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Ultraproducts and Their Applications

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Ultraproducts and Their Applications © COPYRIGHT by Amanda Purcell 2013 ALL RIGHTS RESERVED ULTRAPRODUCTS AND THEIR APPLICATIONS BY Amanda Purcell ABSTRACT An ultraproduct is a mathematical construction used primarily in abstract algebra and model theory to create a new structure by reducing a product of a family of existing structures using a class of objects referred to as filters. This thesis provides a rigorous construction of ultraproducts and investigates some of their applications in the fields of mathematical logic, nonstandard analysis, and complex analysis. An introduction to basic set theory is included and used as a foundation for the ultraproduct construction. It is shown how to use this method on a family of models of first order logic to construct a new model of first order logic, with which one can produce a proof of the Compactness Theorem that is both elegant and robust. Next, an ultraproduct is used to offer a bridge between intuition and the formalization of nonstandard analysis by providing concrete infinite and infinitesimal elements. Finally, a proof of the Ax- Grothendieck Theorem is provided in which the ultraproduct and other previous results play a critical role. Rather than examining one in depth application, this text features ultraproducts as tools to solve problems across various disciplines. ii ACKNOWLEDGMENTS I would like to give very special thanks to my advisor, Professor Ali Enayat, whose expertise, understanding, and patience, were invaluable to my pursuit of a degree in mathematics. His vast knowledge and passion for teaching inspired me throughout my collegiate and graduate career and will continue to do so. iii TABLE OF CONTENTS ABSTRACT .................................................................................................................................... ii ACKNOWLEDGMENTS ............................................................................................................. iii Chapter 1. INTRODUCTION .......................................................................................................... 1 2. FILTERS, ULTRAFILTERS, AND REDUCED PRODUCTS ..................................... 3 3. ULTRAPRODUCTS AND COMPACTNESS ............................................................ 16 4. ULTRAPRODUCTS AND NONSTANDARD UNIVERSES .................................... 31 5. THE AX-GROTHENDIECK THEOREM ................................................................... 47 BIBLIOGRAPHY ......................................................................................................................... 59 iv CHAPTER 1 INTRODUCTION The ultraproduct construction is a method used primarily in model theory and abstract algebra that creates a new structure by reducing a Cartesian product of existing structures. Its value stems from the ability for one to determine properties of the ultraproduct based solely on the properties of the structures within the reduced product and the equivalence relation by which it is reduced. The ultraproduct is attractive in its algebraic nature and the fact that it can be constructed using only basic set theoretic concepts. This thesis begins (Chapter 2) with an outline of the set theory required for the construction of ultraproducts beginning with filters (specific sets of sets). It includes important observations on the existence and uses of certain filters before providing the step-by-step construction of reduced products and describing their elements. This chapter provides the foundation for all applications of ultraproducts discussed ahead. To be able to define the notion of an ultraproduct, and to be able to understand its properties, it is necessary to provide a brief introduction to model theory, which comes in the beginning of Chapter 3. Chapter 3 then defines an ultrapoduct of models and the elements, functions, and relations on this new object. Next, this chapter instructs how to interpret sentences of first order logic in the ultraproduct model by proving the ever-important and applicable Łos’s´ Theorem which will recur throughout the text. Ultimately, Chapter 3 provides a proof of the Compactness Theorem, one of the most fundamental results in first order logic. Chapter 4 then changes gears to provide an examination of basic nonstandard analysis using the tools and elements of ultraproducts developed in earlier sections. The ultraproduct construction allows us to build ordered "nonarchimedean" fields, i.e., fields 1 2 that contain, as concrete objects, infinitely small quantities as well as infinitely large ones. Moreover, these nonarchimedean fields are shown to satisfy the same first order sentences as those that are true in the ordered field of real numbers. In chapter 4 we employ the aforementioned nonarchimedean fields to provide a new way of establishing a number of classical results in Calculus in an intuitively clear manner, with the explicit, unabashed use of infinitesimals. The final chapter (5) offers a proof of the Ax-Grothendieck Theorem, alternative to the proof from Walter Rudin [1] utilizing techniques of complex analysis. This final part highlights the rather metamathematical method of proof used a great deal in disciplines of advanced mathematics: the method in which one takes a problem from some specific area, maps the relevant pieces to a different universe in which the problem can be solved, and maps back to complete the proof. In the end, it should be clear to the reader that the ultraproduct model is not only interesting in and of itself, but is a very powerful tool that can provide answers, or methods to obtain them, in many other disciplines outside of model theory. 3 CHAPTER 2 FILTERS, ULTRAFILTERS, AND REDUCED PRODUCTS Set Theory and Construction of Filters We begin with a few basic set theoretic concepts that will be useful in our construction of filters, and ultimately in the construction of ultraproducts. • The powerset of a set X, denoted P(X), is the set of all possible subsets of X. For example, for X = fA;Bg, the powerset of X would be written P(X) = f/0;fAg;fBg;fA;Bgg, where /0denotes the “empty set,” the unique set containing no elements. • A is a subset of a set X, or X contains A (written A ⊆ X), if every element of A is also an element of X. • The union of two sets, A and B, denoted A [ B, is the collection of elements that are in A, in B, or in both A and B. • The intersection of two sets A and B, denoted A \B, is the collection of elements that are in both A and B. • The complement of a set A in relation to a background set X where A ⊆ X, denoted XnA, is the set of all elements of X that are not in the subset A. For purposes that arise in a later section, it is also important to understand the concept of cardinality. The cardinality of a set X, denoted jXj, is the number of elements in the set; numbers representing the cardinality of sets are called “cardinal numbers.” For example, the cardinality of the natural numbers (N) is ℵ0, the first infinite cardinal number. The ℵ ℵ cardinality of the real numbers (R) is 2 0 , where 2 0 > ℵ0 by a classical theorem of Cantor. 4 Armed with these basic set theoretic concepts, we are now equipped to discuss the first class of objects important to our construction of the ultraproduct: filters. Definition 1. Let X be a nonempty set. A filter F over X is a nonempty family of sets F ⊆ P(X) such that: (i) if A; B 2 F , then A \ B 2 F , and (ii) if A 2 F and A ⊆ B ⊆ X, then B 2 F . In other words, F is a set of subsets of X that is closed under intersections and supersets. For example, F = P(X) is a filter over X; this is called the “improper filter.” The set containing only X itself, fXg, also forms a filter, called the “trivial filter.” All other filters are referred to as “proper nontrivial filters.” Proposition 2. A filter F is proper if and only if /0 2= F . Proof. We prove the ) direction by way of contradiction. Suppose that F is a proper filter over X and that /0 2 F . As all subsets A ⊆ X contain /0, (/0 ⊆ A ⊆ X) ) (A 2 F ); for all A ⊆ X, and every possible subset of X is a member of F by upward closure. Therefore F = P(X), and F must be improper, contradicting our assumption. Hence, for any proper filter F over a set X, we have /0 2= F . The proof of the ( direction is trivial. Also note that for all filters F over X, the background set X will be a member of the filter F . Given that F is nonempty by definition, let A 2 F for some A ⊆ X. By upward closure, we have (A ⊆ X) ) (X 2 F ); 5 and X 2 F for all filters F over X. Some examples of proper filters include: • The principal filter: Fix some element i0 2 X, and consider the family of subsets fA ⊆ X ji0 2 Ag. This defines a filter over X, and is referred to as the principal filter generated by i0. Any filter that is not principal is referred to as free. • The Fréchet filter: Let X be an infinite set, and let F = fA ⊆ X jXnA is finiteg; i.e., the set of all co-finite subsets of X. This filter is called the Fréchet filter, a very important filter that will be utilized later. Before we can discuss more interesting properties about filters, we must first define the following property: Definition 3. A set F of subsets of X is said to have the finite intersection property, or f.i.p., if the intersection of any finite number of those subsets is nonempty, i.e., A1 \ A2 \ ::: \ An 6= /0 for Ai 2 F , 1 ≤ i ≤ n, and n 2 N. We claim that any proper filter has f.i.p., and for any set of subsets that has f.i.p., there exists a proper filter containing that set. Before we prove our claim, consider our examples of proper filters above. The principal filter generated by i0 clearly has f.i.p. as i0 will be a member of every set in the filter, and thus a member of any finite intersection.
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