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Model Theory Mo del Theory J van Oosten Department of Mathematics Utrecht University Spring Contents Intro duction and Notations Homomorphisms Emb eddings and Diagrams Elementary Emb eddings and Elementary Diagrams Directed Systems of Lstructures Theorems of Robinson Craig and Beth Preservation Theorems Filters and Ultrapro ducts Quantier Elimination and Mo del Completeness Quantier Elimination Real Closed Fields Mo del Completeness and Mo del Companions Mo del Completions Amalgamation Quantier Elimination Countable Mo dels End extensions of mo dels of PA Atomic Theories and Atomic Mo dels Saturated Mo dels Stone Duality Literature Intro duction and Notations In the course Foundations of Mathematics we have made our acquaintance with the notions of a language L in predicate logic and a structure A for the lan guage L A language is a collection of symb ols divided into three groups we have constants function symbols and relation or predicate symbols Given a language L we have dened the class of Lterms A A structure A for the language L is a pair A A where A is a A nonempty set and is an interpretation function dened on the symb ols in A L such that for every constant c of L c is an element of A for every nplace A n predicate symb ol R R is a subset of the set A of ntuples of elements of A A n and for every nary function symb ol F of L F is a function from A to A The set A is called the domain universe or underlying set of the structure A A A In practice we shall often omit brackets writing F instead of F and ditto for R In these notes structures will b e denoted by Gothic symb ols A B their domains will b e denoted by the corresp onding Latin characters A B Given a structure A for the language L we consider the language L for A every element a of A we add a constant a to the language and we assume that the new constants a are dierent from constants that are already in the A language L The structure A b ecomes a L structure by putting a a for A each a A We have also dened the relation A j for L sentences rst for closed A A L terms t we dene their meaning t in A Then we dened by recursion on A the L sentence A A A A j t s if and only if t s A A A A j Rt t if and only if t t R n n A j if and only if A j and A j A j if and only if A j or A j or b oth A j if and only if A j A j xx if and only if for some a A A j a A j xx if and only if for all a A A j a Furthermore we have learnt the notions of L or L theory this is a collection A of L or L sentences A is a model of the theory T if A j for every T A The relation A j is pronounced as A satises is true in A or holds in A We also recall one of the most imp ortant theorems we proved in the course Foundations of Mathematics Theorem Compactness Theorem A theory T has a model if and only if every nite subset of T has a model In this course we shall give an indep endent pro of of Theorem that is indep endent of the Completeness Theorem Often we shall say that a theory is consistent in this course this is synony mous with has a mo del We shall also use the notation T j where T is an Ltheory and and Lsentence this means that is true in every mo del of T A theory T is complete if for every sentence in the language of T either T j or T j holds Homomorphisms Emb eddings and Diagrams The purp ose of Mo del Theory is the study of theories by means of their classes of mo dels A very useful to ol of Mo del Theory is the p ossibility of varying the language Supp ose we have two languages L L If B is an L structure then restricting its interpretation function to L gives an Lstructure A We say that A is the Lreduct of B and that B is an L expansion of A In our denition of A j we have already seen the expansion of A to L A We start by giving some structure on the class of Lstructures Let A A B A and B B b e Lstructures A function f A B is called a homomorphism of Lstructures if it commutes with the interpretation functions That is A B For every constant c of L f c c for every nary function symb ol F of L and every ntuple a a of n A B elements of A we have f F a a F f a f a n n for every nplace predicate symb ol R of L and ntuple a a of el n A ements of A we have if a a R then f a f a n n B R Examples If L is the language of groups and A and B are groups a homo morphism of Lstructures is nothing but a homomorphism of groups If L is the language fg of partial orders and A B are partial orders a homomorphism is nothing but an orderpreserving map Similar for rings graphs etcetera We note immediately that if f A B and g B C are homomorphisms of Lstructures then so is the comp osition g f A C Moreover the identity function A A is always a homomorphism of Lstructures A homomorphism f A B is an isomorphism if there is a homomorphism g B A inverse to f so the comp ositions g f and f g are the identity functions on A and B resp ectively Exercise Prove if f A B is an isomorphism of Lstructures then for any Lformula x x and any ntuple a a from A we have n n A j a a B j f a f a n n A consequence of this exercise is that isomorphic Lstructures satisfy the same Lsentences check this Two Lstructures that satisfy the same Lsentences are called elementarily equivalent Notation A B The notation for isomor phic structures is Summarizing A B A B The converse implication do es not hold Exercise Prove an Ltheory T is complete if and only if every pair of mo dels of T is elementarily equivalent One sees that the notion of completeness can b e characterized by a prop erty of the class of mo dels of the theory A homomorphism f A B is called injective if the function f is f is an embedding if f is injective and moreover for every relation symb ol R of B L and every ntuple a a from A if f a f a R then n n A a a R Clearly if L contains no relation symb ols every injec n tive homomorphism is an emb edding but in the general case the notions are dierent A substructure or submodel of an Lstructure B is an Lstructure A such that A is a subset of B and the interpretation function of A is the restriction of the one of B to A Hence an emb edding f A B denes an isomorphism b etween A and a submo del of B If B is an Lstructure and X B there is a least subset A of B which B contains X and the elements c c a constant of L and is closed under the interpretations in B of the function symb ols of L if A A is the domain of a submo del A of B the submo del generated by the set X which we denote by h X i Exercise Show that the domain of hX i is the set B ft x x j n IN tv v an Lterm x x X g n n n A structure B is nitely generated if B hX i for some nite subset X of B Recall that a formula is called atomic if it contains no connectives or quan tiers in other words if it is of from t t or Rt t n Exercise Let f A B an emb edding Then for every atomic Lformula x x and every ntuple a a from A n n A j a a B j f a f a n n Prove also that this equivalence in fact holds for every quantierfree formula Exercise Let L fg b e the language of partial orders Find out when a monotone map b etween partial orders is an injective Lhomomorphism and when it is an emb edding Give an example of an injective homomorphism which is not an emb edding Let A b e an Lstructure with its natural expansion to L We dene a few A L theories in connection to A A The positive diagram of A is the collection of all atomic L sentences A A that are true in A The diagram of A is the collection of all atomic L sentences and all A A negations of atomic L sentences that are true in A A The notion of positive diagram generalizes the idea of multiplication table of a group Exercise Show that giving an Lhomomorphism A B is equivalent to giving an expansion of B to L which is a mo del of the p ositive diagram of A A Show also that giving an emb edding A B is equivalent to giving an expansion of B to L which is a mo del of the diagram of A A We now give an example in order to apply our denitions to a little mathemat ical theorem Theorem Order Extension Principle Let A be a partial ly or dered set Then there is a linear order on A which extends equiva lently there is an injective monotone function from A into a linearly ordered set B Pro of The equivalence stated in the theorem is clear b ecause if B is a linear order and f A B an injective orderpreserving map then dening on A by x y i f x f y gives a linear order on A which extends We prove the theorem in two steps a First we do it for A nite This is easy induction on the numb er of elements of A If jAj we are done Now if A fa a g cho ose a A such n i that a is maximal wrt the order on A By induction hyp othesis Anfa g has i i a linear order which extends the one of A relativized to Anfa g put a back in i i as greatest element b The general case Let L fg We use the idea of diagrams and the Compactness Theorem Let T b e the L theory where A is the partial order A A seen as Lstructure consisting of the axioms The p ositive diagram of A The axioms of a linear order The axioms a b for a b A In every nite subset D of T there is only a nite numb er of constants from the set A say a a Let fa a g b e the subpartial order of A on these n n elements This has by a an extension
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