Abstract.
Theories of classification distinguish classes with some good structure theorem from those for which none is possible. Some classes (dense linear orders, for instance) are non-classifiable in general, but are classifiable when we consider only countable members. This paper explores such a notion for classes of computable structures by working out several examples. One motivation is to see whether some classes whose set of countable members is very complex become classifiable when we consider only computable members. We follow recent work by Goncharov and Knight in using the degree of the isomorphism problem for a class to distinguish classifiable classes from non-classifiable. For real closed fields we show that the isomorphism problem is Δ1 1 complete (the maximum possible), and for others we show that it is of relatively low complexity. We show that the isomorphism problem for algebraically closed fields, Archimedean real closed fields, or vector spaces is Π0 3 complete.
This is a preview of subscription content, log in via an institution to check access.
Similar content being viewed by others
References
Baker, A.: Transcendental number theory. Cambridge: Cambridge University Press, 1975
Becker, H., Kechris, A. S.: The descriptive set theory of polish group actions. London Mathematical Society Lecture Note Series, vol. 232, Cambridge University Press, 1996
Besicovitch, A. S.: On the linear independence of fractional powers of integers. J. London Math. Soc. 15, 3–6 (1940)
Calvert, W.: The isomorphism problem for classes of computable Abelian groups. Preprint, 2003
Friedman, H., Stanley, L.: A Borel reducibility theory for classes of countable structures. J. Symbolic Logic 54, 894–914 (1989)
Goncharov, S. S., Knight, J. F.: Computable structure and non-structure theorems. Algebra and Logic 41, 351–373 (2002)
Hirschfeldt, D., Khoussainov, B., Shore, R., Slinko, A. M.: Degree spectra and computable dimensions in algebraic structures. Ann. Pure Appl. Logic 115, 71–113 (2002)
Hjorth, G., Kechris, A. S., Louveau, A.: Borel equivalence relations induced by actions of the symmetric group. Annals of Pure and Applied Logic 92, 63–112 (1998)
Hodges, W.: What is a structure theory?. Bull. London Math. Soc. 19, 209–237 (1987)
Metakides, G., Nerode, A.: Recursively enumerable vector spaces. Ann. Math. Logic 11, 146–171 (1977)
Morozov, A. S.: Functional trees and automorphisms of models. Algebra and Logic 32, 28–38 (1993)
Nies, A.: Undecidable fragments of elementary theories, Algebra Universalis 35, 8–33 (1996)
Rabin, M. O., Scott, D.: The undecidability of some simple theories. Preprint
Shelah, S.: Classification of first order theories which have a structure theorem. Bull. Amer. Math. Soc. (N.S.) 12, 227–232 (1985)
Soare, R. I.: Recursively enumerable sets and degrees, Springer-Verlag, 1987
Author information
Authors and Affiliations
Corresponding author
Rights and permissions
About this article
Cite this article
Calvert, W. The isomorphism problem for classes of computable fields. Arch. Math. Logic 43, 327–336 (2004). https://doi.org/10.1007/s00153-004-0219-1
Received:
Published:
Issue Date:
DOI: https://doi.org/10.1007/s00153-004-0219-1