##### Citation preview

Fundamentals of Model Theory William Weiss and Cherie D'Mello Department of Mathematics University of Toronto

c 1997 W.Weiss and C. D'Mello

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Introduction

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William Weiss, based upon suggestions from several graduate students. The electronic version of this book may be downloaded and further modied by anyone for the purpose of learning, provided this paragraph is included in its entirety and so long as no part of this book is sold for prot.

Contents Chapter 0. Models, Truth and Satisfaction Formulas, Sentences, Theories and Axioms Prenex Normal Form Chapter 1. Notation and Examples Chapter 2. Compactness and Elementary Submodels Compactness Theorem Isomorphisms, elementary equivalence and complete theories Elementary Chain Theorem Lowenheim-Skolem Theorems The L os-Vaught Test Every complex one-to-one polynomial map is onto Chapter 3. Diagrams and Embeddings Diagram Lemmas Every planar graph can be four coloured Ramsey's Theorem The Leibniz Principle and innitesimals Robinson Consistency Theorem Craig Interpolation Theorem Chapter 4. Model Completeness Robinson's Theorem on existentially complete theories Lindstrom's Test Hilbert's Nullstellensatz Chapter 5. The Seventeenth Problem Positive denite rational functions are the sums of squares Chapter 6. Submodel Completeness Elimination of quantiers The Tarski-Seidenberg Theorem Chapter 7. Model Completions Almost universal theories Saturated models Blum's Test Bibliography Index 3

4 4 9 11 14 14 15 16 19 21 23 24 25 25 26 26 27 31 32 32 35 38 39 39 45 45 49 50 52 54 55 61 62

CHAPTER 0 Models, Truth and Satisfaction

We will use the following symbols: logical symbols: { the connectives ^ ,_ , : , ! , \$ called \and", \or", \not", \implies" and \i" respectively { the quantiers 8 , 9 called \for all" and \there exists" { an innite collection of variables indexed by the natural numbers N v0 ,v1 , v2 , : : : { the two parentheses ), ( { the symbol = which is the usual \equal sign" constant symbols : often denoted by the letter c with subscripts function symbols : often denoted by the letter F with subscripts each function symbol is an m-placed function symbol for some natural number m  1 relation symbols : often denoted by the letter R with subscripts each relational symbol is an n-placed relation symbol for some natural number n  1. We now dene terms and formulas. Definition 1. A term is dened as follows: (1) a variable is a term (2) a constant symbol is a term (3) if F is an m-placed function symbol and t1  : : :  tm are terms, then F (t1 : : : tm ) is a term. (4) a string of symbols is a term if and only if it can be shown to be a term by a nite number of applications of (1), (2) and (3). Remark. This is a recursive denition. Definition 2. A formula is dened as follows : (1) if t1 and t2 are terms, then t1 = t2 is a formula. (2) if R is an n-placed relation symbol and t1  : : :  tn are terms, then R(t1 : : : tn ) is a formula. (3) if ' is a formula, then (:') is a formula (4) if ' and  are formulas then so are (' ^ ), (' _ ), (' ! ) and (' \$ ) (5) if vi is a variable and ' is a formula, then (9vi )' and (8vi )' are formulas (6) a string of symbols is a formula if and only if it can be shown to be a formula by a nite number of applications of (1), (2), (3), (4) and (5). Remark. This is another recursive denition. :' is called the negation of '. ' ^  is called the conjunction of ' and . ' _  is called the disjunction of ' and . Definition 3. A subformula of a formula ' is dened as follows: 4

0. MODELS, TRUTH AND SATISFACTION

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(1) is a subformula of (2) if (: ) is a subformula of then so is (3) if any one of ( ^ ), ( _ ), ( ! ) or ( \$ ) is a subformula of , then so are both and (4) if either (9 ) or (8 ) is a subformula of for some natural number , then is also a subformula of (5) A string of symbols is a subformula of , if and only if it can be shown to be such by a nite number of applications of (1), (2), (3) and (4). Definition 4. A variable is said to occur bound in a formula i for some subformula of either (9 ) or (8 ) is a subformula of . In this case each occurance of in is said to be a bound occurance of . Other occurances of which do not occur bound in are said to be free. Exercise 1. Using the previous denitions as a guide, dene the substitution of a term for a variable in a formula . Example 1. 2 ( 3 4 ) is a term. '

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((8 3 )( 3 = 3 ^ 0 = 1 ) _ (9 0 ) 0 = 0 ) is a formula. In this formula the variable 3 occurs bound, the variable 1 occurs free, but the variable 0 occurs both bound and free. Exercise 2. Reconsider Exercise 1 in light of substituting the above term for 0 in the formula. Definition 5. A language L is a set consisting of all the logical symbols with perhaps some constant, function and/or relational symbols included. It is understood that the formulas of L are made up from this set in the manner prescribed above. Note that all the formulas of L are uniquely described by listing only the constant, function and relation symbols of L. We use ( 0 ) to denote a term all of whose variables occur among . 0 We use ( 0 ) to denote a formula all of whose free variables occur among 0 . Example 2. These would be formulas of any language : For any variable : = for any term ( 0 ) and other terms 1 and 2 : )= ( 0 ) 1= 2! ( 0 ;1 1 +1 ;1 2 +1 for any formula ( 0 ) and terms 1 and 2 : )\$ ( 0 ) 1= 2! ( 0 ;1 1 +1 ;1 2 +1 Note the simple way we denote the substitution of 1 for . Definition 6. A model (or structure) A for a language L is an ordered pair hA Ii where A is a set and I is an interpretation function with domain the set of all constant, function and relation symbols of L such that: 1. if is a constant symbol, then I ( ) 2 A I ( ) is called a constant v

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0. MODELS, TRUTH AND SATISFACTION

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is an m-placed function symbol, then I ( ) is an m-placed function on F

3. if is an n-placed relation symbol, then I ( ) is an n-placed relation on A. A is called the universe of the model A. We generally denote models with Gothic letters and their universes with the corresponding Latin letters in boldface. One set may be involved as a universe with many dierent interpretation functions of the language L. The model is both the universe and the interpretation function. Remark. The importance of Model Theory lies in the observation that mathematical objects can be cast as models for a language. For instance, the real numbers with the usual ordering and the usual arithmetic operations, addition + and multiplication along with the special numbers 0 and 1 can be described as a model. Let L contain one two-placed (i.e. binary) relation symbol 0 , two two-placed function symbols 1 and 2 and two constant symbols 0 and 1 . We build a model by letting the universe A be the set of real numbers. The interpretation function I will map 0 to , i.e. 0 will be interpreted as . Similarly, I ( 1 ) will be +, I ( 2 ) will be , I ( 0 ) will be 0 and I ( 1 ) will be 1. So hA Ii is an example of a model. We now wish to show how to use formulas to express mathematical statements about elements of a model. We rst need to see how to interpret a term in a model. Definition 7. The value  0 q ] of a term ( 0 q ) at 0 q in the model A is dened as follows: 1. if is i then  0 q ] is i , 2. if t is the constant symbol c, then  0 q ] is I ( ), the interpretation of in A, 3. if is ( 1 m ) where is an m-placed function symbol and 1 m are terms, then  0 q ] is ( 1  0 q] m 0 q ]) where is the m-placed function I ( ), the interpretation of in A. Definition 8. Suppose A is a model for a language L. The sequence 0 q satis es the formula ( 0 q ) all of whose free and bound variables are among 0 q , in the model A, written A j=  0 q ] provided we have: 1. if ( 0 q ) is the formula 1 = 2 , then A j= 1 = 2  0 q ] means that 1  0 q ] equals 2  0 q ], 2. if ( 0 q ) is the formula ( 1 n ) where is an n-placed relation symbol, then A j= ( 1 n ) 0 q ] means ( 1  0 q] n 0 q ]) where is the n-placed relation I ( ), the interpretation of in A, 3. if is (: ), then A j=  0 q ] means not A j=  0 q ], 4. if is ( ^ ), then A j=  0 q ] means both A j=  0 q ] and A j=  0 q ], 5. if is ( _ ) then A j=  0 q ] means either A j=  0 q ] or A j=  0 q ], R

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