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Model Theory – Draft Math 225A – Model Theory – Draft Speirs, Martin Autumn 2013 Contents Contents 1 0 General Information 5 1 Lecture 16 1.1 The Category of τ-Structures....................... 6 2 Lecture 2 11 2.1 Logics..................................... 13 2.2 Free and bound variables......................... 14 3 Lecture 3 16 3.1 Diagrams................................... 16 3.2 Canonical Models.............................. 19 4 Lecture 4 22 4.1 Relations defined by atomic formulae.................. 22 4.2 Infinitary languages............................ 22 4.3 Axiomatization............................... 24 The Arithmetical Hierarchy........................ 24 1 Contents 2 5 Lecture 5 25 5.1 Preservation of Formulae......................... 26 6 Lecture 6 31 6.1 Theories and Models............................ 31 6.2 Unnested formulae............................. 34 6.3 Definitional expansions.......................... 36 7 Lecture 7 38 7.1 Definitional expansions continued.................... 38 7.2 Atomisation/Morleyisation........................ 39 8 Lecture 8 42 8.1 Quantifier Elimination........................... 42 8.2 Quantifier Elimination for (N, <) ..................... 42 8.3 Skolem’s theorem.............................. 45 8.4 Skolem functions.............................. 46 9 Lecture 9 47 9.1 Skolemisation................................ 47 9.2 Back-and-Forth and Games........................ 49 9.3 The Ehrenfeucht-Fraïssé game...................... 50 10 Lecture 10 52 10.1 The Unnested Ehrenfeucht-Fraïssé Game................ 52 11 Lecture 11 59 11.1 Games and Products............................ 59 12 Lecture 12 60 12.1 Quantifier Elimination for Th(Z) as an Ordered Group........ 60 12.2 Automorphisms............................... 63 13 Lecture 13 65 13.1 Automorphism Groups of Structures................... 65 14 Lecture 14 70 14.1 Interpretations................................ 70 15 Lecture 15 75 15.1 Elimination of Imaginaries......................... 75 15.2 Multi-Sorted Structures, Aeq ....................... 78 Contents 3 16 Lecture 16 79 16.1 Compactness................................. 79 16.2 Upward Löwenheim-Skolem....................... 83 17 Lecture 17 84 17.1 Compactness Applied........................... 84 17.2 Types..................................... 86 18 Lecture 18 89 18.1 Type Spaces................................. 89 19 Lecture 19 95 19.1 Amalgamation................................ 95 20 Lecture 20 101 20.1 Heirs and Coheirs.............................. 101 20.2 Preservation Theorems........................... 104 21 Lecture 21 107 21.1 Interpolation................................. 107 21.2 Indiscernibles................................ 109 22 Lecture 22 114 22.1 Finite Ramsey Theory........................... 114 22.2 Fraïssé Constructions and Ages...................... 115 23 Lecture 23 117 23.1 Fraïssé’s Theorem.............................. 117 24 Lecture 24 124 24.1 Omitting Types............................... 124 24.2 Atomic and Prime Models......................... 127 25 Lecture 25 130 25.1 Countable Categoricity........................... 130 26 Lecture 26 134 26.1 Saturation.................................. 134 26.2 Existence of Saturated Models...................... 139 26.3 Categoricity of Saturated Models..................... 141 27 Lecture 27 144 Contents 4 27.1 Quantifier Elimination........................... 144 27.2 Examples of Quantifier Elimination................... 146 Bibliography 148 General Information These notes are based on a course in Metamathematics taught by Professor Thomas Scanlon at UC Berkeley in the Autumn of 2013. The course focused on Model Theory and the course book was Hodges’ a shorter model theory (see [1]). As with any such notes, these may contain errors and typos. I take full responsi- bility for such occurrences. If you find any errors or typos (no matter how trivial!) please let me know at [email protected]. Thanks to Derric Chien for informing me of several errors and unclear state- ments. 5 Lecture 1 Model Theory is the study of the interrelation between structures and syntax. We shall start by defining the structures. To define a structure we need the data of a signature and then an interpretation of the signature. 1.1 The Category of τ-Structures Definition. A signature τ consists of three (disjoint) sets Cτ, Fτ, Rτ together with a function arity : Fτ [ Rτ Z+ C F R The sets τ, τ, τ will contain the constant! symbols, function symbols and relation symbols, respectively. The arity function assigns to each function symbol and each relation symbol some positive integer thought of as the number of ar- guments that the function (respectively, the relation) takes. Note that we do not allow arities of functions and relations to be zero. Definition. A τ-structure A is given by a set A and interpretations of the elements of Cτ [ Fτ [ Rτ, i.e.: A • each c 2 Cτ is interpreted as an element c in A. A arity(f) • each f 2 Fτ is interpreted as a function f : A A. A arity(R) • each R 2 Rτ is interpreted as a set R ⊆ A . ! The set A is called the domain of A and also denoted dom(A). We also use the notation R(A) for RA in anticipation of definable sets. For a 2 RA we may also write R(a). Remark. In this course we do not require that A be nonempty! Note that in order for ? to be a structure there can be no constant symbols (i.e. Cτ = ?). Remark. There is also a notion of a sorted signature, in which we would have another set Sτ and in which “arity” would be replaced by giving the sort of each 6 1.1 The Category of τ-Structures 7 constant symbol, the domain and target of each function symbol and the field of each relation symbol. This is relevant in many situations for example when describing a vector space over a field (so we need two sorts: vectors and scalars) and in computer science. Example (Groups). A group G may be regarded as a structure. The signature is in this case Cτ = f1g, Fτ = f·g and Rτ = ?, and arity(·) = 2. As an interpretation we might let 1G be the identity element of G and ·G : G × G G the group multiplication. Example (Graphs). A graph G is a triple (V, E, I) of vertices, edges and an incidence relation,! such that for e 2 E and v, w 2 V we have I(v, w, e) is (v, w) 2 e (i.e. if e is an edge between v and w). There are two natural signatures to use that do give different notions of graphs as structures. • Let τ be given by Cτ = ?, Fτ = ? and Rτ = fV, E, Ig where arity(V) = arity(E) = 1 and arity(I) = 3. With this signature we can now set dom(G) = V [ E, VG = V, EG = E and IG = f(v, w, e)j(v, w) 2 eg. • Let σ be the signature given by Cσ = ?, Fσ = ? and Rσ = fEg with arity(E) = 2. Now dom(G) = V and EG = f(v, w) j 9e 2 E such that (v, w) 2 eg. Now both signatures can be used to describe graphs but they are different. In the first case there can be multiple edges between the same vertices, while in the second there cannot. However in the first case we can have an edge e 2 E which is not connected to any vertices. So it makes a difference which language one uses! We want to turn the collection of τ-structures into a category. For this we need morphisms. Definition. A homomorphism f :A B of τ-structures A and B, is given by a function f : dom(A)! dom(B) which respects all the “extra structure”. More precisely ! A B • for all c 2 Cτ we have f(c ) = c • for all g 2 Fτ (with say n = arity(g)) and a1,..., an 2 dom(A) then A B f(g (a1,..., an)) = g (f(a1),..., f(an)). 1.1 The Category of τ-Structures 8 A • for all R 2 Rτ (with say n = arity(R)) if (a1,..., an) 2 R then B (f(a1),..., f(an)) 2 R . Note that the notion of homomorphism depends on the choice of signature. For instance when defining homomorphisms of rings if we use a signature which has a constant symbol for the unit element then we get unit-preserving homo- morphisms. If the signature does not have a constant symbol for the unit then homomorphisms of rings need not preserve the unit. Proposition. If f :A B and g :B C are homomorphisms of τ-structures then g ◦ f is a homomorphism from A to C. Furthermore the identity map 1A : A A gives a homomorphism of τ-structures! 1A :A A!. ! Proof. Exercise (purely formal). ! Thus, the collection of τ-structures together with homomorphisms between them form a category, called Str(τ). Definition. For A and B τ-structures (with dom(A) = A and dom(B) = B) then A is a substructure of B, written A ⊆ B if, A ⊆ B and A B • for all c 2 Cτ, c = c . A B • for all f 2 Fτ, f = f jAn where n = arity(f). A B n • for all R 2 Rτ, R = R \ A where n = arity(R). Proposition. For τ-structures A and B. If A ⊆ B then the inclusion map ι : dom(A) dom(B) is a homomorphism. ! Proof. We have a function from dom(A) to dom(B). We check all three conditions A A B • let c 2 Cτ then ι(c ) = c = c since A is a substructure of B. • ι commutes with all interpretations of the function symbols since ι = 1BjA. A • if (a1,..., an) 2 R then by the substructure property (ι(a1),..., ι(an)) = B (a1,..., an) 2 R . so ι is indeed a homomorphism. Warning!. The converse of the above proposition is not true in general! There exist τ-structures A and B such that dom(A) ⊆ dom(B) and such that the inclusion map is a homomorphism and yet A is not a substructure of B. As an example, let τ be the signature given by Cτ = Fτ = ? and Rτ = fPg with arity(P) = 1. Let B = R and A = R and consider these as τ-structures where B A P = R and P = ?, respectively.
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