[PDF] Ultraproducts And Their Application To Model Theory eBook

Author : Amanda Purcell
Publisher :
Page : 59 pages
File Size : 39,23 MB
Release : 2013
Category : First-order logic
ISBN : 9781303604058

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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.

Author : John Lane Bell
Publisher :
Page : 344 pages
File Size : 32,91 MB
Release : 1971
Category : Mathematics
ISBN :

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The aim of this book is to provide an elementary exposition of some of the basic concepts of model theory. Model theory, which can be described briefly as the study of the relationship between formal languages and abstract structures, covers a very wide field and it is not possible to compress it into one volume. We have chosen as our theme the ultraproducts construction. We hope this book we be of use to undergraduate and practicing mathematicians.

Author : Hans Schoutens
Publisher : Springer Science & Business Media
Page : 215 pages
File Size : 44,90 MB
Release : 2010-07-31
Category : Mathematics
ISBN : 3642133673

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Exploring ultraproducts of Noetherian local rings from an algebraic perspective, this volume illustrates the many ways they can be used in commutative algebra. The text includes an introduction to tight closure in characteristic zero, a survey of flatness criteria, and more.

Author : William Wistar Comfort
Publisher : Springer
Page : 518 pages
File Size : 33,30 MB
Release : 1974
Category : Mathematics
ISBN :

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An ultrafilter is a truth-value assignment to the family of subsets of a set, and a method of convergence to infinity. From the first (logical) property arises its connection with two-valued logic and model theory; from the second (convergence) property arises its connection with topology and set theory. Both these descriptions of an ultrafilter are connected with compactness. The model-theoretic property finds its expression in the construction of the ultraproduct and the compactness type of theorem of Los (implying the compactness theorem of first-order logic); and the convergence property leads to the process of completion by the adjunction of an ideal element for every ultrafilter-i. e., to the Stone-Cech com- pactification process (implying the Tychonoff theorem on the compact- ness of products). Since these are two ways of describing the same mathematical object, it is reasonable to expect that a study of ultrafilters from these points of view will yield results and methods which can be fruitfully crossbred. This unifying aspect is indeed what we have attempted to emphasize in the present work.

Author : Chen Chung Chang
Publisher : Princeton University Press
Page : 180 pages
File Size : 45,26 MB
Release : 1966-06-21
Category : Mathematics
ISBN : 0691079293

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This is a study of the theory of models with truth values in a compact Hausdorff topological space.