دانلود کتاب مبانی منطقی تحلیل غیراستاندارد با کاربردهایی در نظریه اقتصادی

It has been my intention in writing this book to achieve two goals: on the one hand, to provide a complete chain of demonstrations that lead from the solid foundation of class theory to the main results of nonstandard analysis along with some of its applications to economic theory; on the other hand, to minimize the length of the chain by restricting it to the strictly necessary links. I have used a unified language that is standard in logical theory and more digestible for philosophers, yet also attractive for mathematicians and economists. The novelty of the book is primarily in the organization and minimalist completeness of the subject, as well as in a logical foundation for nonstandard economics, based on Debreu’s [1] concept of social system. I have aimed to provide my own demonstrations of well-known theorems, simpler and more consistent with the adopted notation. Ackermann [2] provided a very elegant and austere foundation for class theory that I have taken as point of departure. The original system had to be complemented with additional axioms that I introduce in order to get a complete system for class and set theory. The present work leads the reader from this system, step by step, to a universe rich and complex enough to provide the foundations for nonstandard analysis and develop it sufficiently for its applications in economic theory. The idea has been to introduce the minimum theory and results required t o lead the reader (particularly one with philosophical training) from Cantor’s Paradise to the Garden of Delights of nonstandard analysis and its applications in economic theory. Almost all of the theorems presented are well known, but the sequence in which they are ordered and some of the demonstrations are not. In particular, finding the minimal sequence leading to saturation results turned out to be an exacting task. Eager to develop or apply nonstandard analysis, mathematicians generally dedicate little space to the complete and systematic development of its foundations. On the other hand, logicians barely sketch the construction of the nonstandard universe (not even [3] fully develops the demonstrations) and fall short of showing that such a construction satisfies the requirements demanded by analysis (as, e.g., Goldblatt formulates them in 1988, pp. 158–162, who, by the way, also does not show it). This work aims to introduce the necessary apparatus to present a complete and integral foundation of nonstandard analysis. After reviewing the necessary set theory for the task, I present a fairly complete account of Boolean algebra theory, including demonstrations of Tarski’s theorems on fixed points of lattices and a detailed proof of Stone’s Representation Theorem. The constructions in this chapter will be used in Chap. 3 to present a somewhat different proof of Gödel’s Completeness Theorem. I develop topics—aimed at completeness—that are not strictly necessary for the development of nonstandard analysis but were within reach of developments that are. They may be omitted to stay strictly within the minimalist chain, but they may be useful for a course in Boolean algebras. In Chap. 3, I review first-order logic and prove the existence of “non- Archimedean” spaces using Anatoli Maltsev’s Lemma (Compactness Theorem). Later, in Chap. 4, using the ultraproduct construction, I proceed to develop the theory of ultraproducts up to the natural embedding. I place particular emphasis on the construction of saturated models, as saturated universes have nice properties that allow for the adequate development of the topological spaces required by nonstandard economics. In Chap. 5, I construct the required nonstandard space and prove that this space satisfies the requirements demanded by analysis, as Goldblatt formulates them. I include a discussion of internal entities with some methods to deal with them. In Chap. 6, I develop in some detail the theory of nonstandard numbers, also called “hyperreal numbers.” The development is sufficient to provide, in Chap. 7, an introduction to infinitesimal calculus that is sufficient for a host of applications in economics. Chapter 8 is dedicated to an exposition of the nonstandard topologies required by economic theory, but not before providing an introduction to the standard topology needed to transition to the nonstandard topological notions. In Chap. 9, I introduce nonstandard versions of the classical theorems of the Separating Hyperplane and the Fixed Point of FanGlicksberg, as well as Kakutani’s, which turns out to be a corollary of the latter. I also address the use of the Lyapunov- Loeb Theorem in economic theory. I conclude t-he last chapter, Chap. 10, by developing in detail Richter’s [4] Theorem on the existence of nonstandard representations of utility. Then, I introduce a nonstandard formulation of Debreu’s [1] concept of a social system, prove the existence of equilibria for social systems, and conclude the book with an application of the concept to economic theory, specifically to the theory of pure exchange. Although relatively self-contained, and dealing with almost all topics in textbooks, due to its monographic nature, this work does not intend to be a textbook on set theory and logic, although it can be used as such in a graduate course. I have aimed to leave to the reader some easy demonstrations as exercises, or the completion of others.