模型论入门
    无穷小微积分是模型论的小分支。当今，国际上，模型论发展很快，我们必须补课。
    站在模型论的角度，极限微积分的未来就是无穷小微积分。这是事物发展的逻辑。
    我们苦苦求索，终于找到一篇关于模型论的入门教材。请读者参阅本文附件。
袁萌  陈启清   12月16日
附：
Introduction to model theory
Wilfrid Hodges Queen Mary, University of London
This course is an introduction in two senses. First, it is for people who haven’t studied model theory before, though I trust most people in the class will have heard of it. I have tried to make most of the material accessible to people coming from any of the main disciplines where model theory is used: mathematics, philosophy, computer science, linguistics, cognitive psychology. And second, I have tried to start where model theory starts. This course is an introduction to the basic notions rather than to the most impressive achievements. Since we are discussing the starting points, I dip back into history perhaps more than is usual in introductory courses.
Unfortunately I haven’t yet found time to put together complete notes. What you have here is a provisional schedule of topics, and some historical and other material that I expect to refer to when we discuss these topics.
1
DAY ONE: Schemas, formulas, models, structures
The texts quoted here illustrate important steps in the early development of the notion of a model. From David Hilbert, Foundations of geometry (1899), [6] §9: Consider a pair of numbers (x,y) from the field Ω [the field of algebraic numbers] as a point and the ratios (u : v : w) of any three numbers from Ω as a line provided u,v are not both zero. Furthermore, let the existence of the equation
ux + vy + w =0
mean that the point (x,y) lies on the line (u : v : w). Thereby, as is easy to see, Axioms I, 1-3 and IV are immediately satisfied. [Axiom I,1 said ‘For every two points A,B there exists a line a that contains each of the points A,B’.]
From Oswald and Young, Projective Geometry II (1918) [17]:
This is represented in fig. 19 and may be realized in a model by cutting out a rectangular strip of paper, giving it a half twist, and pasting together the two ends. ... To complete the model it would be necessary to bring the two edges labeled β in fig. 18 into coincidence. This, however, is not possible in a finite three-dimensional figure without letting the surface cut itself. (Footnote: Plaster models showing this surface are manufactured by Martin Schilling of Leipzig.)
2
From Alfred Tarski, Foundations of the geometry of solids (1929) [13] pp. 28f.:
THEOREM A. The postulate system of the geometry is categorical. This means that, given any two models of this postulate system, we can establish a one-to-one correspondence between solids of the first model and those of the second ...
From Richard Dedekind’s appendices to his edition of Dirichlet’s Zahlentheorie (1871) [4] p. 424:
Unter einem K¨orper wollen wir jedes System von unendlich vielen reelllen oder complexen Zahlen verstehen, welches in sich so abgeschlossen und vollst¨andig ist, dass die Addition, Subtraction, Multiplication und Division von je zwei dieser Zahlen immer wieder eine Zahl desselben Systems hervorbringt.
And on p. 462 of the same work:
Das System E aller Hauptideale besitzt folgende fundamentale Eigenschaften. I. Jedes Product aus zwei Idealen in E ist wieder ein Ideal in E. ... II. Sind e und ee0 Ideale in E, so ist auch e0 ein Ideal in E. ... III. Ist a ein beliebiges Ideal, so giebt es immer ein Ideal m der Art, dass am ein Ideal in E wird.
3
DAY TWO: Truth definitions
From Tarski and Vaught, Arithmetical extensions of relational systems (1957) [16]: A relational system is a sequence R = hA,R0,...,Rξ,...iξ<α in which A is a non-empty set and each Rξ is a relation among the elements of A; α is called the order of R. Instead of “relational system” we shall sometimes say simply “system”. Elements of the set A are referred to as elements of the system R; the system R is called infinite if the set A is infinite, and we speak of the power of R meaning the power of A. In most discussions it is tacitly assumed that all the systems hA,R0,...,Rξ,...iξ<α involved are similar, i.e., that they all have the same order α and, for each ξ<α, all relations Rξ have the same rank nξ. ...Inorder to simplify the notation we shall henceforth restrict ourselves to an explicit discussion of relational systems hA,Ri formed by a non-empty set A and a single ternary relation R. ... The notions of satisfaction and truth will play an essential part in our discussion and therefore will be defined here in a formal way. The definition of satisfaction is given in a recursive form (cf. [Tarski’s paper on the Concept of Truth, [13]]): DEFINITION 1.1. We say that x satisfies φ in a relational system R = hA,Ri if x ∈ A(ω), φ is a formula, and one of the following five conditions holds: (i) φ is of the form vm ≡ vn, where m and n are natural numbers, and xm = xn; (ii) φ is of the form P(vm,vn,vp), where m,n, and p are natural numbers, and hxm,xn,xpi∈R; (iii) φ is of the form ∼ ψ, where ψ is a formula which is not satisfied by x; (iv) φ is of the form ψ0∧ψ00, where ψ0 and ψ00 are formulas which are both satisfied by x; (v) φ is of the formVvkψ, where k is a natural number, ψ is a formula, and there is an element a ∈ A such that x(k/a) satisfies ψ.
4
From Rudolf Carnap, Introduction to Symbolic Logic and its Applications (1958), [2] p. 99:
Rules of designation
Primitive sign Intension Extension ‘a’ (the individual concept) (the thing) moon moon ‘b’ (the individual concept) (the thing) sun sun ‘c’ (the individual concept) (the thing) Africa Africa ‘P’ the property the class of being spherical of spherical things ‘Q’ the property the class of being blue of blue things ‘R’ the relation the class of pairs x,y such greater than that x is greater than y
5
DAY THREE: Model-theoretic entailment
From Alfred Tarski, On the concept of logical consequence (1936) [13] p. 414ff:
Certain considerations of an intuitive nature will form our startingpoint. Consider any class K of sentences and a sentence X which follows from the sentences of this class. From an intuitive standpoint it can never happen that both the class K consists only of true sentences and the sentences X is false. Moreover, since we are concerned here with the concept of logical, i.e. formal, consequence, and thus with a relation which is to be uniquely determined by the form of the sentences between which it holds, this relation cannot be influenced in any way by empirical knowledge, and in particular by knowledge of the objects to which the sentence X or the sentences of the class K refer. The consequence relation cannot be affected by replacing the designations of the objects referred to in these sentences by the designations of any other objects. The two circumstances just indicated, which seem to be very characteristic and essential for the proper concept of consequence, may be jointly expressed in the following statement:
(F) If, in the sentences of the class K and in the sentence X, the constants—apart from purely logical constants—are replaced by any other constants (like signs being everywhere replaced by like signs), and if we denote the class of sentences thus obtained from K by ‘K0’, and the sentence obtained from X by ‘X0’, then the sentence X0 must be true provided only that all sentences of the class K0 are true.
... The condition (F) could be regarded as sufficient for the sentence X to follow from the class K only if the designations of all possible objects occurred in the language in question. This assumption, however, is fictitious and can never be realised .... We must therefore look for some means of expressing the intentions of the condition (F) which will be completely independent of that fictitious assumption.
6
Such a means is provided by semantics. ...One of the concepts which can be defined in terms of the concept of satisfaction is the concept of model. Let us assume that in the language we are considering certain variables correspond to every extra-logical constant, and in such a way that every sentence becomes a sentential function if the constants in it are replaced by the corresponding variables. Let K be any class of sentences. We replace all extra-logical constants which occur in the sentences belonging to L by corresponding variables, like constants being replaced by like variables, and unlike by unlike. In this way we obtain a class L0 of sentential function. An arbitrary sequence of objects which satisfies every sentential function of the class L0 will be called a model or realization of the class L of sentences (in just this sense one usually speaks of models of an axiom system of a deductive theory). If, in particular, the class L consists of a single sentence X, we shall also refer to a model of the class L as a model of the sentence X. In terms of these concepts we can define the concept of logical consequence as follows:
The sentence X follows logically from the sentences of the class K if and only if every model of the class K is also a model of the sentence X. ...
It seems to me that everyone who understands the content of the above definition must admit that it agrees quite well with common usage.
From Tarski, Mostowski and Robinson, Undecidable Theories (1953) [15] p. 8:
A sentence Φ is said to be a logical consequence of a set A of sentences if it is satisfied in every realization R in which all sentences of A are satisfied; it is called logically true if it is satisfied in every possible realization.
7
The next two examples illustrate model-theoretic and other approaches to defining and manipulating algebraic structures. Both authors are defining boolean algebras.
From E. V. Huntington: Postulates for the algebra of logic (1904). (To remove a conflict with modern notations, I have used 0,1 in place of Huntington’s ∧,∨. I have also used a symbol ≤ in place of a symbol which is not available in my TEX.) In §1 we take as the fundamental concepts a class, K, with two rules of combination, ⊕ and ¯; and as the fundamental propositions, the following ten postulates: Ia. a⊕b is in the class whenever a and b are in the class. Ib. a¯b is in the class whenever a and b are in the class. IIa. There is an element 0 such that a⊕0=a for every element a. IIb. There is an element 1 such that a¯1=a for every element a. IIIa. a⊕b = b⊕a whenever a,b,a⊕b, and b⊕a are in the class. IIIb. a¯b = b¯a whenever a,b,a¯b, and b¯a are in the class. IVa. a⊕(b¯c)=( a⊕b)¯(a⊕c) whenever a,b,c,a⊕b,a⊕ c,b¯c,a⊕(b¯c), and (a⊕b)¯(a⊕c) are in the class. IVb. a¯(b⊕c)=( a¯b)⊕(a¯c) whenever a,b,c,a¯b,a¯ c,b⊕c,a¯(b⊕c), and (a¯b)⊕(a¯c) are in the class. V. If the elements 0 and 1 in postulates IIa and IIb exist and are unique, then for every element a there is an element ¯ a such that a⊕¯ a = 1 and a¯¯ a = 0. VI. There are at least two elements, x and y, in the class such that x 6= y. ... DEFINITION. We shall write a ≤ b (or b ≥ a) when and only when a⊕b = b. ... Any system (K,⊕,¯,≤) which obeys the laws of the algebra of logic may be called a logical field ...
8
From Peter J. Cameron, Combinatorics (1994), [1] p˙193ff.
A partial order on X is a relation R on X which is • reflexive:(x,x) ∈ R for all x ∈ X; • antisymmetric:(x,y),(y,x) ∈ R imply x = y; and • transitive:(x,y),(y,z) ∈ R imply (x,z) ∈ R. The pair (X,R) is called a partially ordered set, orposet for short. ... A lattice is a poset in which each pair of elements has a unique greatest lower bound and a unique least upper bound. ...We use the notation x∧y and x∨y for the g.l.b. and l.u.b. of x and y in a lattice. ... (12.1.2) Proposition. Let X be a set, ∧ and ∨ two binary operations defined on X, and 0 and 1 two elements of X. Then (X,∧,∨,0,1) is a lattice if and only if the following axioms are satisfied: • Associative laws: x∧(y∧z)=(x∧y)∧z and x∨(y∨z)= (x∨y)∨z); • Commutative laws: x∧y = y∧x and x∨y = y∨x; • Idempotent laws: x∧x = x∨x = x; • x∧(x∨y)=x = x∨(x∧y); • x∧0 = 0,x∨1 = 1. ... A lattice L is distributive if it satisfies the two distributive laws x∨(y∧z)=(x∨y)∧(x∨z), x∧(y∨z)=(x∧y)∨(x∧z). ... Among distributive lattices, a special class are the Boolean lattices. These are the distributive lattices L possessing a unary operation x 7→ x0 called complementation, satisfying
9
• (x∨y)0 = x0∧y0, (x∧y)0 = x0∨y0; • x∨x0 =1,x ∧x0 =0 . (12.3.3) Theorem. A finite Boolean lattice is isomorphic to the lattice of all subsets of a finite set X, with x0 interpreted as X \x.
PROOF. Let L be a finite Boolean lattice. We have an embedding of L into P(X), where X is the set of join-indecomposable elements of L. To show that L = P(X), we show that any two join-indecomposable elements are incomparable—then any set of join-indecomposable elements is a down-set. So suppose that a and b are distinct join-indecomposable elements with a ≤ b. Then a∨(b∧a0)=( a∨b)∧(a∨a0)=b∧1=b. Since b is join-indecomposable and a 6= b, we must have b = b∧a0 ≤ a0. ...
The remaining two quotations illustrate the view that there is such a thing as semantic reasoning.
From Johnson-Laird and Byrne, Deduction (1991) [9] p. 212:
[Against the view that ‘there is little or no difference between mental models and formal rules of inference’:]
[It is a] mistaken assumption that the semantic method of truth tables is not fundamentally different from the syntactic method of proof in the propositional calculus .... Infact, there is a profound difference between the two sorts of psychological theories. They postulate different sorts of mental representation, and different sorts of procedures: rule theories employ representations that are language-like and that contain variables, whereas mental models are remote from the structure of sentences and do not contain variables. At the heart of deduction for rule theories is the application of formal rules of inference, such as modus ponens, to representations of the logical forms of sentences. At the heart of deduction for model theories is a search for alternative models of the premises. The search makes no use of modus ponens or any other formal rules of inference.
10
From Hans Kamp and Uwe Reyle, From Discourse to Logic (1993) [10] p. 17ff.
...in the second part of the 19th century, the logician John Venn (1834–1923) developed a method, usually referred to as the method of Venn-diagrams, for “modelling” the premises and conclusions of syllogistic inferences in a systematic way. To determine the validity of a given pattern it suffices to construct the relevant Venn-diagram or diagrams for it. The answer can then be read off these diagrams mechanically. In a Venn-diagram the classes that are given in our syllogistic notation by the letters P,Q,R,...are represented by circles (or other closed curves) whose relations of inclusion and intersection conform to the premises of the pattern under consideration. If every such diagram also verifies the conclusion of the pattern, then the pattern will be valid, and otherwise not. ...Venn’s method may be called a “semantic” method inasmuch as it involves, through its diagrams, the concept of an interpreting structure. The method has an analogue for the much richer notation of predicate logic ...Both in the case of syllogistic and in that of predicate logic the semantic method for analysing validity must be distinguished from another, in which a small number of inference patterns are selected as basic and the validity of other patterns is established by chaining two or more applications of the basic patterns together. This second method ...is known as the proof-theoretic or deductive method. It captures an aspect of deduction which the semantic definition of validity does not touch: in many cases where we reason from given premises to a certain conclusion it is only by moving in small steps that we succeed in arriving at the final conclusion; in others, where we may see the inference more or less directly, it may nevertheless be necessary to break it up into a chain of simple inferences to persuade others that our conclusion really follows.
11
DAY FOUR: Classes of structures
From Alfred Tarski, Contributions to the theory of models. I (1954), [14]:
Within the last years a new branch of metamathematics has been developing. It is called the theory of models and can be regarded as a part of the semantics of formalized theories. The problems studied in the theory of models concern mutual relations between sentences of formalized theories and mathematical systems in which these sentences hold. Every set Σ of sentences determines uniquely a class K of mathematical systems; in fact, the class of all those mathematical systems in which every sentence of Σ holds. Σ is sometimes referred to as a postulate system for K; mathematical systems which belong to K are called models of Σ. Among questions which naturally arise in the discussion of these notions, the following may be mentioned: Knowing some structural (formal) properties of a set Σ of sentences, what conclusions can we draw concerning mathematical properties of the correlated class K of models? Conversely, knowing some mathematical properties of a class K of mathematical systems, what can we say about structural properties of a set Σ which constitutes a postulate system for K? Among publications in this field we may point out the articles and monographs [of Birkhoff, Henkin, Abraham Robinson and Tarski].
12
DAY FIVE: Telling one structure from another
From Anand Pillay, Geometric stability theory (1996), [12] Introduction:
Morley proved in the early 1960s that if T is a complete countable first-order theory which has exactly one model (up to isomorphism) of some given uncountable cardinality κ, then T has exactly one model of cardinality λ for all uncountable cardinals λ. Such a theory is called uncountably categorical, or often just ω1-categorical. The classical examples are the theory of algebraically closed fields (of a fixed characteristic) and the theory of vector spaces (over a fixed field). Shelah, in the late 1960s, then initiated a far-reaching programme of attempting, for an arbitrary first-order theory T, either to ‘classify’ the models of T (up to isomorphism) or to show such a classification to be impossible. This was ‘classification theory’, and at least for countable theories, it reached a successful conclusion in the early 1980s. ‘Classifying’ the models of T amounted to, from Shelah’s viewpoint, describing models by certain nice trees of cardinal invariants. If this could be done (for a given theory T), the class of models of T was said to have a ‘structure theorem’. (Uncountably categorical theories have the ‘best’ structure theorem.) The ‘impossibility’ of such a classification was usually taken to amount to showing that T has 2λ models of cardinality λ for all large λ. This was called a ‘non-structure theorem’. (There have been in the meantime refinements of the ‘structure/non-structure’ conceptual dichotomy, involving for example the question of whether models can be characterized by their theories in certain infinitary or generalized logics.) In order to implement his programme Shelah developed a series of ‘fundamental dichotomies’ (on the class of first-order theories). The most important of these was ‘stability’. Roughly speaking, T is said to be stable if no model of T contains an infinite set of tuples on which some formula defines a linear ordering. ... ...the present period is a very exciting one for model theory, in which there are both grand unifying trends within model theory (in which geometric stability theory has an important role) and new applications to and connections with core areas of mathematics. In fact, it is becoming increasingly clear that the scope
13
of the ideas discussed above goes beyond stable theories, and that, in a sense, the prior concentration on stability was the result of the particular role of stability within Shelah’s programme. But discussion and elaboration of these issues should be left for another book.
From Phokion Kolaitis,Combinatorial Games in Database Theory (extracts) [11].
The study of database query languages has occupied a prominent place in database theory during the past twenty years. ... Main Issues: Database Query Language L. • Determine whether or not a particular query Q is expressible in L. • Characterize the class of queries that are expressible in L. ... Question: • How are negative results established? • What are the main techniques used to obtain lower bounds for expressibility?
Note: Similar issues have been studied extensively in mathematical logic and, especially, in model theory. ... Facts: • Combinatorial Games can be used to characterize the expressive power of various logics. • These characterizations remain valid if only finite structures are considered. • Combinatorial Games have been used extensively in Finite Model Theory during the past twenty years.
14
References
[1] Peter J. Cameron, Combinatorics, Cambridge University Press, Cambridge 1994.
[2] Rudolf Carnap, Introduction to Symbolic Logic and its Applications, Dover, New York 1958.
[3] C. C. Chang and H. J. Keisler, Model Theory, North-Holland, Amsterdam 1973.
[4] P. G. Lejeune Dirichlet, Vorlesungen ¨uber Zahlentheorie, herausgegeben und mit Zus¨atzen versehen von R. Dedekind, Vieweg, Braunschweig 1871.
[5] Kees Doets, Basic Model Theory, CSLI, Stanford 1996.
[6] David Hilbert, Foundations of Geometry (translated and revised; original from 1899), Open Court, La Salle 1971.
[7] Wilfrid Hodges, Model Theory, Cambridge University Press, Cambridge 1993.
[8] Edward V. Huntington, Sets of independent postulates for the algebra of logic, Transactions of the American Mathematical Society 5 (1904) 288–309.
[9] P. N. Johnson-Laird and Ruth M. J. Byrne, Deduction, Lawrence Erlbaum, Hove 1991.
[10] Hans Kamp and Uwe Reyle, From Discourse to Logic, Kluwer, Dordrecht 1993.
[11] Phokion G. Kolaitis, Combinatorial games in database theory, notes presented at PODS ’95.
[12] Anand Pillay, Geometric Stability Theory, Clarendon Press, Oxford 1996.
[13] Alfred Tarski, Logic, Semantics, Metamathematics, trans. J. H. Woodger, second edition ed. and introduced by John Corcoran, Hackett, Indianapolis 1983.
[14] Alfred Tarski, Contributions to the theory of models I, Indagationes Mathematicae 16 (1954) 572–581.
15
[15] Alfred Tarski, A. Mostowski and R. M. Robinson, Undecidable theories, North-Holland, Amsterdam 1953.
[16] Alfred Tarski and Robert L. Vaught, Arithmetical extensions of relational systems, Compositio Mathematica 13 (1957) 81–102.
[17] Oswald Veblen and John Wesley Young, Projective Geometry II, Boston 1918.
16