Super-Real Fields Totally Ordered Fields with Additional Structure

by Dales, H. Garth; Woodin, W. Hugh

• ISBN13: 9780198539919
• ISBN10: 0198539916
• Format: Hardcover
• Copyright: 1996-08-01
• Publisher: Clarendon Press

Summary

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 continuousultrapowers. 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 beaccessible 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 modeltheory.

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

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