Super-Real Fields: Totally Ordered Fields with Additional Structure / Edition 1

by H. Garth Dales, W. Hugh Woodin

Overview

Super-fields are a class of totally ordered fields that 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 an important topic in their own right and have many surprising applications in analysis and logic. The authors introduce these exciting new fields to mathematicians, analysts, and logicians, including a natural generalization of the real line R, and resolve a number of open problems. After an exposition of the general theory of ordered fields and a careful proof of some classic theorems, including Kapansky's embedding, they establish important new results in Banach algebra theory, non-standard analysis, and model theory.

Product Details

ISBN-13:	9780198539919
Publisher:	Oxford University Press
Publication date:	08/01/1996
Series:	London Mathematical Society Monographs , #14
Pages:	376
Product dimensions:	9.50(w) x 6.38(h) x 0.97(d)

About the Author

University of Leeds

University of California, Berkeley

Table of Contents

Introduction1. Ordered sets and ordered groups2. Ordered fields3. Completions of ordered groups and fields4. Algebras of continuous functions5. Normability and universality6. The operational calculus and the field R7. Examples8. Non-standard structures for super-real fields and the gap theorem9. R as a hyper-real field10. Models and weak Cauchy completeness11. Rigid fields and solids structures12. Open questions