Super-real fields : totally ordered fields with additional structure
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Bibliographic Information
Super-real fields : totally ordered fields with additional structure
（London Mathematical Society monographs, new ser.,
Clarendon, 1996
Available at / 34 libraries
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Library, Research Institute for Mathematical Sciences, Kyoto University数研
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Description and Table of Contents
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.
Table of Contents
- 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
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