Geodesic
Order positive fields.~I
Algebra i logika, Tome 62 (2023) no. 3, pp. 307-322.

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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, computable real numbers.
Mots-clés : positive structures 
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M. V. Korovina; O. V. Kudinov. Order positive fields.~I. Algebra i logika, Tome 62 (2023) no. 3, pp. 307-322. http://geodesic.mathdoc.fr/item/AL_2023_62_3_a0/

[1] M. O. Rabin, “Computable algebra, general theory and theory of computable fields”, Trans. Am. Math. Soc., 95:2 (1960), 341–360 | MR | Zbl

[2] A. I. Maltsev, “O rekursivnykh abelevykh gruppakh”, Dokl. AN SSSR, 146:5 (1962), 1009–1012 | Zbl

[3] N. G. Khisamiev, “Constructive abelian groups”, Handbook of recursive mathematics, v. 2, Stud. Logic Found. Math., 139, Recursive algebra, analysis and combinatorics, eds. Yu. L. Ershov et al., Elsevier, Amsterdam, 1998, 1177–1231 | DOI | MR | Zbl

[4] Yu. L. Ershov, Problemy razreshimosti i konstruktivnye modeli, Nauka, M., 1980

[5] V. Stoltenberg-Hansen, J. V. Tucker, “Computable rings and fields”, Handbook of computability theory, Stud. Logic Found. Math., 140, ed. E. R. Griffor, Elsevier, Amsterdam, 1999, 363–447 | DOI | MR | Zbl

[6] K. Weihrauch, Computable analysis. An introduction, Texts Theor. Comput. Sci., EATCS Ser., Springer, Berlin, 2000 | DOI | MR | Zbl

[7] S. S. Goncharov, Yu. L. Ershov, Konstruktivnye modeli, Sibirskaya shkola algebry i logiki, Nauchnaya kniga, Novosibirsk, 1999

[8] Yu. L. Ershov, Teoriya numeratsii, Nauka, M., 1977 | MR

[9] R. Miller, V. O. Gonzáles, “Degree spectra of real closed fields”, Arch. Math. Logic, 58:3/4 (2019), 387–411 | DOI | MR | Zbl

[10] E. W. Madison, “A note on computable real fields”, J. Symb. Log., 35:2 (1970), 239–241 | DOI | MR | Zbl

[11] R. M. Smullyan, “Creativity and effective inseparability”, Trans. Am. Math. Soc., 109 (1963), 135–145 | DOI | MR | Zbl