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.

• 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