模型论的入门读物

    数学历史发展已经证明，塔尔斯基、鲁宾逊的模型论思想已经融入现代数学的身躯，包括微积分学。

   现代微积分教员不懂模型理论的基础知识，就是小学生登上大学讲台，……

    为此，我们推荐本文附件文章供读者阅读参考。

 

袁萌  陈启清 7月1日

附件:

Introduction to Model Theory

DAVID MARKER

Abstract. This article introduces some of the basic concepts and results from model theory, starting from scratch. The topics covered are be tailored to the model theory of elds and later articles. I will be using algebraically closed elds to illustrate most of these ideas. The tools described are quite basic; most of the material is due either to Alfred Tarski or Abraham Robinson. At the end I give some general references.

1. Languages and Structures

What is a mathematical structure? Some examples of mathematical structures we have in mind are the ordered additive group of integers, the complex eld, and the ordered real eld with exponentiation. To specify a structure we must specify the underlying set, some distinguished operations, some distinguished relations and some distinguished elements. For example, the ordered additive group of integers has underlying set Z and we distinguish the binary function +, the binary relation < and the identity element 0. For the ordered eld of real numbers with exponentiation we have underlying set R and might distinguish the binary functions + and ×, the unary function exp, the binary relation < and the elements 0 and 1. Here is the formal denition.

Definition 1.1. A structure M is given by the following data.

(i) A set M called the universe or underlying set of M. (ii) A collection of functions {fi : i ∈ I0} where fi : Mni → M for some ni ≥1. (iii) A collection of relations {Ri : i ∈ I1} where Ri ⊆ Mmi for some mi ≥1. (iv) A collection of distinguished elements {ci : i ∈ I2}⊆ M. Any (or all) of the sets I0,I1 and I2 may be empty. We refer to ni and mj as the arity of fi and Rj. Here are some examples:

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16 DAVID MARKER (1) The ordered eld of real numbers has domain R, binary functions +,−,×, relation <, and distinguished elements 0 and 1. (2) Thevaluedeldof p-adicnumbershasdomainQp, binaryfunctions+,−,×, distinguished elements 0 and 1, and a unary relation Zp, for the ring of integers. In mathematical logic we study structures by examining the sentences of rst order logic true in those structures. To any structure we attach a language L where we have an ni-ary function symbol ˆ fi for each fi, an mi-ary relation symbol ˆ Ri for each Ri and constant symbols ˆ ci for each ci. An L-structure is a structure M where we can interpret all of the symbols of L. For example, let L be the language where we have a binary function symbol ˆ × and a constant symbol ˆ1. The following are examples of L-structures: (1) M1 has underlying set Q. We interpret ˆ × as × and ˆ1 as 1.(2) M2 has underlying set Z. We interpret ˆ × as + and ˆ1 as 0. Of course we also could take the natural interpretation of L in Z. (3) M3 has underlying set Z. We interpret ˆ × as × and ˆ1 as 1. Definition. If M and N are L-structures with underlying sets M and N, respectively, an L-embedding σ : M → N is an injective map that preserves the interpretation of all function symbols, relation symbols and constant symbols of L. An L-isomorphism is a bijective L-embedding. We say that M is a substructure of N (and write M ⊂ N) if M ⊂ N and the inclusion map is an L-embedding. Formulas in our language are nite strings made from the symbols of L, the equality relation =, variables x0,x1,..., the logical connectives ∃,∧,∨, quantiers ∃ and ∀ and parentheses. (See the appendix on page 34 for precise denitions.) We interpret ∃,∧,∨ as “not”, “and” and “or” and ∃ and ∀ as “there exists” and “for all”. I will use x,y,z ... and their subscripted varieties as variables and not use the symbol ˆ when no confusion arises. Let Lr be the language of rings, where we have binary function symbols +,−and × and constant symbols 0 and 1. The language of ordered rings, Lor is Lr withanadditionalbinaryrelationsymbol <. (Asusualwewillwrite x+y instead of +(x,y) and x < y for <(x,y).) Here are some examples of Lor-formulas: x1 = 0 ∨ x1 > 0 ∃x2 x2 ×x2 = x1 ∀x1 (x1 = 0 ∨ ∃x2 x2 ×x1 = 1) Intuitively, the rst formula asserts that x1 ≥0, the second asserts that x1 is a square and the third asserts that every nonzero element has a multiplicative inverse. We would like to dene what it means for a formula to be true in a

INTRODUCTION TO MODEL THEORY 17

structure, but these examples already show one diculty. While in any Lorstructure the third formula will either be true or false, the rst two formulas express a property which may or may not be true of elements of the structure. We say that a variable occurs freely in a formula φ if it is not inside the scope of a quantier; otherwise we say it is bound. For example, x1 is free in the rst two formulas and bound in the third, while x2 is bound in both formulas. We call a formula a sentence if it has no free variables. For any L-structure each sentence of L is either true or false. Let φ be a sentence. We say that M is a model of φ, and write M |= φ, if and only if φ is true in M. We often write φ(x1,...,xn) to show that the variables x1,...,xn are free in the formula φ. We think of φ(x1,...,xn) as describing a property of ntuples from M. For example, the Lor-formula ∃x2 x2 ×x2 = x1 has the single free variable x1 and describes the property “x1 is a square”. If a1,...,an are elements of M we say M |= φ(a1,...,an) if the property expressed by φ is true of the tuple (a1,...,an). We say that two L-structures M and N are elementarily equivalent if for all L-sentences M |= φ ⇐⇒ N |= φ. Proposition 1.2. If M and N are isomorphic, then they are elementarily equivalent. Proof. Show by induction on formulas that if φ(x1,...,xn) is a formula, σ : M → N is an isomorphism and a1,...,an ∈ M, then M |= φ(a1,...,an) ⇐⇒ N |= φ(σ(a1),...,σ(an)).

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 We say that an L-embedding f : M → N is elementary if for any a1,...,an ∈ M and any formula φ(x1,...,xn) M |= φ(a1,...,an) ⇐⇒ N |= φ(f(a1),...,f(an)). If M ⊂ N we say that M is an elementary substructure if the inclusion map is elementary. Definition. We say that a set X ⊂ Mn is denable in the L-structure M if there is a formula φ(x1,...,xn+m) and elements b1,...,bm ∈ M such that X ={(a1,...,an) : M |= φ(a1,...,an,b1,...,bm)}. We say that X is A-denable or denable over A, where A ⊆ M, if we can choose that b1,...,bm ∈ A. For example, if m = 0 we say X is ∅-denable. For example, {x : x > π} is denable over R but not ∅-denable, while {x : x > √2} is ∅-denable by the formula x × x > 1 + 1 ∧ x > 0. In the eld (Qp,+,−,×,0,1) if p 6= 2 we can dene the valuation ring Zp by the formula ∃y y2 = px2 +1. We can give a very simple characterization of the denable sets.

18 DAVID MARKER

Proposition 1.3. Suppose that Dn is a collection of subsets of Mn for all n ≥ 1 such that D = (Dn : n ≥ 1) is the smallest collection satisfying the following conditions: (i) Mn ∈ Dn. (ii) For all n-ary functions f of M, the graph of f is in Dn+1. (iii) For all n-ary relations R of M, R ∈ Dn. (iv) For all i,j ≤ n, {(x1,...,xn)∈ Mn : xi = xj}∈ Dn. (v) Each Dn is closed under complement, union and intersection. (vi) If X ∈ Dm and π : Mn → Mm is the projection map (x1,...,xn) 7→ (xi1,...,xim), then π−1(X)∈ Dn. (vii) If X ∈ Dn and π is as above, then π(X)∈ Dm. (viii) If X ∈ Dn+m and b ∈ Mm, then {a ∈ Mn : (a,b)∈ X}∈ Dn. Then X ⊆ Mn is denable if and only if X ∈ Dn.

2. Theori)es

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