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.2010 Mathematics Subject Classification 12L15, 12J15, 12L99 (primary); 03E10, 03E15 (secondary)

Intermediate arithmetic operations on ordinal numbers

“…Remark 4.2. The version of this theorem where hypothesis (5') is used was also proven independently, in slightly stronger form, by Asperó and Tsaprounis [2], using essentially the same means.…”

“…We shall prove this in § . (A version of this theorem was also proven independently by Asperó and Tsaprounis around the same time this paper was being written . Their desiderata for natural exponentiation are slightly different, but the method of proof is essentially the same.…”

Generalized Polish spaces at regular uncountable cardinals

“…The fact that there is no preferred structure for the range of our generalized metrics implies that every possible generalization‐to‐level‐$$ \kappa$$ of the reals yields an example of $$\mathbb {G}$$‐Polish space (as long as this generalization preserves properties like being Cauchy‐complete with respect to its canonical metric over itself). For example, this applies to the long reals introduced by Klaua in [11] and studied by Asperó and Tsaprounis in [2], or to the generalization of $$\mathbb {R}$$ introduced in [7] using the surreal numbers. See also [4] for other examples of $$\mathbb {G}$$‐Polish spaces, as well as methods to construct Cauchy‐complete totally ordered fields.…”

“…In the subsequent results, $$\mathbb {G}$$ is a totally ordered Abelian group with $$\operatorname{Deg}(\mathbb {G})=\kappa$$. Examples of groups of this form are $$\mathbb {Z}^\kappa$$ and $$\mathbb {R}^\kappa$$ (and any other $$(\mathbb {G}^{\prime })^\kappa$$ for an Abelian group $$\mathbb {G}^{\prime }$$) equipped with coordinate‐wise operations and lexicographic orders, the “$$\kappa$$‐versions of the reals” proposed in [2] or in [7], or any nonstandard model of the reals of degree $$\kappa$$.…”

Generalized Polish spaces at regular uncountable cardinals

“…The fact that there is no preferred structure for the range of our generalized metrics implies that every possible generalization-to-level-κ of the reals yields to an example of G-Polish space (as long as this generalization preserves properties like being Cauchy-complete with respect to its canonical metric over itself). For example, this applies to the long reals introduced by Klaua in [Kla60] and studied by Asperó and Tsaprounis in [AT18], or to the generalization of R introduced in [Gal19] using the surreal numbers. See also [DW96] for other examples of G-Polish spaces, as well as methods to construct Cauchy-complete totally ordered fields.…”