organized in a complicated way that makes things hard to find. The author extracted a shorter and more straightforward text entitled A Shorter Model Theory, which is published in an inexpensive paperback edition.
In the early days of the subject (i.e., 1950s and 1960s), Abraham Robinson was the person who did the most to make model theory a useful tool in the main body of mathematics. Along with Alfred Tarski, he created much of modern model theory and gave it its current style and emphasis. He published three books in model theory, and they are still interesting to read: (a) Intro. to Model Theory and the Metamathematics of Algebra, 1963; (b) Complete Theories, 1956; new edition 1976; (c) On the Metamathematics of Algebra, 1951.
The final reference listed here is Handbook of Mathematical Logic, Jon Barwise, editor; this contains expository articles on most parts of logic. Of particular interest to students in model theory are the following chapters: A.1. An introduction to first-order logic, Jon Barwise. A.2. Fundamentals of model theory, H. Jerome Keisler. A.3. Ultraproducts for algebraists, Paul C. Eklof. A.4. Model completeness, Angus Macintyre.
Contents
Introduction 3 1. Ultraproducts and the Compactness Theorem 1 Appendix 1.A: Ultrafilters 6 Appendix 1.B: From prestructures to structures 8 2. Theories and Types 12 3. Elementary Maps 18 4. Saturated Models 25 5. Quantifier Elimination 30 6. L¨owenheim-Skolem Theorems 35 7. Algebraically Closed Fields 39 8. Z-groups 44 9. Model Theoretic Algebraic Closure 49 10. Algebraic Closure in Minimal Structures 52 11. Real Closed Ordered Fields 58 12. Homogeneous Models 62 13. Omitting Types 68 14. ω-categoricity 76 15. Skolem Hulls 80 16. Indiscernibles 82 17. Morley rank and ω-stability 86 18. Morley’s uncountable categoricity theorem 96 19. Characterizing Definability 102 Appendix: Systems of Definable Sets and Functions 110

1. Ultraproducts and the Compactness Theorem
The main purpose of this chapter is to give a proof of the Compactness Theorem for arbitrary first order languages. We do this using ultraproducts. The ultraproduct construction has the virtue of being explicit and algebraic in character, so it is accessible to mathematicians who know little about formal logic. Fix a first order language L. Let I be a nonempty set and let U be an ultrafilter1 on I. Consider a family of L-structures (Ai | i ∈ I). For each i ∈ I let Ai denote the underlying set of the structure Ai and take A =Q(Ai | i ∈ I) to be the cartesian product of the sets Ai. We define an interpretation2 A of L as follows: (i) the underlying set of A is the cartesian product A =Q(Ai | i ∈ I); (ii) for each constant symbol c of L we set cA = (cAi | i ∈ I); (iii) for each n and each n-ary function symbol F of L we let FA be the function defined on An by FA(f1,...,fn) = (FAi(f1(i),...,fn(i))| i ∈ I); (iv) for each n and each n-ary predicate symbol P of L we let PA be the n-ary relation on A defined by PA(f1,...,fn) ⇐⇒ {i ∈ I | PAi(f1(i),...,fn(i))}∈ U; (v) =A is the binary relation on A defined by f =A g ⇐⇒ {i ∈ I | f(i) = g(i)}∈ U. Note that constants and function symbols are treated in this construction in a “coordinatewise” way, exactly as we would do in forming the cartesian product of algebraic structures. Only in defining the interpretations of predicate symbols and = (clauses (iv) and (v)) do we do something novel, and only there does the ultrafilter enter into the definition. For the algebraic part of A we have the following easy fact, proved by a straightforward argument using induction on terms: 1.1. Lemma. For any L-term t(x1,...,xn) and any f1,...,fn ∈ A, tA(f1,...,fn) = (tAi(f1(i),...,fn(i))| i ∈ I). The following result gives the most important model theoretic property of this construction: 1.2. Proposition. For any L-formula ϕ(x1,...,xn) and any f1,...,fn ∈ A A |= ϕ[f1,...,fn] ⇐⇒ {i ∈ I | Ai |= ϕ[f1(i),...,fn(i)]}∈ U. 1See Appendix 1 of this chapter for some basic facts about filters and ultrafilters. 2See Appendix 2 of this chapter for an explanation of the words “interpretation”, “prestructure”, and “structure” and for some basic relations among them. 1
Proof. The proof is by induction on formulas ϕ(x1,...,xn), where x1,...,xn is an arbitrary list of distinct variables. In the basis step of the induction ϕ is an atomic formula of the form P(t1,...,tm), where P is an m-place predicate symbol or the equality symbol =. Our assumptions ensure that any variable occurring in a term tj, j = 1,...,m, is among x1,...,xn; thus we may write each such tj as tj(x1,...,xn). Let (f1,...,fn) range over An; let gj(i) = tAi j (f1(i),...,fn(i)) for each j = 1,...,m and i ∈ I. Note that gj ∈ A for each j = 1,...,m. Then we have: A |= ϕ[f1,...,fn] ⇔ PAtA 1 (f1,...,fn),...,tA m(f1,...,fn)⇔ PA(g1,...,gm) ⇔ i | PAi (g1(i),...,gm(i))    ∈ U ⇔ ni | PAitAi 1 (f1(i),...,fn(i)),...,tAi m (f1(i),...,fn(i))o∈ U⇔ { i | Ai |= ϕ[f1(i),...,fn(i)]}∈ U. (Lemma 1.1 is used in the second equivalence.) In the induction step of the proof we consider three cases: (1) ϕ is ¬ϕ1 for some formula ϕ1; (2) ϕ is (ϕ1∧ϕ2) for some formulas ϕ1,ϕ2 (3); ϕ is ∃yϕ1 for some formula ϕ1 and some variable y. Case (1) ϕ is ¬ϕ1: A |= ϕ[f1,...fn] ⇔ A 6|= ϕ1[f1,...fn] ⇔ {i | Ai |= ϕ1[f1(i),...,fn(i)]}6∈ U ⇔? {i | Ai 6|= ϕ1[f1(i),...,fn(i)]}∈ U ⇔ {i | Ai |=¬ϕ1[f1(i),...,fn(i)]}∈ U ⇔ {i | Ai |= ϕ[f1(i),...,fn(i)]}∈ U In the equivalence ? we use the fact that for any subset A of I, A is not in U if and only if I \A is in U. Case (2) ϕ is (ϕ1 ∧ϕ2): A |= ϕ[f1,...fn] ⇔ A |= (ϕ1 ∧ϕ2)[f1,...fn] ⇔ A |= ϕ1[f1(i),...,fn(i)] and A |= ϕ2[f1(i),...,fn(i)] ⇔ {i | Ai |= ϕ1[f1(i),...,fn(i)]}∈ U and {i | Ai |= ϕ2[f1(i),...,fn(i)]}∈ U ⇔? {i | Ai |= ϕ1[f1(i),...,fn(i)]}∩ {i | Ai |= ϕ2[f1(i),...,fn(i)]}∈ U ⇔ {i | Ai |= ϕ1[f1(i),...,fn(i)] and Ai |= ϕ2[f1(i),...,fn(i)]}∈ U ⇔ {i | Ai |= (ϕ1 ∧ϕ2)[f1(i),...,fn(i)]}∈ U ⇔ {i | Ai |= ϕ[f1(i),...,fn(i)]}∈ U In the equivalence ? we use the fact that for any subsets A and B of I, A and B are in U if and only if A∩B is in U.
2
Case (3) ϕ is the formula ∃yϕ1: We may assume that y is not among the variables x1,...,xn. A |= ϕ[f1,...,fn] ⇔ A |=∃yϕ1[f1,...,fn] ⇔ for some g ∈ A : A |= ϕ1[f1,...,fn,g] ⇔ for some g ∈ A : {i | Ai |= ϕ1[f1(i),...,fn(i),g(i)]}∈ U ⇔† {i |for some a ∈ Ai : Ai |= ϕ1[f1(i),...,fn(i),a]}∈ U ⇔ {i | A |=∃yϕ1[f1(i),...,fn(i)]}∈ U To see the “⇐”-part of the equivalence †, use the Axiom of Choice to obtain a function g such that for any i ∈ {i | for some a ∈ Ai : Ai |= ϕ1[f1(i),...,fn(i),a]} we have Ai |= ϕ1[f1(i),...,fn(i),g(i)]}. For all other values of i the value of g(i) can be arbitrary.   1.3. Corollary. The interpretation A defined above is a prestructure.
Proof. Applying Proposition 1.2 to the equality axioms, we see that they are valid in A.   1.4. Definition (Ultraproduct of a family of L-structures). Let (Ai | i ∈ I) be a family of L-structures and U an ultrafilter on I. Let A be the prestructure for L that is defined above. The ultraproductQU Ai of the given family of L-structures (Ai | i ∈ I) with respect to U is defined to be the L-structure B obtained by taking the quotient of A by the congruence =A as described in Appendix 2 of this chapter. 1.5. Notation. Let (Ai | i ∈ I) and A be as above. For each f ∈QAi we let f/U denote the equivalence class of f under the equivalence relation =A. As f varies, f/U gives an arbitrary element of the underlying set of the ultraproductQU Ai. The ultrapower of the L-structure C with respect to U is the ultraproduct QU Ai with Ai equal to C for every i ∈ I. We denote this structure by CI/U. 1.6. Fact. Let I be a nonempty set and let U be the principal ultrafilter on I that is generated by the singleton set {j}, where j is a fixed element of I. For every family {Ai | i ∈ I} of L-structures, the ultraproductQU Ai is isomorphic to Aj. The next theorem is the main result of this chapter; it is basic to any use of the ultraproduct construction in model theory. This result was originally proved by the Polish logician Jerzy L os. 1.7. Theorem (Fundamental Theorem of Ultraproducts). Let an indexed family of L-structures and an ultrafilter U be given as described above. For any L-formula ϕ(x1,...,xn) and any sequence f/U = (f1/U,...,fn/U), Y U Ai |= ϕ[f/U] if and only if {i ∈ I | Ai |= ϕ[f1(i),...,fn(i)]}∈ U. 3
Proof. This is an immediate consequence of Propositions 1.2 and 1.29.   1.8. Corollary. If σ is an L-sentence, then Y U Ai |= σ if and only if {i ∈ I | Ai |= σ}∈ U. Proof. This is a special case of Theorem 1.7.  
Now we use the ultraproduct construction to prove the Compactness Theorem, which is one of the most important tools in model theory. First we need a basic definition: 1.9. Definition. Let T be a set of sentences in L and let A be an Lstructure. We say that A is a model of T, and write A |= T, if every sentence in T is true in A. 1.10. Theorem (Compactness Theorem). Let T be any set of sentences in L. If every finite subset of T has a model, then T has a model.
Proof. Assume that every finite subset of T has a model. Let I be the set of all finite subsets of T. For each i ∈ I let Ai be any model of i, which exists by assumption. We will obtain the desired model of T as an ultraproduct QU Ai for a suitably chosen ultrafilter U on I. Let S be the family of all the subsets of I of the form Iσ ={i ∈ I : σ ∈ i}, where σ ∈ T. Note that S has the finite intersection property; indeed, each finite intersection Iσ1 ∩ ... ∩ Iσn has {σ1,...,σn} as an element. So there exists an ultrafilter U on I that contains S, by Corollary 1.25. We complete the proof by showing that the ultraproductQU Ai is a model of T. Given σ ∈ T, we see that Ai |= σ whenever σ ∈ i, because of the way we chose Ai. Hence {i : Ai |= σ}⊇ Iσ ∈ U. It follows from Theorem 1.7 that each such σ is true inQU Ai.   1.11. Remark. Note that the preceding proof yields the following result: Let T beasetofsentencesin L andlet C beaclassof L-structures. Suppose each finite subset of T has a model in C. Then T has a model that is an ultraproduct of structures from C.
The Compactness Theorem is a very useful tool for building models of a given set of sentences, and nearly everything we do in this course depends on it in one way or another. We give a number of examples of this in the rest of this chapter. 1.12. Corollary. Let L be a first order language and let κ be an infinite cardinal number. If T is a set of sentences in L such that for each positive integer n there is a model of T with at least n elements, then T has a model with at least κ many elements. (In particular, this holds if T has at least one infinite model.)
4
Proof. Expand L by adding a set of κ many new constant symbols; let L0 be the new language. Let T0 be T together with all sentences ¬(c1 = c2), where c1 and c2 are distinct new constants. Our hypothesis implies that every finite subset of T0 has a model. Therefore T0 itself has a model A, by the Compactness Theorem. Let B be the reduct of A to the original language L; A is a model of T and has at least κ many elements.   1.13. Fact. Let T be a set of sentences in a first order language L and let ϕ(x) be a formula in L. For each L-structure A let ϕA denote the set of tuples a from A such that A |= ϕ(a). Suppose that the set ϕA is finite whenever A is a model of T. Then there is a positive integer N such that ϕA has at most N elements for every model A of T. This can be proved using the Compactness Theorem in a manner similar to the proof of the previous result. 1.14. Remark. The preceding results demonstrate a fundamental limitation on the expressive power of first order logic: only finite cardinalities can be “expressed” by first order formulas. There is no way to express any bound on the sizes of definable sets other than a uniform finite upper bound. We will see later on how to control more precisely the cardinality of models like the one constructed above. In particular, it turns out to be possible to make the model have precisely κ many elements, as long as the number of symbols in the language L is less than or equal to κ. 1.15. Definition. Let Γ be a set of L-formulas and let the family (xj | j ∈ J) include all variables that occur free in some member of Γ. Let A be an L-structure. We say that Γ is satisfiable in A if there exist elements (aj | j ∈ J) of A such that A |= Γ[aj | j ∈ J]. 1.16. Definition. Let T be asetofsentencesin L andΓ asetof L-formulas. We say that Γ is consistent with T if for every finite subsets F of T and G of Γ there exists a model A of F such that G is satisfiable in A.
The next result is a version of the Compactness Theorem for formulas. 1.17. Corollary. Let T be a set of sentences in L and Γ a set of L-formulas, and assume that Γ is consistent with T. Then Γ is satisfiable in some model of T. Proof. Let (xj | j ∈ J) include all variables that occur free in some member of Γ. Let (cj | j ∈ J) be new constants and consider the language L(cj | j ∈ J). Apply the Compactness Theorem to the set T ∪Γ(cj | j ∈ J) of L(cj | j ∈ J)-sentences.  
5
Appendix 1.A: Ultrafilters
Here we present some prerequisites about filters and ultrafilters: 1.18. Definition. (1) Let I be a nonempty set. A filter on I is a collection F of subsets of I that satisfies: (a) ∅6∈ F and I ∈ F; (b) for all A,B ∈ F,A∩B ∈ F; (c) for all A ∈ F and B ⊆ I, if A ⊆ B then B ∈ F. (2) Let F be a filter on I; F is an ultrafilter if it is maximal under⊆among filters on I. F is principal if there exists a subset A of I such that F is exactly the collection of all sets B that satisfy A ⊆ B ⊆ I. 1.19. Definition. Let I be a set and let S be a collection of subsets of I; S has the finite intersection property (FIP) if for every integer n and every finite subcollection {A1,...,An} of S, the intersection A1 ∩...∩An is nonempty. 1.20. Lemma. Let I be a nonempty set and let S be a collection of subsets of I. There exists a filter F on I that contains S if and only if S has the finite intersection property. Proof. (⇒) It is immediate (by induction) from the definition of filter that any filter is closed under finite intersections. Since no filter contains the empty set, this shows that each filter has the FIP; hence the same is true of any subcollection of a filter. (⇐) Let S ⊆ P(I) have the FIP; we want to find a filter F ⊇ S. We know that all supersets of finite intersections of elements of S must be elements of F. Thus we are led to define F ={A | A ⊆ I and there exist A1,...,An ∈ S with A ⊇ A1 ∩...∩An} Since S has the FIP, we see that F does not contain the empty set. It is easy to check that conditions (b) and (c) in the definition of filter are satisfied.  
Remark: The filter defined in the preceding proof is evidently the smallest filter on I containing S. Thus it is called the filter generated by S. 1.21. Facts. Let I be a nonempty set. (a) If S is a collection of subsets of I and S has the FIP, then for any A ⊆ I, either S ∪{A} or S ∪{I \A} has the FIP. (b) Suppose J is an index set and for each j ∈ J, Sj is a collection of subsets of I that has the FIP. Suppose that the family {Sj | j ∈ J} is directed, in the sense that for any j1,j2 ∈ J there is j3 ∈ J such that Sj1 ∪Sj2 ⊆ Sj3. Let S be the union of the family {Sj | j ∈ J}. Then S has the FIP. 6
1.22. Lemma. Let I be a nonempty set and let F be a filter on I. F is an ultrafilter if and only if for each A ⊆ I, either A ∈ F or I \A ∈ F. Proof. (⇒) If F is an ultrafilter on I and A ⊆ I, then by Fact 1.21(a), either F ∪{A} or F ∪{I \ A} has the FIP. Therefore by Lemma 1.20, F ∪{A} or F ∪{I \A} is contained in a filter. But F is maximal, so the only filter it can be contained in must be F itself. Hence F ∪{A}⊆ F or F ∪{I \A}⊆ F. (⇐) Suppose F is a filter on I with the property that for any A ⊆ I either A ∈ F or I\A ∈ F. We have to show that F is maximal among the filters on I under set-theoretic inclusion. If F is not maximal, then there is a filter G on I with F ⊆ G and G 6= F. Take any set A ∈ G\F. Since A / ∈ F wemust have I\A ∈ F ⊆ G. But then A and I\A are in G, and this implies ∅∈ G, which is a contradiction.   1.23. Facts. Let I be a nonempty set and let U be an ultrafilter on I. (a) If A1,...,An are subsets of I and if the set A1 ∪•••∪An is in U, then for some j = 1,...,n the set Aj is in U. (b) If A1,...,An are subsets of I and if the set A1 ∩•••∩An is in U, then for all j = 1,...,n the set Aj is in U. (c) The ultrafilter U is principal iff some element of U is a finite set iff some element of U is a singleton set (a set of the form {i} for some i ∈ I).
In the next proof we are going to use the Axiom of Choice in the form of Zorn’s Lemma, which we formulate as follows: Zorn’s Lemma: If (Λ,≤) is a nonempty partially ordered set with the property that every linearly ordered subset of (Λ,≤) has an upper bound in (Λ,≤), then (Λ,≤) has a maximal element. Remark: An element of (Λ,≤) is maximal if no other element is strictly larger than it. There may be many maximal elements. We will often use Zorn’s Lemma where Λ is a collection of sets and ≤ is the set containment relation ⊆. In that situation the hypothesis of Zorn’s Lemma states that whenever C is a subcollection of Λ and C is a chain under⊆, the union of C is a subset of some element of Λ. (This restricted formulation is easily seen to be equivalent to Zorn’s Lemma; it is known as Hausdorff’s Maximum Principle.) 1.24. Theorem. Let I be a nonempty set. Every filter on I is contained in an ultrafilter on I. Proof. Let F be a filter on I. Let Λ ={G | G is a filter on I and F ⊆ G}. Partially order Λ by set inclusion ⊆. We want to apply Zorn’s Lemma to (Λ,≤). Suppose C is a chain in Λ. It is easy to show that the union of C is a filter, and hence it is in Λ. (Compare Fact 1.21(b).) Zorn’s Lemma yields the existence of a maximal element G in Λ. That is, G is a filter 7
that contains F and G is maximal among all filters on I that contain F. In particular, G is maximal as a filter on I; by definition G is an ultrafilter.   1.25.Corollary. Let I be a nonempty set and let S be a collection of subsets of I. If S has the FIP, then there is an ultrafilter on I that contains S.
Proof. Immediate from Lemma 1.20 and Theorem 1.24.  
Appendix 1.B: From prestructures to structures
Let L be any first order language. 1.26. Definition. An interpretation A of L consists of (i) a nonempty set A, the underlying set of A; (ii) for each constant symbol c of L an element cA of A, the interpretation of c in A; (iii) for each n and each n-ary function symbol F of L a function FA from An to A, the interpretation of F in A; (iv) for each n and each n-ary predicate symbol P of L a subset PA of An, the interpretation of P in A; (v) a subset =A of A2, the interpretation of = in A.
Suppose A is an interpretation of L. For each L-term t(x1,...,xn) we define the interpretation of t in A by induction on t; it is a function from An to A and it is denoted by tA. By induction on formulas we likewise define the satisfaction relation A |= ϕ[a1,...,an] where ϕ(x1,...,xn) is an L-formula and a1,...,an ∈ A. Formally this is identical to what is done for L-structures, with which we assume the reader is familiar. The only difference here is that we are allowing an arbitrary binary relation to be used as the interpretation of =; that is, we are temporarily treating = as if it were a non-logical symbol. 1.27. Definition. A prestructure A for L is an interpretation of L in which the logical equality axioms are valid; that is, (i) =A is an equivalence relation on A; (ii) for any n, any n-ary function symbol F of L, and any elements a1,b1,...,an,bn of A such that a1 =A b1,...,an =A bn one has FA(a1,...,an) =A FA(b1,...,bn); (iii) for any n, any n-ary predicate symbol P of L, and any elements a1,b1,...,an,bn of A such that a1 =A b1,...,an =A bn one has PA(a1,...,an) ⇐⇒ PA(b1,...,bn). 8
When =A is an equivalence relation on A, universal algebraists express conditions (ii) and (iii) by saying that =A is a congruence with respect to the functions FA mentioned in (ii) and the relations PA mentioned in (iii). Note that A is a structure for L if it is an interpretation of L and =A is the identity relation on A, that is a =A b ⇔ a = b for any a,b ∈ A. (In that case, A trivially satisfies the equality axioms and hence it is a prestructure.) When A is a prestructure for L, we define the quotient of A by =A as follows; it is a structure for L. We will denote it here by B. (i) The underlying set B of B is the set of all equivalence classes of =A. We denote the equivalence class of a ∈ A with respect to =A by [a], and we let π: A → B denote the quotient map that takes each a ∈ A to its equivalence class (π(a) = [a] for each a ∈ A). (ii) For each constant symbol c of L the interpretation of c in B is [cA]. (iii) For each n and each n-ary function symbol F of L the interpretation of F in B is the function FB: Bn → B defined by FB([a1],...,[an]) = [FA(a1,...,an)] for every a1,...,an ∈ A. The fact that =A is a congruence for FA ensures that the right hand side of this definition depends only on the equivalence classes [a1],...,[an] and not on their representatives a1,...,an. (iv) For each n and each n-ary predicate symbol P of L the interpretation of P in B is the n-ary relation PB on B defined by PB([a1],...,[an]) ⇐⇒ PA(a1,...,an) for every a1,...,an ∈ A. The fact that =A is a congruence for PA ensures that the right hand side of this definition depends only on the equivalence classes [a1],...,[an] and not on their representatives a1,...,an. Since B is to be a structure, the interpretation =B of = in B must be the identity relation on B. Note that we have [a] =B [b] ⇐⇒ a =A b for all a,b ∈ A. Hence the identity interpretation of = in B is the same as the one we would get if we treated = as another predicate symbol of L and applied clause (iv) of this construction. Our definition of the quotient structure B can be summarized by saying that the quotient map π from A onto B is a strong homomorphism of A onto B. The following Lemma is easily proved by induction on terms. 1.28. Lemma. For any L-term t(x1,...,xn) and any a1,...,an ∈ A, tB([a1],...,[an]) = [tA(a1,...,an)].
The following result gives the main content of this quotient construction from a model theoretic point of view. It says that no difference between 9
a prestructure A and its quotient structure B can be expressed in first order logic. It justifies the usual practice of only considering structures in model theory. (However, prestructures are often used, at least implicitly, in the construction of structures; this happens in the usual proof of the completeness theorem for first order logic, for example.) 1.29. Proposition. Let A be a prestructure for L and B its quotient structure as described above. For any L-formula ϕ(x1,...,xn) and any a1,...,an ∈ A B |= ϕ[[a1],...,[an]] ⇐⇒ A |= ϕ[a1,...,an]. Proof. By induction on the formula ϕ. When ϕ is an atomic formula, this equivalence follows from the preceding Lemma and the fact that π is a strong homomorphism. The induction step is an immediate consequence of the definition of |= and (for quantifiers) the fact that π is surjective.   Exercises 1.30. Let I be a nonempty set, U an ultrafilter on I, and J an element of U. Define V to be the set of X ⊆ J such that X ∈ U. • Show that V is an ultrafilter on J. • Show that if (Ai | i ∈ I) is a family of L-structures, then ΠU(Ai | i ∈ I) is isomorphic to ΠV (Aj | j ∈ J) 1.31. Let I be an index set and U an ultrafilter on I. Let (Ai | i ∈ I) and (Bi | i ∈ I) be families of L-structures. If Ai can be embedded in Bi for all i ∈ I, show that ΠUAi can be embedded in ΠUBi. 1.32. Let A be any L-structure. Show that A can be embedded in some ultraproduct of a family of finitely generated substructures of A. 1.33. Let L be the first order language whose only nonlogical symbol is the binary predicate symbol <. Let A = (N,<) and let B = AI/U be an ultrapower of A where I is countably infinite and U is a nonprincipal ultrafilter on I. • Show that B is a linear ordering. • Show that the range of the diagonal embedding of A into B is a proper initial segment of B. Give an explicit description of an element of B that is not in the range of this embedding. •Show that B is not a well ordering; that is, describe an infinite descending sequence in B. 1.34. Let L be the first order language whose nonlogical symbols consist of a binary predicate symbol <, a binary function symbol + and a constant symbol 0. Let Z be the ordered abelian group of all the integers, considered as an L-structure. Let I be any countable infinite set and let U be a nonprincipal ultrafilter on I. Consider the ultrapower ZI/U. 10
• Show that ZI/U is an ordered abelian group. • Find a natural embedding of Z into this group so that the image of the embedding is a convex subgroup. • Show that ZI/U contains a nonzero element b that is divisible in ZI/U by every positive integer n. (This means that for each n ≥1 there exists a in ZI/U that satisfies b = a +•••+ a (n times).) Such an element can be produced explicitly.
11
2. Theories and Types
In this chapter we discuss a few basic topics in model theory; they are closely tied to the Compactness Theorem. A theory consists of a first order language L together with a set T of sentences in L; often the language is determined by the context. We may refer to T as an L-theory to indicate which language is intended. 2.1. Definition. Let L be a first order language, let T,T0 be L-theories, let σ be an L-sentence, and let K be a class of L-structures. • T is satisfiable if it has at least one model; Mod(T) denotes the class of all models of T. • σ is a logical consequence of T (and we write T |= σ) if σ is true in every model of T. • T is complete if it is satisfiable and for every L-sentence σ, either T |= σ or T |=¬σ. • T and T0 are equivalent if they have the same logical consequences in L; this is the same as saying that each sentence in T is a logical consequence of T0 and each sentence in T0 is a logical consequence of T. When T and T0 are equivalent we will also say that T is axiomatized by T0. • The theory of K is defined by Th(K) ={σ | σ is an L-sentence and A |= σ for all A ∈ K}. If K ={A}we write Th(A) instead of Th({A}). We say K is axiomatizable if K = Mod(T) for some theory T.
The following results are easy consequences of the definitions. 2.2. Facts. Let K, K1 and K2 be classes of L-structures, and let T,T1, and T2 be L-theories. (1) K1 ⊆ K2 ⇒Th(K1)⊇Th(K2); (2) T1 ⊆ T2 ⇒Mod(T1)⊇Mod(T2); (3) T1 and T2 are equivalent iff Mod(T1) = Mod(T2); (4) Mod(Th(K))⊇ K, with equality if K is axiomatizable; (5) Th(K) contains its logical consequences; (6) Th(Mod(T))⊇ T and T axiomatizes Th(Mod(T)). (7) T is of the form Th(A), where A is an L-structure, iff T is finitely satisfiable and it is ⊆-maximal among finitely satisfiable sets of Lsentences. (8) Among L-theories containing T, the complete theories are those equivalent to theories of the form Th(A), where A is a model of T. 2.3. Definition. If A and B are two structures for the same language L, we say that A and B are elementarily equivalent, and write A ≡ B, if Th(A) = Th(B).
12
2.4. Fact. Let T be a theory. Then T is complete iff T has a model and any two models of T are elementarily equivalent. 2.5. Definition. Let L1 ⊆ L2 be two first order languages and let Ti be a theory in Li for i = 1,2. We say that T2 is an extension of T1 (and, equivalently, that T1 is a subtheory of T2), if T1 is contained in the set of logicalconsequencesof T2. Further, T2 issaidtobeaconservative extension of T1, if, in addition, T2 |= σ ⇒ T1 |= σ for every sentence σ of L1. 2.6.Fact. If T2 isanextensionof T1 andeverymodelof T1 hasanexpansion to a model of T2, then T2 is a conservative extension of T1.
The following result, which is just a restatement of the Compactness Theorem, expresses a fundamental property of logical consequence in first order logic. It shows that the relation T |= σ has finitary character, and thus (in principle) can be analyzed by some sort of (possibly abstract) “proof system” with the property that only finitely many sentences appear in each “proof.” Each presentation of G¨odel’s Completeness Theorem (which we do not need for model theory) gives such a proof system. 2.7. Corollary. If T |= σ, then there is a finite set T0 ⊆ T with T0 |= σ. Proof. Assume T |= σ, so T∪{¬σ}has no model. Hence there exists a finite T0 ⊆ T ∪{¬σ} such that T0 has no model, by the Compactness Theorem. There exists a finite T0 ⊆ T with T0 ⊆ T0 ∪{¬σ}. Evidently T0 ∪{¬σ} cannot have a model, and therefore T0 |= σ.   The following result is a variation on the same theme: 2.8. Corollary. Let L be a first order language and let S and T be sets of sentences in L. Suppose that for every model A of T there exists γ ∈ S such that A |= γ. Then there exists a finite subset {γ1,...,γm} of S such that T |= γ1 ∨...∨γm. Proof. Apply the Compactness Theorem to T ∪{¬γ | γ ∈ S}.   2.9. Definition. Let T be a satisfiable L-theory. We denote by S0(T) the set of all theories of the form Th(A), where A is a model of T.
Note that we may regard S0(T) as the set of complete L-theories that extend T, up to equivalence of theories. We think of it as the space of completions of T. We put a natural topology on S0(T) as follows: for each L-sentence σ, let [σ] ={T0 ∈ S0(T)| σ ∈ T0}={T0 ∈ S0(T)| T0 |= σ}. Note that the family F ={[σ]| σ is an L-sentence} 13
is closed under finite intersections and unions; indeed, for any L-sentences σ and τ, we see that [σ]∩[τ] = [σ∧τ] and [σ]∪[τ] = [σ∨τ]. The logic topology on S0(T) is the topology for which F is the family of basic open sets. That is, for each T0 ∈ S0(T), the basic open neighborhoods of T0 are the sets [σ] where σ ∈ T0. Evidently this is a Hausdorff topology. Moreover, each set of the form [σ] is closed as well as open, since S0(T)\[σ] = [¬σ] for all L-sentences σ. Furthermore, the logic topology on S0(T) is compact; this is an immediate consequence of Corollary 2.8. Note also that there is a close relation between closed sets in S0(T) and L-theories T1 that extend T. For such a theory T1, define K(T1) ={T0 ∈ S0(T)| T1 ⊆ T0}=\{[σ]| σ ∈ T1}.Then K(T1) is closed, because it is the intersection of a family of clopen sets. Conversely, if K is a closed set in S0(T), then there is a set Σ of L-sentences such that the open set S0(T)\K is equal to the union of the basic open sets [σ] with σ ∈ Σ. Taking T1 = T ∪{¬σ | σ ∈ Σ} we have that T1 extends T and K(T1) = K. Note that K(T1) is nonempty iff T1 is satisfiable. 2.10. Proposition. Let T be a satisfiable L-theory. The space S0(T) with the logic topology is a totally disconnected, compact Hausdorff space. Its closed sets are the sets of the form {T0 ∈ S0(T)| T1 ⊆ T0} where T1 is a set of L-sentences containing T. Moreover, the clopen subsets of S0(T) in this topology are exactly the sets of the form [σ], where σ is an L-sentence. Proof. It remains only to prove that each clopen set C ⊆ S0(T) is of the form [σ] for some sentence σ. Since C is open, it is the union of a family of basic open sets. Since C is closed, hence compact, this family can be taken to be finite. In S0(T), a union of finitely many basic open sets is itself a basic open set.   2.11. Fact. Let T be a satisfiable L-theory and let σ,τ be L-sentences. Then [σ] = [τ] iff σ and τ are equivalent over T (i.e., σ ↔ τ is a logical consequence of T).
These results show that the topological space S0(T) by itself characterizes the relation of equivalence of L-sentences over the theory T.
14
Types Next we introduce types; they provide a way of describing the first order expressible properties of elements of a structure. Fix n ≥1 and let x1,...,xn be a fixed sequence of distinct variables. 2.12. Definition. Let A be an L-structure and consider a1,...,an ∈ A. • The type of (a1,...,an) in A is the set of L-formulas {ϕ(x1,...,xn)| A |= ϕ[a1,...,an]}; we denote this set by tpA(a1,...,an) or simply by tp(a1,...,an) if the structure A is understood. • An n-type in L is a set of formulas of the form tpA(a1,...,an) for some L-structure A and some a1,...,an ∈ A. A partial n-type in L is a subset of an n-type in L. • If Γ(x1,...,xn) is a partial n-type in L, we say (a1,...,an) realizes Γ in A if every formula in Γ is true of a1,...,an in A. • If Γ(x1,...,xn) is a partial n-type in L and A is an L-structure, we say that Γ is realized in A if there is some n-tuple in A that realizes Γ in A. If no such n-tuple exists, then we say that A omits Γ. 2.13. Facts. Let Γ(x1,...,xn) be a set of formulas in L, all of whose free variables are among x1,...,xn. • Γ(x1,...,xn) is an n-type in L if and only if it is a maximal (finitely) satisfiable subset of the set of all formulas in L whose free variables are among x1,...,xn. •Γ(x1,...,xn)isapartial n-typein L ifandonlyifitis(finitely)satisfiable. 2.14. Definition. Let T be a theory in L and let Γ = Γ(x1,...,xn) be a partial n-type in L. • Γ is consistent with T if T ∪Γ is finitely satisfiable. This is equivalent to saying that Γ is realized in some model of T. • The set of all n-types that contain T is denoted by Sn(T). These are exactly the n-types in L that are consistent with T.
Let c1,...,cn be distinct new constant symbols and let Ln be the language L(c1,...,cn) extending T. Let Tn denote the theory whose set of sentences is T but whose language is Ln. The simple observation we give next allows us to identify Sn(T) with S0(Tn) using the bijection that takes an n-type Γ(x1,...,xn) to the set of Ln-sentences Γ(c1,...,cn). 2.15. Lemma. Let A,B be L-structures, let a be an n-tuple in A and let b be an n-tuple in B. The n-type tpA(a) can be identified with the complete theory Th(A,a1,...,an). In particular, tpA(a) = tpB(b) if and only if (A,a1,...,an)≡(B,b1,...,bn). 15
Proof. Set Γ(x1,...,xn) = tpA(a), and consider the set of formulas Γ(c1,...,cn) ={ϕ(c1,...,cn)| ϕ(x1,...,xn)∈Γ(x1,...,xn)} in the language Ln. Evidently Γ(c1,...,cn) ⊆ Th(A,a1,...,an). Moreover, it is an easy exercise in changing bound variables to show that every sentence in Th(A,a1,...,an) is logically equivalent to a sentence in Γ(c1,...,cn).  
We define the logic topology on the space of n-types Sn(T) so that the bijection by which we identify Sn(T) with S0(Tn) is a homeomorphism, when we put the logic topology defined above on S0(Tn). That is, the basic open sets for the logic topology on Sn(T) are the sets of the form [ϕ(x1,...,xn)] ={Γ(x1,...,xn)∈ Sn(T)| ϕ ∈Γ} where ϕ(x1,...,xn) is any L-formula whose free variables are among x1,...,xn. The following result is immediate from Corollary 2.10. 2.16. Proposition. Let T be a satisfiable L-theory and n ≥ 0. The space Sn(T) with the logic topology is a totally disconnected, compact Hausdorff space. Its closed sets are the sets of the form {Γ0 ∈ Sn(T)|Γ⊆Γ0} where Γ is a set of L-formulas whose free variables are among x1,...,xn such that Γ⊇ T. Moreover, the clopen subsets of Sn(T) in this topology are exactly the sets of the form [ϕ(x1,...,xn)], where ϕ(x1,...,xn) is an L-formula whose free variables are among x1,...,xn. Furthermore, two L-formulas ϕ(x1,...,xn) and ψ(x1,...,xn) are equivalent over T iff the basic open sets [ϕ(x1,...,xn)] and [ψ(x1,...,xn)] are equal. Types over a set of parameters Later we will need the formalism of n-types over X, where X is a subset of a model A of an L-theory T. In such a situation, we take TX to be Th((A,a)a∈X); thus TX is a complete L(X)-theory. It specifies the elementary properties of elements of X within a model A of T. (The model A is arbitrary except that X ⊆ A and (A,a)a∈X |= TX. Note that any model of TX is isomorphic to an L(X)-structure of the form (A,a)a∈X, where A |= T and X ⊆ A.) 2.17. Definition. An n-type over X for the theory T is an n-type in L(X) that is consistent with TX. The space of all n-types over X for the theory T, namely the space Sn(TX), will be denoted by Sn(X) if the theory T and model A containing X are understood. 2.18. Fact. Let T be an L-theory, A a model of T, and X a subset of A. Let Γ(x1,...,xn) be an n-type in L(X). Then Γ ∈ Sn(X) iff Γ is finitely satisfiable in the given structure (A,a)a∈X. 16
An application of type spaces To close this chapter we give an application of the topology of type spaces that will be used later (for example, when we consider Quantifier Elimination). Let T be a satisfiable L-theory and n ≥ 0. Let Σ be a nonempty set of L-formulas whose free variables are among x1,...,xn. Assume that Σ is closed under disjunction and conjunction (up to equivalence over T). 2.19. Proposition. Let ϕ(x1,...,xn) be an L-formula. The following are equivalent: (1) T |= ϕ or T |=¬ϕ or T |= ϕ ↔ σ for some formula σ ∈Σ. (2) For every T1,T2 ∈ Sn(T), if ϕ ∈ T1 and ¬ϕ ∈ T2, then there exists σ ∈Σ such that σ ∈ T1 and ¬σ ∈ T2. Proof. (1 ⇒ 2): If ϕ ∈ T1 and ¬ϕ ∈ T2, then neither T |= ϕ nor T |= ¬ϕ hold. Thus there exists σ ∈Σ such that T |= ϕ ↔ σ. It follows that σ ∈ T1 and ¬σ ∈ T2. (2 ⇒ 1): Assume that condition (2) holds and that neither T |= ϕ nor T |= ¬ϕ. Let K denote the clopen set [ϕ] in Sn(T), with its complement denoted by Kc. Note that both K and Kc are nonempty. Let S be the family of basic open sets of the form [σ] where σ ∈Σ. We will first show that K is the union of a family of basic open sets from S. Fix T1 ∈ K; condition (1) implies that there is a subset Σ0 of Σ such thatS{[¬σ0]| σ0 ∈Σ0} contains Kc as a subset and does not have T1 as an element. Since Kc is compact, the set Σ0 can be taken to be finite. Since Σ is closed under conjunction, there is a single formula σ0 from Σ such that Kc ⊆[¬σ0] and T1 ∈[σ0]. That is, T1 ∈[σ0]⊆ K. Therefore K is the union of a family of basic open sets from S. Since K is compact, it is a finite union of such basic open sets. Since Σ is closed under disjunction, there must be a single formula σ ∈ Σ such that K = [σ], and therefore T |= ϕ ↔ σ, as desired.  
Exercises 2.20. Show that the Compactness Theorem (Theorem 1.10) can be derived from Corollary 2.7 by a trivial argument. 2.21. Let T be an L-theory and let K be the set of all L-structures that are not models of T. Show that T is equivalent to a finite L-theory iff K is axiomatizable.
17
3. Elementary Maps
3.1. Definition. Let A,B be L-structures and let f be a function from a subset X of A into B. We say f is elementary (with respect to A,B) if for every L-formula ϕ(x1,...,xm) and every a1,...,am ∈ X A |= ϕ[a1,...,am]⇔ B |= ϕ[f(a1),...,f(am)]. If the domain of the function f is all of A and f is elementary with respect to A,B, then f is called an elementary embedding from A into B. 3.2. Fact. Let A,B,X,f be as in the Definition. The function f is elementary with respect to A,B if and only if (A,a)a∈X ≡ (B,f(a))a∈X. In particular, if there exists a function f: X → B that is elementary with respect to A,B for some subset X of A (including the empty set), then A ≡ B. 3.3. Fact. If f is an elementary function then f must be 1-1. Moreover, if f is an elementary embedding of A into B, then f is an embedding of A into B. 3.4. Definition. Let A,B be L-structures and suppose A ⊆ B. We say A is an elementary substructure of B and write A B if the inclusion map is an elementary embedding from A into B. In this case we also refer to B as an elementary extension of A and write B  A. The importance of elementary extensions for model theoretic arguments is indicated by the following remark. 3.5. Remark. Let L be a first order language and let A,B be L-structures that satisfy A  B. An important property of elementary extensions is that each relation R on A that is definable in A has a canonical extension R0 to a relation on B that is definable in B. To obtain this extension, take any L-formula ϕ(x1,...,xm,y1,...,yn) and any b1,...,bn ∈ A such that R ={(a1,...,am)∈ Am | A |= ϕ[a1,...,am,b1,...,bn]} and set R0 ={(a1,...,am)∈ Bm | B |= ϕ[a1,...,am,b1,...,bn]} It is an easy exercise to show that R0 does not depend on the specific L(A)-formula ϕ(x1,...,xm,b1,...,bn) used in defining R. Note that the parameters needed to define R0 in B are exactly the same as the parameters used to define R in A. This correspondence between (certain) relations R on A and R0 on B preserves all structural properties that can be expressed in first order logic. For example: it is an isomorphism with respect to Boolean operations and projections; also, if R is the graph of a function, then so is R0. 18
3.6. Facts. (a) Let g be an isomorphism from A onto B, and let f be any restriction of g to a subset X of A. Then f is elementary with respect to A,B. (b) If g is elementary (with respect to A,B) and f is elementary (with respect to B,C), and if the range of g is contained in the domain of f, then the composition f ◦g is elementary (with respect to A,C). (c) If f is elementary (with respect to A,B), then f−1 is elementary (with respect to B,A). 3.7. Fact. Let I be an index set and U be an ultrafilter on I. Fix a first order language L and an L-structure A. Consider the ultrapower AI/U of A. Define a function δ on A by setting δ(a) = ga/U, where ga is the constant function with ga(i) = a for all i ∈ I. Then δ is an elementary embedding from A into AI/U. (This is called the diagonal embedding; often one identifies a with δ(a) for each a ∈ A and thereby regards A as an elementary substructure of AI/U.) The following result gives a useful tool for showing that A is an elementary substructure of B. Note that the condition in this Theorem refers to truth of formulas only in the structure B. 3.8. Theorem (Tarski-Vaught Test for ). Let B be an L-structure and suppose A ⊆ B. Then A is the underlying set of an elementary substructure of B if and only if for every formula ψ(x1,...,xm,y) in L and every sequence a1,...,am in A, if B |= ∃yψ[a1,...,am], then there exists b ∈ A such that B |= ψ[a1,...,am,b]. Proof. (⇒) This follows immediately from the definition of elementary substructure. (⇐): Suppose A and B satisfy the given conditions. We first need to show that A is the underlying set of a substructure of B. If c is a constant symbol in L, apply the given conditions on A,B to the formula ψ(y) equal to y = c; this shows that cB ∈ A. If F is an m-ary function symbol in L, apply the given conditions on A,B to the formula ψ(y) equal to F(x1,...,xm) = y; this shows that A is closed under the function FB. Hence there exists A ⊆ B whose underlying set is A. We need to show that for any formula ϕ(x1,...,xm) and any a1,...,am ∈ A, A |= ϕ[a1,...,am]⇔ B |= ϕ[a1,...,am]. The proof is by induction on the formula ϕ. By changing bound variables if necessary, we may restrict attention to formulas ϕ(a1,...,xm) that have no bound occurrences of any xj, j = 1,...,m. In the basis step ϕ is an atomic formula; the displayed equivalence follows from the assumption that A is a substructure of B. In the induction step, the cases of propositional connectives are trivial. In the remaining case ϕ is of the form∃yψ(x1,...,xm,y), where the statement 19
to be proved is assumed to be true for ψ and y is not among x1,...,xm. Then we have: A |= ϕ ⇔ A |=∃yψ[a1,...,am] ⇔ A |= ψ[a1,...,am,b] for some b ∈ A ⇔ B |= ψ[a1,...,am,b] for some b ∈ A ⇔ B |=∃yψ[a1,...,am] ⇔ B |= ϕ[a1,...,am] In the third equivalence we used the induction hypothesis and in the fourth we used the hypothesis of the implication we are proving as well as the fact that y is distinct from all of x1,...,xm.   3.9.Facts (UnionsofChains). Let(I,≤)bealinearlyorderedset. Foreach i ∈ I let Ai be an L-structure, and suppose this indexed family of structures is a chain. That is, for each i,j ∈ I, we suppose i ≤ j ⇒ Ai ⊆ Aj. (1) There is a well defined structure whose universe is the union of the sets Ai and which is an extension of each Ai; moreover, such a structure is unique. (For obvious reasons, this structure is called the union of the given chain of structures.) (2) If, in addition, Ai  Aj holds whenever i,j ∈ I and i ≤ j, then the union of this chain of structures is an elementary extension of each Ai. (In thissituationwereferto(Ai | i ∈ I)asanelementary chain of L-structures. A useful way of proving that functions are elementary is the back-and-forth method, which we now describe. 3.10. Definition. Let A,B be L-structures and let F be a nonempty family of functions. We say F is a local isomorphism from A onto B if it has the following properties: • Each function in F is an embedding from a substructure of A into B. • (“back”) For each f ∈ F and each b ∈ B there is some g ∈ F such that g extends f and b is in the range of g. • (“forth”) For each f ∈ F and each a ∈ A there is some g ∈ F such that g extends f and a is in the domain of g. We say A is locally isomorphic to B if there is a local isomorphism from A onto B.
To work effectively with local isomorphisms, we need some facts about maps between substructures. 3.11. Lemma. Let A,B be L-structures and let f be an embedding of a substructure of A into B. Then (1) The range of f is a substructure of B. (2) For each L-term t(x1,...,xn) and each a1,...,an in the domain of f, tB(f(a1),...,f(an)) = f(tA(a1,...,an)).
(3) For each quantifier-free L-formula ϕ(x1,...,xn) and each a1,...,an in 20
the domain of f, B |= ϕ[f(a1),...,f(an)] ⇔ A |= ϕ[a1,...,an]. Proof. (1) We need to show that cB is in the range of f for any constant symbol c of L and that the range of f is closed under the application of FB for any function symbol F of L. If c is a constant symbol of L, then cA is in the domain of f and we have cB = f(cA). If F is an n-ary function symbol of L and a1,...,an are in the domain of f (so f(a1),...,f(an) are arbitrary elements of the range of f), we have FB(f(a1),...,f(an)) = f(FA(a1,...,an)), which is in the range of f. (2) This is proved by induction on terms. (3) This is proved by induction on formulas. Part (2) yields the base case, in which atomic formulas are treated. The induction steps for propositional connectives are trivial.   3.12. Proposition. Let A,B be L-structures and let F be a local isomorphism from A onto B. Then each function in F is elementary with respect to A,B. In particular, A ≡ B. Proof. Let ϕ(x1,...,xn) be an L-formula, f a function in F, and a1,...,an elements of the domain of f. We must prove A |= ϕ[a1,...,an] ⇐⇒ B |= ϕ[f(a1),...,f(an)]. This is done by induction on ϕ(x1,...,xn). In the base case of the induction ϕ is an atomic formula, and the desired equivalence is contained in Lemma 3.11(3). The induction steps for propositional connectives are trivial. The induction steps for quantifiers follow from the “back-and-forth” properties satisfied by F. The final statement follows, because F is nonempty.  
When constructing local isomorphisms, the following notation and result are often useful. 3.13. Notation. Let A be an L-structure and X a nonempty subset of A. We denote by hXiA the substructure of A that is generated by X. 3.14. Lemma. Let A,B be L-structures. Let J be a nonempty set and consider two functions α: J → A, β: J → B. Let (xj | j ∈ J) be a family of distinct variables. Suppose that for any quantifier-free formula ϕ(xj | j ∈ J) whose variables are among (xj | j ∈ J) we have A |= ϕ[α(j)| j ∈ J] ⇔ B |= ϕ[β(j)| j ∈ J]. Then there exists an embedding f from h{α(j)| j ∈ J}iA into B such that f(α(j)) = β(j) for all j ∈ J. Moreover, f is unique with these properties and its range is h{β(j)| j ∈ J}iB. 21
Proof. The underlying set of h{α(j) | j ∈ J}iA consists exactly of those elements of A that can be written in the form tA(α(j) | j ∈ J) where t(xj | j ∈ J) is any L-term whose variables are among (xj | j ∈ J). If t1,t2 are two such terms and tA 1 (α(j) | j ∈ J) = tA 2 (α(j) | j ∈ J), thenour assumptions yield that tB 1 (β(j) | j ∈ J) = tB 2 (β(j) | j ∈ J). (Considerthe quantifier-free formula t1 = t2.) Thus we may define a function f on h{α(j)| j ∈ J}iA by f(tA(α(j)| j ∈ J)) = tB(β(j)| j ∈ J) where t ranges over the L-terms whose variables are among (xj | j ∈ J). It is routine to show that this f has the desired properties.   Theory of dense linear orderings without endpoints We illustrate the use of these ideas by treating the theory of dense linear orderings without endpoints. Let L denote the language whose only nonlogical symbol is a binary predicate symbol <. Let DLO denote the theory of dense linear orderings without maximum or minimum element, formulated as a (finite) set of L-sentences. 3.15. Example. Each L-formula is equivalent in DLO to a quantifier-free L-formula.
Proof. We apply Proposition 2.19. Fix an L-formula ϕ(x1,...,xm). Let Σ be the set of quantifier-free L-formulas whose free variables are among x1,...,xm. We will verify condition (2) of Proposition 2.19. To that end, consider two dense linear orderings without endpoints (A,<) and (B,<) and elements a1,...,am ∈ A,b1,...,bm ∈ B. We assume that (A,<) |= ϕ[a1,...,am] and that every quantifier-free L-formula satisfied in (A,<) by a1,...,am is satisfied in (B,<) by b1,...,bm. We need to show (B,<) |= ϕ[b1,...,bm]. Let F be the set of all order preserving functions from a finite subset of A into B. An easy argument shows that A is a local isomorphism from (A,<) onto (B,<). Our assumptions ensure that there exists f ∈ F such that f is defined on {a1,...,am} and satisfies f(ai) = bi for all i = 1,...,m. By Proposition 3.12, the function f is elementary with respect to (A,<) and (B,<).  
Note that we have proved in passing that every two models of DLO are elementarily equivalent, since there is a local isomorphism from one onto the other. Hence DLO is complete. Theory of equality We complete this chapter by analyzing the simplest logical theory, which is the theory of equality. Let L denote the language of =, without any nonlogical symbols. For each n ≥0 let σn be a sentence in L that expresses the statement that the universe has at most n elements (so ¬σ0 is logically 22
valid). For each n ≥1 let Tn be the theory in L axiomatized by σn∧¬σn−1 and let T∞ be the theory axiomatized by the set {¬σn | n ≥ 1}. Thus Tn is the theory of sets of size n (n ≥1) and T∞ is the theory of infinite sets. 3.16. Example (Theories in the language of equality). (i) Each formula in the pure language of = is logically equivalent to a Boolean combination of quantifier free formulas and the sentences σn for n ≥1. (ii) The complete theories in the language of = are equivalent to T∞ and Tn for n ≥ 1. For each such theory T, every formula in the language of = is equivalent in T to a quantifier free formula.
Proof. (i) We apply Proposition 2.19. Fix a formula ϕ(x1,...,xm) in the language of =. Let Σ be the set of all Boolean combinations of quantifier free formulas whose variables are among x1,...,xm and the sentences σn for n ≥ 1. We want to verify condition (2) of Proposition 2.19. To that end, consider two sets A,B and elements a1,...,am ∈ A,b1,...,bm ∈ B. We assume that A |= ϕ[a1,...,am] and that every formula in Σ that is satisfied by a1,...,am in A is satisfied by b1,...,bm in B. We need to show B |= ϕ[b1,...,bm]. Our hypotheses ensure that for all n ≥ 1 we have A |= σn ⇐⇒ B |= σn. Therefore, either A and B have the same finite cardinality or both A,B are infinite. Moreover, we also have that for each 1 ≤ i < j ≤ m, ai = aj ⇐⇒ bi = bj. Therefore there is a bijection f from {a1,...,an} onto {b1,...,bn} such that f(ai) = bi for all i = 1,...,n. Let F be the set of all 1-1 functions g that extend f and map a finite subset of A into B. It is easy to check that F is a local isomorphism from (A,a1,...,an) onto (B,b1,...,bn). By Proposition 3.12, (A,a1,...,an) ≡ (B,b1,...,bn), and hence B |= ϕ[b1,...,bm]. (ii) For finite n, any two models of Tn are isomorphic, hence elementarily equivalent, so Tn is complete in these cases. On the other hand, T∞ has only infinite models; the back-and-forth argument used to prove (i) shows that any two infinite sets are elementarily equivalent, which proves that T∞ is also complete. If T is any complete theory in the language of equality, and A is one of its models, then A is a model of T∞ or of Tn for some n ≥ 1, depending on the cardinality of A. Therefore T is equivalent to Th(A), which contains one of these theories, say Tj where j ≥1 or j =∞. But we showed that each such Tj is complete, from which it follows easily that T and Tj are equivalent.   The previous result allows one to show that if T is the empty theory in the language L of equality, then the space S0(T) consists of a sequence of points (Tn | n ≥ 1) that are isolated, together with a point T∞ to which this sequence converges.
23
Exercises 3.17. Let A be an L-structure and X a nonempty subset of A. The diagram of X in A, denoted by DiagX(A), is the set of all quantifier-free L(X)sentences that are true in (A,a)a∈X. Suppose X is a set of generators for A and B is another L-structure. Show that there is a 1-1 correspondence between embeddings of A into B and expansions of B that are models of DiagX(A). 3.18. Let A be an L-structure and X a nonempty subset of A. The elementary diagram of X in A, denoted by EDiagX(A), is the set of all L(X)sentences that are true in (A,a)a∈X. Suppose X is a set of generators for A and B is another L-structure. Show that there is a 1-1 correspondence between elementary embeddings of A into B and expansions of B that are models of EDiagX(A). 3.19. Let I be an index set and U an ultrafilter on I. Let (Ai | i ∈ I) and (Bi | i ∈ I) be families of L-structures. If Ai can be elementarily embedded in Bi for all i ∈ I, show that ΠUAi can be elementarily embedded in ΠUBi. 3.20. Let A be an infinite L-structure and κ an infinite cardinal. Show that there exists an ultrapower of A that has cardinality at least κ. (Compare Corollary1.12.) Itfollowsthat A hasanelementaryextensionofcardinality at least κ. 3.21. Let A ⊆ B be L-structures. Suppose that for every finite sequence a1,...,am ∈ A and every b ∈ B there is an automorphism of B that fixes each element of a1,...,am and moves b into A. Show that A  B. 3.22. Let K be a field and let L be the first order language of vector spaces over K; the nonlogical symbols of L are a constant 0, a binary function symbol +, and a unary function symbol Fa for each a ∈ K. Given a Kvector space V , we regard V as an L-structure in the obvious way: 0 is interpreted by the identity element of V , + is interpreted by the addition of V , and each Fa is interpreted by the operation of scalar multiplication by a. Suppose W ⊆ V are infinite dimensional K-vector spaces. Use the previous exercise to prove that W  V . Use this result to show that any two infinite K-vector spaces are elementarily equivalent. 3.23. Let L be the language whose only nonlogical symbol is a binary predicate symbol <. Let A be an L-structure that is a dense linear ordering without endpoints. Let ϕ(x,y1,...,yn) be any L-formula (with x a single variable) and let a1,...,an ∈ A. Show that the definable set {a ∈ A | A |= ϕ[a,a1,...,an]} is the union of a finite number of open intervals (whose endpoints are in A∪{−∞,+∞}) and a finite subset of A. 3.24. Let L be the pure language of =, so L has no nonlogical symbols, and let σ be any L-sentence. Show that if σ is satisfiable, then σ is true in some finite set.
24
4. Saturated Models
In this chapter we prove that every satisfiable theory T has models that are rich, in a certain sense. This is the first of several such notions that turn out to be useful in model theory. (See Section 12.) 4.1. Definition. Let A be an L-structure and let κ be an infinite cardinal. We say that A is κ-saturated if the following condition holds: if X is any subset of A having cardinality < κ and Γ(x) is any 1-type in L(X) that is finitely satisfiable in (A,a)a∈X, then Γ(x) itself is satisfiable in (A,a)a∈X. 4.2. Facts. Let A be an L-structure and κ an infinite cardinal. (a) If A is infinite and κ-saturated, then the underlying set of A has cardinality at least κ. (b) If A is finite, then A is κ-saturated for every κ. (c) If A is κ-saturated and X is a subset of A having cardinality < κ, then the expansion (A,a)a∈X is also κ-saturated. Definition 4.1 refers only to realizing 1-types. The following result shows that κ-saturated structures realize partial types in a very rich way. 4.3. Theorem. Let κ be an infinite cardinal and suppose A is a κ-saturated L-structure. Suppose X ⊂ A has cardinality < κ. Let Γ(xj | j ∈ J) be a set of L(X)-formulas, where J has cardinality ≤ κ. If Γ is finitely satisfiable in (A,a)a∈X, then Γ is satisfiable in (A,a)a∈X. Proof. Let X,J, and Γ(xj | j ∈ J) be as in the statement of the Theorem. Extend Γ so that it is maximal among sets of L(X)-formulas with free variables among (xj | j ∈ J) that are finitely satisfiable in (A,a)a∈X. Let < be a well ordering of J such that the order type of (J,<) is the cardinal of J. As a consequence, each proper initial segment of (J,<) has cardinality < κ. For each k ∈ J let Γ≤k be the set of formulas in Γ whose free variables are among (xj | j ≤ k). Note that the maximality of Γ ensures that if ϕ is any L(X)-formula whose free variables are among (xj | j ≤ k), then either ϕ ∈ Γ≤k or ¬ϕ ∈ Γ≤k. Moreover, Γ≤k is closed under conjunction and under application of the existential quantifier ∃xk. We need to obtain a family (aj | j ∈ J) of elements of A that satisfies Γ in (A,a)a∈X; we do this by induction over the well ordering (J,<). Fix k ∈ J and suppose we have already obtained (aj | j < k) that satisfy all the formulas from Γ that have free variables among (xj | j < k). Let Γ0 be the result of substituting aj for all free occurrences of xj in Γ≤k, for all j < k. We see that Γ0 is a 1-type (with xk the allowed free variable) in L(X ∪{aj | j < k}) that is finitely satisfiable in (A,a)a∈X∪{aj|j<k}. Since X ∪{aj | j < k} has cardinality < κ and A is κ-saturated, there exists ak in A that satisfies Γ0 in (A,a)a∈X∪{aj|j<k}. It follows that the family (aj | j ≤ k) satisfies Γ≤k in (A,a)a∈X. 25
The result of this construction is a family (aj | j ∈ J) of elements of A such that for ea