Super-Real Fields

Totally Ordered Fields with Additional Structure
Erscheinungsjahr 1996
ISBN: 978-0-19-853991-9
Verlag: OUP Oxford

Buch, Englisch, 376 Seiten, Print PDF, Format (B × H): 161 mm x 240 mm, Gewicht: 728 g

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.

Autoren/Hrsg.

Dales, H. Garth

Woodin, W. Hugh

Fachgebiete

• Mathematik | Informatik Mathematik Algebra Homologische Algebra
• Mathematik | Informatik Mathematik Mathematik Allgemein Mathematische Logik
• Mathematik | Informatik Mathematik Mathematische Analysis Reelle Analysis

Inhaltsverzeichnis

- Introduction

- 1: Ordered sets and ordered groups

- 2: Ordered fields

- 3: Completions of ordered groups and fields

- 4: Algebras of continuous functions

- 5: Normability and universality

- 6: The operational calculus and the field R

- 7: Examples

- 8: Non-standard structures for super-real fields and the gap theorem

- 9: R as a hyper-real field

- 10: Models and weak Cauchy completeness

- 11: Rigid fields and solids structures

- 12: Open questions