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, we show 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.
This is a preview of subscription content, log in via an institution to check access.
Similar content being viewed by others
References
M. O. Rabin, “Computable algebra, general theory and theory of computable fields,” Trans. Am. Math. Soc., 95, No. 2, 341-360 (1960).
A. I. Mal’tsev, “On recursive Abelian groups,” Dokl. Akad. Nauk SSSR, 146, No. 5, 1009-1012 (1962).
N. G. Khisamiev, “Constructive Abelian groups,” in Handbook of Recursive Mathematics, Vol. 2, Recursive Algebra, Analysis and Combinatorics, Stud. Log. Found. Math., 139, Yu. L. Ershov et al. (eds.), Elsevier, Amsterdam (1998), pp. 1177-1231.
Yu. L. Ershov, Problems of Decidability and Constructive Models [in Russian], Nauka, Moscow (1980).
V. Stoltenberg-Hansen and J. V. Tucker, “Computable rings and fields,” in Handbook of Computability Theory, Stud. Logic Found. Math., 140, Elsevier, Amsterdam (1999), pp. 363-447.
K. Weihrauch, Computable Analysis. An Introduction, Texts Theor. Comput. Sci., EATCS Ser., Springer-Verlag, Berlin (2000).
S. S. Goncharov and Yu. L. Ershov, Constructive Models, Siberian School of Algebra and Logic [in Russian], Nauch. Kniga, Novosibirsk (1999).
Yu. L. Ershov, Numeration Theory [in Russian], Nauka, Moscow (1977).
R. Miller and V. O. Gonz´ales, “Degree spectra of real closed fields,” Arch. Math. Log., 58, Nos. 3/4, 387-411 (2019).
E. W. Madison, “A note on computable real fields,” J. Symb. Log., 35, No. 2, 239-241 (1970).
R. M. Smullyan, “Creativity and effective inseparability,” Trans. Am. Math. Soc., 109, 135-145 (1963).
Author information
Authors and Affiliations
Corresponding author
Additional information
Translated from Algebra i Logika, Vol. 62, No. 3, pp. 307-322, May-June, 2023. Russian DOI:https://doi.org/10.33048/alglog.2023.62.301.
The study was carried out within the framework of the state assignment to Ershov Institute of Informatics Systems SB RAS, project FWNU-2021-0003. (M. V. Korovina)
O. V. Kudinov is supported by RSCF grant No. 23-11-00170 (https://rscf.ru/project/23-11-00170) and by the Program of Fundamental Research RAS, project FWNF-2022-0011.
Rights and permissions
Springer Nature or its licensor (e.g. a society or other partner) holds exclusive rights to this article under a publishing agreement with the author(s) or other rightsholder(s); author self-archiving of the accepted manuscript version of this article is solely governed by the terms of such publishing agreement and applicable law.
About this article
Cite this article
Korovina, M.V., Kudinov, O.V. Order Positive Fields. I. Algebra Logic 62, 203–214 (2023). https://doi.org/10.1007/s10469-024-09738-1
Received:
Accepted:
Published:
Issue Date:
DOI: https://doi.org/10.1007/s10469-024-09738-1