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We recall and refine some definitions from the entries on classical logic and model theory. A signature is a set of individual constants, predicate symbols and function symbols; each of the predicate symbols and function symbols has an arity (for example it is binary if its arity is 2). Each signature K gives rise to a firstorder language, by building up formulas from the symbols in the signature together with logical symbols (including =) and punctuation.
If K is a signature, then a structure of signature K, say A, consists of the following items:
For example the field of real numbers forms a structure R whose elements are the real numbers, with signature consisting of the individual constant 0 to name the number zero, a 1ary function symbol  for minus, and two 2ary function symbols + and . for plus and times. At first sight we can't add a symbol to express 1/x, since all the named functions have to be defined on the whole domain of the structure, and there is no such real number as 1/0. But on second thoughts this is not a serious problem; any competent mathematician puts the condition ‘x is not zero’ before dividing by x, and so it never matters what the value of 1/0 is, and we can harmlessly take it to be 42. But most model theorists are uncomfortable with any kind of division by zero, so they stick with plus, times and minus.
If L is the firstorder language of signature K, then Tarski's modeltheoretic truth definition tells us when a sentence of L is true in A, and when an assignment of elements of A to variables satisfies a formula of L in A. Instead of talking of assignments satisfying a formula, model theorists often speak of the set of ntuples of elements of A that is defined by a formula φ (v_{1},…,v_{n}); the connection is that an ntuple (a_{1},…,a_{n}) is in the defined set if and only if the assignment taking each v_{i} to a_{i} satisfies the formula.
If φ is a sentence, we write
A φto mean that φ is true in A. If φ(v_{1},…,v_{n}) is a formula with free variables as shown, we write
A φ[a]to mean that the ntuple a is in the set defined by φ. (The entry on classical logic uses the notation ‘A,s φ’, where s is any assignment to all the variables of L that assigns to each variable v_{i} free in φ the ith element in the ntuple a.)
Two Lstructures that are models of exactly the same sentences of L are said to be elementarily equivalent. Elementary equivalence is an equivalence relation on the class of all Lstructures. The set of all the sentences of L that are true in the Lstructure A is called the complete theory of A, in symbols Th(A). A theory that is Th(A) for some structure A is said to be complete. (By the completeness theorem for firstorder logic, for which see the entry on classical logic, a theory is complete if and only if it is maximal syntactically consistent.) The two structures A and B are elementarily equivalent if and only if Th(A) = Th(B).
To continue the example of the field R of real numbers: It is often not at all obvious whether two given structures are or are not elementarily equivalent. One of the greatest achievements of the prehistory of model theory was Tarski's description in 1930 of Th(R) (which he published in full only after the war; see his book in the Bibliography below). This description implied among other things that the structures elementarily equivalent to R are exactly the realclosed fields, a class of fields which was already known to the algebraists in its own right.
When mathematicians introduce a class of structures, they like to define what they count as the basic maps between these structures. The basic maps between structures of the same signature K are called homomorphisms, defined as follows. A homomorphism from structure A to structure B is a function f from dom(A) to dom(B) with the property that for every atomic formula φ(v_{1},…,v_{n}) and any ntuple a = (a_{1},…,a_{n}) of elements of A,
A φ [a] ⇒ B φ [b]where b is (f(a_{1}),…,f(a_{n})). If we have ‘⇔’ in place of ‘⇒’ in the quoted condition, we say that f is an embedding of A into B. Since the language includes =, an embedding of A into B is always onetoone, though it need not be onto the domain of B. If it is onto, then the inverse map from dom(B) to dom(A) is also a homomorphism, and both the embedding and its inverse are said to be isomorphisms. We say that two structures are isomorphic if there is an isomorphism from one to the other. Isomorphism is an equivalence relation on the class of all structures of a fixed signature K. If two structures are isomorphic then they share all modeltheoretic properties; in particular they are elementarily equivalent.
If A and B are structures of signature K with dom(A) a subset of dom(B), and the interpretations in A of the symbols in K are just the restrictions of their interpretations in B, then we say that A is a substructure of B and conversely B is an extension of A. If moreover B has some elements that are not in A, we say that A is a proper substructure of B and B is an proper extension of A. If B is a structure and X is a nonempty subset of dom(B), then there is a unique smallest substructure of B whose domain contains all of X. It is known as the substructure of B generated by X, and we find it by first adding to X all the elements c^{B} where c are individual constants, and then closing off under the functions F^{B} where F are function symbols.
For example the substructure of the field R generated by the number 1 consists of 1, 0 (since it is named by the constant 0), 1+1, 1+1+1 etc., 1, 2 etc., in other words the ring of integers. (There is no need to close off under multiplication too, since the set of integers is already closed under multiplication.) If we had included a symbol for 1/x too, the substructure generated by 1 would have been the field of rational numbers. So the notion of substructure is sensitive to the choice of signature.
A φ(a_{1},…,a_{n}) ⇒ B φ(e(a_{1}),…,e(a_{ n})).We say that e is an elementary embedding of A into B if e is an elementary map and its domain is the whole domain of A. As the name implies, elementary embeddings are always embeddings.
If there is an elementary embedding from A to B then A and B are elementarily equivalent. On the other hand an embedding between elementarily equivalent structures, or even between isomorphic structures, need not be elementary. (For example, writing Z for the abelian group of the integers with signature consisting of 0 and +, the embedding from Z to Z that takes each integer n to 2n is an embedding, and of course Z is isomorphic to itself; but this embedding is not elementary, since 1 satisfies the formula ¬∃y(y + y = v_{1}), but 2 doesn't.)
We say that A is an elementary substructure of B, and B is an elementary extension of A, if A is a substructure of B and the inclusion map is an elementary embedding. It's immediate from the definitions that an elementary extension of an elementary extension of A is again an elementary extension of A.
Elementary embeddings are natural maps to consider within firstorder model theory. Around 1950 Abraham Robinson was impressed that maps between algebraic structures in general seem hardly ever to be elementary, whereas some important maps (such as embeddings between two algebraically closed fields, or between two realclosed fields) turn out to be elementary. He was also surprised to find that this fact about algebraically closed fields is another way of stating a celebrated theorem called the Hilbert Nullstellensatz. These observations of Robinson have had a huge effect on the development of model theory. In Robinson's terminology, a firstorder theory is modelcomplete if every embedding between models of the theory is elementary. This notion has found many uses, and it often appears in applications of model theory in algebra.
Elementary embeddings have a number of properties that make them useful. We have space for four.
The downward LoewenheimSkolem theorem:There is a proof of this in the entry on classical logic, using Skolem hulls. Note that λ must be infinite since every firstorder language has infinitely many formulas.
Suppose L is a firstorder language which has κ formulas, A is an Lstructure and λ is a cardinal which is at least κ but less than the cardinality of A. Suppose also that X is a set of at most λ elements of A. Then A has an elementary substructure which has cardinality exactly λ and contains all the elements in X.
The elementary chain theorem:The latter is a consequence of the compactness theorem in the next section.
Suppose that L is a firstorder language and A_{0}, A_{1}, … is a sequence (of any length) of Lstructures such that any structure in the sequence is an elementary substructure of all the later structures in the sequence. Then there is a unique smallest Lstructure B which contains all the structures in the sequence as substructures; this structure B is an elementary extension of all the structures in the sequence.The elementary amalgamation theorem:
Suppose L is a firstorder language, A is an Lstructure and B, C are two elementary extensions of A. Then there are an elementary extension D of B and an elementary embedding e of C into D such that (i) for each element a of A, e(a) = a, and (ii) if c is an element of C but not of A, then e(c) is not in B.
The upward LoewenheimSkolem theorem:There is a proof of this in the entry on classical logic. The name of the theorem is a little unfortunate, since the theorem was first proved by Tarski, and Skolem didn't even believe it (because he didn't believe in uncountable cardinals).
Suppose L is a firstorder language which has κ formulas, A is an Lstructure whose cardinality is an infinite cardinal μ, and λ is a cardinal which is at least as great as both κ and μ. Then A has an elementary extension whose cardinality is λ.
There is another proof using the elementary amalgamation theorem and the elementary chain theorem. The compactness theorem and the diagram lemma (see below) allow us to prove that A has a proper elementary extension A′. Now use A′ and again A′ for the structures B and C in the elementary amalgamation theorem. Then D as in the theorem is an elementary extension of A, and by (ii) in the theorem, it must contain elements that are not in A, so that it is a proper elementary extension. Repeat to get a proper elementary extension of D, and so on until you have an infinite elementary chain. Use the elementary chain theorem to find an elementary extension of A that sits on top of this chain. Keep repeating these moves until you have an elementary extension of A that has cardinality at least λ. Then if necessary use the downward LoewenheimSkolem theorem to pull the cardinality down to exactly λ. This kind of argument is very common in firstorder model theory.
If T is a firstorder theory, and every finite subset of T has a model, then T has a model.There is a proof of this theorem in the entry on classical logic. The theorem has several useful paraphrases. For example it is equivalent to the following statement:
Suppose T is a firstorder theory and φ is a firstorder sentence. If T φ then there is a finite subset U of T such that U φ.(See the entry Model Theory for the notion of modeltheoretic consequence. To derive the second statement from the first, note that ‘T φ’ is true if and only if there is no model of the theory T ∪ {¬ φ}.)
Anatolii Mal'tsev first gave the compactness theorem in 1938 (for firstorder logic of any signature), and used it in 1940/1 to prove several theorems about groups; this seems to have been the first application of model theory within classical mathematics. Leon Henkin rediscovered the theorem a few years later and gave some further applications. The theorem fails badly for nearly all infinitary languages.
If B′ is a model of the diagram of A, and B is B′ with the new constants removed from the signature, then there is an embedding of A into B.Namely, if an element of A is named by a new constant c, then map that element to the element of B′ named c. A variant of this lemma is used in the proof of the elementary amalgamation theorem.
Suppose L and M are firstorder languages, L ∪ M is the smallest firstorder language containing both L and M, and L ∩ M is the language consisting of all the formulas which are in both L and M. Suppose T is a theory in L, U is a theory in M, and no (L ∪ M)structure is both a model of T and a model of U. Then there is a sentence φ of L ∩ M which is true in all models of T and false in all models of U. (This sentence φ is called the interpolant.) Moreover every predicate symbol with a positive occurrence in φ has a positive occurrence in some sentence of T and a negative occurrence in some sentence of U, and conversely every predicate symbol with a negative occurrence in φ has a negative occurrence in some sentence of T and a positive occurrence in some sentence of U.There are several proofs of this theorem, and not all of them are modeltheoretic. Without the last sentence, the theorem is known as Craig's interpolation theorem, since William Craig proved this version a few years before Roger Lyndon found the full statement in 1959. As Craig noted at the time, his interpolation theorem gives a neat proof of Evert Beth's definability theorem, which runs as follows.
Suppose that L is a firstorder language and M is the firstorder language got by adding to L a new predicate symbol R. Suppose also that T is a theory in M. We say that T implicitly defines R if it is false that there are two Mstructures which are models of T, have the same elements and interpret all the symbols of L in the same way but interpret the symbol R differently. We say that T defines R explicitly if there is a formula φ(x_{1},…,x_{n}) of L such that in every model of T, the formulas φ and R(x_{1},…,x_{n}) are satisfied by exactly the same ntuples (a_{1},…,a_{n}) of elements. It is easy to see that if T defines R explicitly then it defines R implicitly. (This fact is known as Padoa's method; Padoa used the failure of implicit definability as a way of proving the failure of explicit definability.) Beth's theorem is the converse:
Suppose that L, M, R and T are as above. If T defines R implicitly then T defines R explicitly.
Suppose L is a firstorder language which has countably many formulas. Suppose T is a complete theory in L, and Φ is a set of formulas of L which all have the free variable x. Finally suppose that every model of T contains an element which satisfies all the formulas in Φ. Then there is a formula ψ(x) of L such that in every model of T there is an element satisfying ψ, and every element that satisfies ψ in any model of T also satisfies all the formulas in Φ. (In other words, there is a finite reason why the ‘type’ Φ can't be ‘omitted’ in any model of T.)
This theorem, which goes back to the mid 1950s, very definitely depends on the language being firstorder and countable. It has several useful generalisations, for example modeltheoretic forcing, which is an analogue of the forcing construction in set theory. In fact the games used for modeltheoretic forcing (see the entry on logic and games) can be adapted to prove the omitting types theorem too. There are similar but more complicated theorems for uncountable firstorder languages; some of these can be paraphrased as omitting types theorems for infinitary languages.
Let T be a theory consisting of strict universal Horn sentences. Then T has a model A with the property that for every model B of T there is a unique homomorphism from A to B. (Such a model A is called an initial model of T. It is unique up to isomorphism.)This theorem is a generalisation, due to Mal'tsev, of a grouptheoretic construction called construction by generators and relations. It is the main idea behind algebraic specification, which is one approach to the specification of systems in computer science. The required behaviour of the system is written down as a set of strict universal Horn sentences, and then the initial model of these sentences is an abstract version of the required system.
(a,b) is in P^{C} if and only if a is in P^{A} and b is in P^{B}.The structures A and B are called the factors of A X B. In the same way we can form products of any number of structures. If all the factors of a product are the same structure A, the product is called a power of A. A theorem called the FefermanVaught theorem tells us how to work out the complete theory of the product from the complete theories of its factors.
This construction has some variants. We can define an equivalence relation on the domain of a product C, and then take a structure D whose elements are the equivalence classes; the predicate symbols are interpreted so as to make the natural map from dom(C) to dom(D) a homomorphism. In this case the structure D is called a reduced product of the factors of C. It is a reduced power of A if all the factors are equal to A; in this case the diagonal map from A to D is the one got by taking each element a to the equivalence class of the element (a,a,…).
Suppose we use a set I to index the factors in a product C. An ultrafilter over I is a set U of subsets of I with the properties
If U is an ultrafilter then the diagonal map from A to Uprod A is an elementary embedding.If the ultrafilter U is nonprincipal, i.e. contains no finite sets, then the diagonal map is not onto the domain of Uprod A, and in fact Uprod A is generally much larger than A. So we have a way of constructing large elementary extensions. The axiom of choice guarantees that every infinite set has many nonprincipal ultrafilters over it. Ultrapowers are an essential tool for handling large cardinals in set theory (see the entry on set theory).
A remarkable (but in practice not terribly useful) theorem of Saharon Shelah tells us that a pair of structures A and B are elementarily equivalent if and only if they have ultrapowers that are isomorphic to each other.
B φ[b,d] ⇔ B φ[c,d].We say that A is saturated if whenever X is a set of elements of A, of cardinality less than that of A, and B is any elementary extension of A, we always have that every element of B has the same type over X as some element already in A.
This rather heavy definition gives little clue how useful saturated structures are. If every structure had a saturated elementary extension, many of the results of model theory would be much easier to prove. Unfortunately the existence of saturated elementary extensions depends on features of the surrounding universe of sets. There are technical ways around this obstacle, for example using weakenings of the notion of saturation. We have two main ways of constructing saturated elementary extensions. One is by ultrapowers, using cleverly constructed ultrafilters. The other is by taking unions of elementary chains, generalising the proof we gave for the upward LoewenheimSkolem theorem.
The existence of partially saturated elementary extensions of the field R of real numbers is the main technical fact behind Abraham Robinson's nonstandard analysis. See Section 4 of the entry on model theory for more information on this. Though model theory provided the first steps in nonstandard analysis, this branch of analysis rapidly became a subject in its own right, and its links with firstorder model theory today are rather slim.
Now there is a heuristic principle that many people have used, though it seems to have no simple formulation. I suggest ‘Few is beautiful’. The principle says that if a firstorder theory T constrains its models (of a particular cardinality) to be all similar to each other, this can only be because the models of T have few irregularities and asymmetries. So there should be a good structural description of these models. One should expect that they are ‘good structures’ from the point of view of classical mathematics. As a first step, one easily sees from the upward and downward LoewenheimSkolem theorems that if T is κcategorical for some κ at least as large as the number of formulas in the language of T, then T must be a complete theory. From now on, T is a complete theory with infinite models; and for simplicity we shall assume that the language of T is countable.
In 1954 Jerzy Los announced that he could only find three kinds of example of theories T that are κcategorical. Namely:
This question of Los was a tremendous stimulus to research, and it led to a classic paper of Michael Morley in 1965 which showed that Los's three possibilities are in fact the only ones. One central idea of Morley's analysis was that models of an uncountably categorical theory have the smallest possible number of types of element; this led directly to the branch of model theory called stability theory, which studies theories that have a limited number of element types. These theories have the remarkable property that every infinite indiscernible sequence in any of their models is indiscernible under any linear ordering whatever; so these sequences are a kind of generalisation of bases of vector spaces. Another idea implicit in Morley's work, but much clarified by later work of John Baldwin and Alistair Lachlan, was that in any model of an uncountably categorical theory there is a central core (called a strongly minimal set) which carries a dependence relation obeying similar laws to linear dependence in vector spaces. In terms of this dependence relation one can define a dimension for the model, and what remains of the model outside the core is so closely tied to the core that the dimension determines the model up to isomorphism.
Saharon Shelah developed Morley's ideas with tremendous resourcefulness and energy. His main aim was to stretch the ‘Few is beautiful’ idea by showing that there are clear dividing lines between kinds of theory T. On one side of a dividing line are theories with some good structural property that forces the number of nonisomorphic models of a given cardinality to be small. On the other side, every theory has (for example) two models of the same cardinality that are not isomorphic but are extremely hard to tell apart. Shelah coined the name classification theory for this research, though the name never caught on. The text of Lascar listed below is an elegant introduction to this whole programme, from Los to Shelah. Meanwhile Shelah himself has extended it far beyond firstorder logic. Even in the firstorder case, Shelah had to invent new settheoretic techniques (such as proper forcing) to carry out his constructions.
Partly because of the difficulty of communications between Siberia and the West, these results of Zil'ber took some time to digest, and in part they had to be rediscovered in the West. But when the message did finally get through, the result was a new branch of model theory which has come to be known as geometric model theory. The programme is broadly to classify structures according to (a) what groups or fields are interpretable in them (in the sense sketched in the entry on model theory) and (b) whether or not the structures have ‘modular geometries’; and then to use this classification to solve problems in model theory and geometry. Since the mid 1980s the leader of this research has been Ehud Hrushovski. In the early 1990s, using joint work with Zil'ber, Hrushovski gave a modeltheoretic proof (the first complete proof to be found) of the geometric MordellLang conjecture in all characteristics; this was a conjecture in classical diophantine geometry. The book edited by Bouscaren in the references below is devoted to Hrushovski's proof and the necessary background in model theory. Both (a) and (b) are fundamental to Hrushovski's argument.
Every set of elements definable by a firstorder formula is a finite union of open intervals with named endpoints, together with some finite set of elements.A linearly ordered structure with this property is said to be ominimal. (The idea of the name is that ominimality is an analogue of Morley's ‘strong minimality’, in a form that makes sense for structures that carry a linear ordering, whence ‘o’ for ordering.)
In 1982 Lou van den Dries showed that the fact that the field of real numbers is ominimal gives a large amount of useful information about the definable sets of higher dimension, such as the family of definable subsets of the real plane. Soon after this, Julia Knight, Anand Pillay and Charles Steinhorn noticed that if a structure A is ominimal, then so is any structure elementarily equivalent to A, and that Van den Dries' analysis of higherdimensional definable set applies to all these structures. These results led to much activity on the frontier between model theory and function theory. Several old problems from model theory and function theory were solved. Alex Wilkie showed that the field of real numbers with a symbol for exponentiation is ominimal and has a modelcomplete complete theory, and thereby gave a positive answer to Tarski's old problem of whether this structure allows a quantifier elimination (though his method was very far from the syntactic analysis that Tarski had in mind). We now know a wide range of ways of adding interesting features to the field of real numbers in such a way that the resulting structure is still ominimal (and hence in some sense mathematically tractable). Van den Dries has urged that ominimal structures provide a good setting for developing the ‘tame topology’ programme of Alexander Grothendieck.
Wilfrid Hodges W.Hodges@qmw.ac.uk 
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