Super-real Fields: Totally Ordered Fields with Additional Structure

Hardcover | March 1, 1978

byH. Garth Dales, W. Hugh Woodin

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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.

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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 thei...

H. Garth Dales is at University of Leeds. W. H. Woodin is at University of California, Berkeley.

other books by H. Garth Dales

Banach Algebras and Automatic Continuity
Banach Algebras and Automatic Continuity

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$439.80 online$555.00list price(save 20%)
Format:HardcoverDimensions:376 pages, 9.21 × 6.14 × 0.87 inPublished:March 1, 1978Publisher:Oxford University Press

The following ISBNs are associated with this title:

ISBN - 10:0198539916

ISBN - 13:9780198539919

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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

Editorial Reviews

`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.'Zentrallblat fur Mathematik, vol. 856, 1997