Super-Real Fields
Totally Ordered Fields with Additional Structure
Series: London Mathematical Society Monographs; 14;
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Product details:
- Publisher OUP Oxford
- Date of Publication 16 May 1996
- ISBN 9780198539919
- Binding Hardback
- No. of pages376 pages
- Size 241x162x22 mm
- Weight 682 g
- Language English
- Illustrations line figures 0
Categories
Short description:
This advanced text expounds the established theory of ordered fields, and continues to develop a quite original theory of super-real fields. This theory has important applications in analysis and logic.
MoreLong description:
Super-real fields are a class of large totally ordered fields. These fields are larger than the real line. They arise from quotients of the algebra of continuous functions on a compact space by a prime ideal, and generalize the well-known class of ultrapowers, and indeed the continuous ultrapowers. These fields are of interest in their own right and have many surprising applications, both in analysis and logic. The authors introduce some exciting new fields, including a natural generalization of the real line R, and resolve a number of open problems. The book is intended to be accessible to analysts and logicians. After an exposition of the general theory of ordered fields and a careful proof of some classic theorems, including Kaplansky's embedding theorems , the authors establish important new results in Banach algebra theory, non-standard analysis, an model theory.
This book is an interesting research monograph which, because of its several fine introductory chapters, should be accessible to a large number of readers ... The breadth and depth of the analysis and synthesis of the "additional structure", used to prove some of these theorems, is impressive .
Table of Contents:
Introduction
Ordered sets and ordered groups
Ordered fields
Completions of ordered groups and fields
Algebras of continuous functions
Normability and universality
The operational calculus and the field R
Examples
Non-standard structures for super-real fields and the gap theorem
R as a hyper-real field
Models and weak Cauchy completeness
Rigid fields and solids structures
Open questions
