Dissertation/Thesis Abstract

Ultraproducts and their applications
by Purcell, Amanda, M.A., American University, 2013, 64; 1549128
Abstract (Summary)

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.

Indexing (document details)
Advisor: Enayat, Ali
Commitee: Casey, Stephen, Lansky, Joshua
School: American University
Department: Mathematics and Statistics
School Location: United States -- District of Columbia
Source: MAI 52/04M(E), Masters Abstracts International
Subjects: Mathematics
Publication Number: 1549128
ISBN: 978-1-303-60405-8
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