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Elementary Model Theory Notes for MATH 762

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George F. McNulty Elementary Model Theory Notes for MATH 762 drawings by the author University of South Carolina Fall 2011 Model Theory MATH 762 Fall 2011 TTh 12:30p.m.–1:45 p.m. in LeConte 303B Office Hours: 2:00pm to 3:30 pm Monday through Thursday Instructor: George F. McNulty Recommended Text: Introduction to Model Theory By Philipp Rothmaler What We Will Cover After a couple of weeks to introduce the fundamental concepts and set the context (material chosen from the first three chapters of the text), the course will proceed with the development of first-order model theory. In the text this is the material covered beginning in Chapter 4. Our aim is cover most of the material in the text (although not all the examples) as well as some material that extends beyond the topics covered in the text (notably a proof of Morley’s Categoricity Theorem). The Work Once the introductory phase of the course is completed, there will be a series of problem sets to entertain and challenge each student. Mastering the problem sets should give each student a detailed familiarity with the main concepts and theorems of model theory and how these concepts and theorems might be applied. So working through the problems sets is really the heart of the course. Most of the problems require some reflection and can usually not be resolved in just one sitting. Grades The grades in this course will be based on each student’s work on the problem sets. Roughly speaking, an A will be assigned to students whose problems sets eventually reveal a mastery of the central concepts and theorems of model theory; a B will be assigned to students whose work reveals a grasp of the basic concepts and a reasonable competence, short of mastery, in putting this grasp into play to solve problems. Students are invited to collaborate on the problem sets. Some of the problems will be designated as individual efforts. Students intending to use this course as part of the Comprehensive Exams should make particularly diligent efforts on the problems sets. The Final As the material is ill-suited to a sit down three hour writing effort, in place of a final exam- ination we will instead have a party at my house. Everyone (and their partners) is invited. The party does have a little exit exam. I plan to offer a MATH 748V in the spring 2012 semester. The topic of that course will be Varieties of Aglebras, which is an algebraic counterpart to a portion of MATH 762. MATH 762 and MATH 748V would form a course sequence upon which a Ph.D. Compre- hensive Exam could be based. Please feel free to drop by my office at any time. My office is 302 LeConte. Contents Syllabus ii Lecture 0 Mathematical Structures and Mathematical Formulas 1 0.1 Symbols for Operations, Symbols for Relations, and Signatures 2 0.2 Mathematical Structures of a Given Signature 3 0.3 Terms, Formulas, and Sentences 5 0.4 Satisfaction and Truth 8 0.5 Problem Set 0 11 Lecture 1 Models, Theories, and Logical Consequence 12 1.1 The Galois Connection Established by Truth 12 1.2 A Short Sampler of Elementary Classes 15 1.3 Problem Set 1 20 Lecture 2 The Compactness Theorem 21 2.1 The Compactness Theorem 21 2.2 A Sampler of Applications of the Compactness Theorem 26 2.3 Problem Set 2 31 Lecture 3 Putting Structures Together with Ultraproducts 32 3.1 Filters and Ultrafilters 32 3.2 Direct Products and Reduced Products 33 3.3 The Fundamental Theorem for Ultraproducts 35 3.4 An Ultraproduct Proof of the Compactness Theorem 38 3.5 Problem Set 3 39 Lecture 4 Elementary Embeddings 40 4.1 The Downward Löwenheim-Skolem-Tarski Theorem 40 4.2 Necessary and Sufficient Conditions for Elementary Embeddings 43 4.3 The Upward Löwenheim-Skolem-Tarski Theorem 46 iii Contents iv Lecture 5 Elementary Chains and Amalgamation of Structures 51 5.1 Elementary Chains and Amalgamation 51 5.2 Multiple Signatures: Joint Consistency, Interpolation, and Definability 54 Lecture 6 Sentences Preserved under the Formation of Substructures 59 6.1 Classes That Are Relativized Reducts of Elementary Classes 59 6.2 The Łoś-Tarski Theorem 60 6.3 Characterizations of Universal Classes 62 Lecture 7 Denumerable Models of Complete Theories 66 7.1 Realizing and Omitting Types 66 7.2 Meager Structures 69 7.3 Countably Saturated Structures 73 7.4 The Number of Denumerable Models of a Complete Theory 76 7.5 Problem Set 4 79 Lecture 8 Categoricity in Uncountable Cardinalities 81 8.1 Morley’s Categoricity Theorem 81 8.2 Keisler’s Two Cardinal Theorem 84 8.3 Order Indiscernibles and the First Modesty Theorem 88 8.4 Saturated Models of ω-Stable Theories 91 8.5 The Second Modesty Theorem 93 Appendix A Ramsey’s Theorem 98 Bibliography 102 Index 105 LECTURE 0 Mathematical Structures and Mathematical Formulas A host of systems have attracted the attention of mathematicians. There are groups, graphs, and ordered sets. There are geometries, ordered fields, and topological spaces. There are categories, affine varieties, and metric spaces. The list is seemingly enormous, as is the list of concepts and theorems emerging from the efforts mathematicians have spent in their investi- gation. It is the ambition of model theory to attach to such mathematical systems some formalized linguistic apparatus appropriate for framing concepts and stating theorems. This linguistic or syntactical apparatus, once it is given an unambiguous definition, can be subjected itself to a mathematical analysis. Roughly speaking, model theory is that branch of mathematics that exploits the connections between such a syntax and the appropriate mathematical systems. This means that model theory is a kind of mathematical semantics: it is centered on the meanings that syntactical expressions achieve in particular mathematical systems. The ambition of model theory to make and to exploit such connections stands in a common mathematical tradition: by adjoining extra structure and developing its mathematics one hopes to be able to throw light on old problems and also to open up new parts of mathematics to further development. It is surely evident, for example, that the geometric, analytic, and topological aspects of the complex plane have greatly informed our understanding of ordinary addition and multiplication of whole numbers—an example of associating extra structure. First observe that a linguistic apparatus suitable for the theory of groups need not be suitable for the theory of rings and may be quite unsuitable for the theory of ordered sets. So we envision the need for a wide assortment formal syntactical systems. Next observe that groups, graphs, ordered sets, ordered fields, and certain geometries have the form of nonempty sets equipped with some system of operations and relations, each having some fixed finite number of places. For example a graph G = hG, Ei can be construed as a set G of vertices and a set E of edges—so that E is a symmetric irreflexive two-place relation on the set G of vertices. On the other hand, a topological space is a system hX, Ti, where X is a set and T is a collection of subsets of X that satisfy certain properties making T into a topology of X. In the part of model theory we will develop, we will allow mathematical systems like groups, graphs, ordered fields, and many others, but we will exclude systems like topological spaces, as well as many other very interesting mathematical systems. 1 0.1 Symbols for Operations, Symbols for Relations, and Signatures 2 0.1 Symbols for Operations, Symbols for Relations, and Signatures Let us consider an example: the set of real numbers endowed with addition, multiplication, negation, zero, one, and the usual ordering. We could denote this structure by hR, +, ·.−, 0, 1, ≤i. We mean here, in part, that + : R × R → R is a function from the set of pairs of real numbers into the set of real numbers and that ≤⊆ R × R is a two-place relation on the set of real numbers. Consider another example: the set of real numbers that are algebraic over the rationals endowed with addition, multiplication, negation, zero, one, and the usual ordering. We could denote this structure by hA, +, ·, −, 0, 1, ≤i. In this example + : A × A → A is again a two-place function but it differs from the operation + of the first example—the domains are not the same. A sharper illustration of this point might use the set of all continuous functions from the unit interval into the set of real numbers, endowed with addition, multiplication, negation, the constantly zero function, the constantly one function, and the usual pointwise ordering. In mathematical practice, we trust to context to resolve any ambiguities that arise in this way. At least tacitly, what happens is that we treat + as a symbol that can be subjected to many interpretations requiring only that in every such interpretation that + denote a two- place operation. In the same way, we treat ≤ as a symbol open to many interpretations, requiring only the in every such interpretation that ≤ denote a two-place relation. We start our project by formalizing this common piece of mathematical practice. In the examples above, we used two 2-place operation symbols, one 1-place operation symbol (it was −), two 0-place operation symbols (these were 0 and 1) and one 2-place relation symbol. It is important for the development of the theory of the ordererd field of real number not to confuse addition and multiplication. Our arrangements must take that into account.
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