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Computable dimension for ordered fields

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Abstract

The computable dimension of a structure counts the number of computable copies up to computable isomorphism. In this paper, we consider the possible computable dimensions for various classes of computable ordered fields. We show that computable ordered fields with finite transcendence degree are computably stable, and thus have computable dimension 1. We then build computable ordered fields of infinite transcendence degree which have infinite computable dimension, but also such fields which are computably categorical. Finally, we show that 1 is the only possible finite computable dimension for any computable archimedean field.

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References

  1. Ash C., Knight J., Manasse M., Slaman T.: Generic copies of countable structures. Ann. Pure Appl. Log. 42(3), 195–205 (1989). doi:10.1016/0168-0072(89)90015-8

    Article  MathSciNet  MATH  Google Scholar 

  2. Chisholm J.: Effective model theory vs. recursive model theory. J. Symb. Log. 55(3), 1168–1191 (1990). doi:10.2307/2274481

    Article  MathSciNet  MATH  Google Scholar 

  3. Csima, B.F., Khoussainov, B., Liu, J.: Computable categoricity of graphs with finite components. In: Logic and Theory of Algorithms. 4th Conference on Computability in Europe, CiE 2008, Athens, Greece, June 15–20, 2008. Proceedings, pp. 139–148. Springer, Berlin (2008). doi:10.1007/978-3-540-69407-6-15

  4. Goncharov S.: Autostability of models and Abelian groups. Algebra Log. 19, 13–27 (1980). doi:10.1007/BF01669101

    Article  MathSciNet  MATH  Google Scholar 

  5. Goncharov S., Dzgoev V.: Autostability of models. Algebra Log. 19, 28–37 (1980). doi:10.1007/BF01669102

    Article  MathSciNet  MATH  Google Scholar 

  6. Goncharov S., Molokov A., Romanovskij N.: Nilpotent groups of finite algorithmic dimension. Sib. Math. J. 30(1), 63–68 (1989). doi:10.1007/BF01054216

    Article  MathSciNet  MATH  Google Scholar 

  7. Goncharov, S.S.: Limit equivalent constructivizations. In: Mathematical Logic and the Theory of Algorithms, Trudy Inst. Mat., vol. 2, pp. 4–12. “Nauka” Sibirsk. Otdel., Novosibirsk (1982)

  8. Goncharov S.S., Lempp S., Solomon R.: The computable dimension of ordered abelian groups. Adv. Math. 175(1), 102–143 (2003). doi:10.1016/S0001-8708(02)00042-7

    Article  MathSciNet  MATH  Google Scholar 

  9. Hungerford, T.W.: Algebra. Holt, Rinehart and Winston, Inc., New York (1974)

  10. Jacobson, N.: Lectures in Abstract Algebra. Theory of Fields and Galois Theory, vol. III. D. Van Nostrand Co. Inc., Princeton, New Jersey (1964)

  11. Lang, S.: Algebra, Graduate Texts in Mathematics, vol. 211, 3rd edn. Springer, New York (2002). doi:10.1007/978-1-4613-0041-0

  12. Lempp S., McCoy C., Miller R., Solomon R.: Computable categoricity of trees of finite height. J. Symb. Log. 70(1), 151–215 (2005). doi:10.2178/jsl/1107298515

    Article  MathSciNet  MATH  Google Scholar 

  13. Levin, O.: Computability Theory, Reverse Mathematics, and Ordered Fields. Ph.D. thesis, University of Connecticut, Storrs, CT (2009)

  14. Madison E.W.: A note on computable real fields. J. Symb. Log. 35, 239–241 (1970)

    Article  MathSciNet  MATH  Google Scholar 

  15. Marker D.: Model Theory, Graduate Texts in Mathematics, vol. 217. Springer, New York (2002)

    Google Scholar 

  16. Metakides G., Nerode A.: Effective content of field theory. Ann. Math. Log. 17(3), 289–320 (1979). doi:10.1016/0003-4843(79)90011-1

    Article  MathSciNet  MATH  Google Scholar 

  17. Miller R.: d-computable categoricity for algebraic fields. J. Symb. Log. 74(4), 1325–1351 (2009). doi:10.2178/jsl/1254748694

    Article  MathSciNet  MATH  Google Scholar 

  18. Miller R., Schoutens H.: Computably categorical fields via Fermat’s last theorem. Computability 2(1), 51–65 (2013)

    MathSciNet  MATH  Google Scholar 

  19. Nurtazin, A.T.: Strong and weak constructivizations, and enumerable families. Algebra i Logika 13, 311–323, 364 (1974)

  20. Prestel A.: Lectures on Formally Real Fields, Lecture Notes in Mathematics, vol. 1093. Springer, Berlin (1984)

    Google Scholar 

  21. Remmel J.B.: Recursively categorical linear orderings. Proc. Am. Math. Soc. 83(2), 387–391 (1981)

    Article  MathSciNet  MATH  Google Scholar 

  22. Soare, R.I.: Recursively Enumerable Sets and Degrees. A Study of Computable Functions and Computably Generated Sets. Perspectives in Mathematical Logic. Springer, Berlin (1987). doi:10.1007/978-3-662-02460-7

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  1. School of Mathematical Sciences, University of Northern Colorado, 501 20th Street, Campus Box 122, Greeley, CO, 80639, USA

    Oscar Levin

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  1. Oscar Levin
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Levin, O. Computable dimension for ordered fields. Arch. Math. Logic 55, 519–534 (2016). https://doi.org/10.1007/s00153-016-0478-7

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  • DOI: https://doi.org/10.1007/s00153-016-0478-7

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