Long reals

Abstract

The familiar continuum R of real numbers is obtained by a well-known procedure which, starting with the set of natural numbers N = ω , produces in a canonical fashion the field of rationals Q and, then, the field R as the completion of Q under Cauchy sequences (or, equivalently, using Dedekind cuts). In this article, we replace ω by any infinite suitably closed ordinal κ in the above construction and, using the natural (Hessenberg) ordinal operations, we obtain the corresponding field κ–R, which we call the field of the κ–reals. Subsequently, we study the properties of the various fields κ–R and develop their general theory, mainly from the set-theoretic perspective. For example, we investigate their connection with standard themes such as forcing and descriptive set theory.