Order positive fields. I

M. V. Korovina^a, O. V. Kudinov<sup>b

a</sup> A.P. Ershov Institute of Informatics Systems, Siberian Branch of the Russian Academy of Sciences, Novosibirsk
^b Novosibirsk State University

Abstract: The notion of a computable structure based on numberings with decidable equality is well established with a number of prominent results. Nevertheless, applied to strictly ordered fields, it fails to capture some natural properties and constructions for which decidability of equality is not assumed. For example, the field of primitive recursive real numbers is not computable, and there exists a computable real closed field with noncomputable maximal Archimedean subfields. We introduce the notion of an order positive field which aims to overcome these limitations. A general criterion is presented which decides when an Archimedean field is order positive. Using this criterion, it is shown that the field of primitive recursive real numbers is order positive and that the Archimedean parts of order positive real closed fields are order positive. We also state a program for further research.

Keywords: strictly ordered fields, positive structures, computable real numbers.

UDC: 510.665:512.623

Received: 21.04.2023
Revised: 10.04.2024

DOI: 10.33048/alglog.2023.62.301

 English version: Algebra and Logic, 2023, 62:3, 203–214