Degree Spectra of Real Closed Fields

TL;DR

This paper explores the spectra of various real closed fields, constructing examples with specific computability properties and demonstrating the diversity of their computational complexity profiles.

Contribution

It introduces new constructions of real closed fields with prescribed computability spectra, including nonarchimedean and archimedean cases, expanding understanding of their computational classifications.

Findings

Constructed a noncomputable, computably enumerable subfield of computable reals with no computable copy.

Built an archimedean real closed field with no computable copy but a computable Dedekind cut enumeration.

Developed a nonarchimedean real closed field whose residue field lacks a computable presentation.

Abstract

Several researchers have recently established that for every Turing degree $\boldsymbol{c}$, the real closed field of all $\boldsymbol{c}$-computable real numbers has spectrum $\{\boldsymbol{d}~:~\boldsymbol{d}'\geq\boldsymbol{c}"\}$. We investigate the spectra of real closed fields further, focusing first on subfields of the field $\mathbb{R}_{\boldsymbol{0}}$ of computable real numbers, then on archimedean real closed fields more generally, and finally on non-archimedean real closed fields. For each noncomputable, computably enumerable set $C$, we produce a real closed $C$-computable subfield of $\mathbb{R}_{\boldsymbol{0}}$ with no computable copy. Then we build an archimedean real closed field with no computable copy but with a computable enumeration of the Dedekind cuts it realizes, and a computably presentable nonarchimedean real closed field whose residue field has no computable presentation.