Periodica Mathematica Hungarica

, Volume 73, Issue 1, pp 1–15 | Cite as

A representation of recursively enumerable sets through Horn formulas in higher recursion theory

  • Juan A. Nido Valencia
  • Julio E. Solís Daun
  • Luis M. Villegas SilvaEmail author


We extend a classical result in ordinary recursion theory to higher recursion theory, namely that every recursively enumerable set can be represented in any model \(\mathfrak {A}\) by some Horn theory, where \(\mathfrak {A}\) can be any model of a higher recursion theory, like primitive set recursion, \(\alpha \)-recursion, or \(\beta \)-recursion. We also prove that, under suitable conditions, a set defined through a Horn theory in a set \(\mathfrak {A}\) is recursively enumerable in models of the above mentioned recursion theories.


Horn theory Primitive recursive set functions Recursively enumerable set \(\alpha \)-Recursion theory Primitive recursively closed ordinals Admissible recursion \(\beta \)-Recursion 

Mathematics Subject Classification

Primary 03C55 03C70 03D20 03D60 03D65 03E45 Secondary 03C62 03D75 



This research was partially supported by CONACYT grant 400200-5-32267-E. This work started during the third author’s sabbatical leave at Institut für Mathematik, Humboldt Universität, 10099 Berlin, Germany. We would like to thank the Referee for his/her interest in our work and for his/her helpful comments that will greatly improve the manuscript.


  1. 1.
    J. Barwise, Infinitary logic and admissible sets. J. Symb. Logic 34(2), 226–252 (1969)MathSciNetCrossRefzbMATHGoogle Scholar
  2. 2.
    J. Barwise, Admissible Sets and Structures (Springer, Berlin, 1975)CrossRefzbMATHGoogle Scholar
  3. 3.
    K. Devlin, Constructibility (Springer, Berlin, 1984)CrossRefzbMATHGoogle Scholar
  4. 4.
    K. Doets, From Logic to Logic Programming (MIT Press, Cambridge, 1994)zbMATHGoogle Scholar
  5. 5.
    M. Fitting, Fundamentals of Generalized Recursion Theory (North Holland, Amsterdam, 1981)zbMATHGoogle Scholar
  6. 6.
    M. Fitting, Incompletness in the Land of Sets (College Publications, London, 2007)zbMATHGoogle Scholar
  7. 7.
    S. Friedman, \(\beta \)-Recursion theory. Trans. Am. Math. Soc. 255, 173–200 (1979)MathSciNetzbMATHGoogle Scholar
  8. 8.
    J. Hamkins, A. Lewis, Infinite time turing machines. J. Symb. Logic 65(2), 567–604 (2000)MathSciNetCrossRefzbMATHGoogle Scholar
  9. 9.
    W. Hodges, Logical Features of Horn Clauses, in Handbook of Logic in Artificial Intelligence and Logic Programming, ed. by D. Gabbay, C. Hogger, J. Robinson (Clarendon Press, Oxford, 1993), pp. 449–503Google Scholar
  10. 10.
    R.B. Jensen, C. Karp, Primitive Recursive Set Functions, in In Axiomatic Set Theory: Proc. Symp. Pure Math, ed. by D. Scott (American Mathematical Society, Providence, 1971), pp. 143–167CrossRefGoogle Scholar
  11. 11.
    R.B. Jensen, The fine structure of the constructible hierarchy. Ann. Math. Logic 4, 229–308 (1972)MathSciNetCrossRefzbMATHGoogle Scholar
  12. 12.
    Jensen, R. B. Subcomplete Forcing and \(L\) -forcing, Lectures given at the All Summer Scholl in Singapore (2012). Available in
  13. 13.
    Koepke, P. \(\alpha \)-Recursion Theory and Ordinal Computability, preprint. Available in
  14. 14.
    P. Koepke, B. Seyffert, Ordinal machines and admissible recursion theory. Ann. Pure Appl. Logic 160(3), 310–318 (2009)MathSciNetCrossRefzbMATHGoogle Scholar
  15. 15.
    K. Kunen, The Fundations of Mathematics (College Publications, London, 2009)Google Scholar
  16. 16.
    A. Nerode, R. Shore, Logic for Applications, 2nd edn. (Springer, Berlin, 1997)CrossRefzbMATHGoogle Scholar
  17. 17.
    G. Sacks, Higher Recursion Theory (Springer, Berlin, 1993)zbMATHGoogle Scholar
  18. 18.
    R. Schindler, M. Zeman, Fine Structure, in In Handbook of Set Theory, ed. by M. Foreman, A. Kanamori (Springer, Berlin, 2010)Google Scholar
  19. 19.
    R. Smullyan, Theory of Formal Systems, Rev. edn. (Princeton University Press, Princeton, 1961)CrossRefzbMATHGoogle Scholar
  20. 20.
    R.I. Soare, Recursively Enumerable Sets and Degrees (Springer, Berlin, 1987)CrossRefzbMATHGoogle Scholar

Copyright information

© Akadémiai Kiadó, Budapest, Hungary 2016

Authors and Affiliations

  • Juan A. Nido Valencia
    • 1
  • Julio E. Solís Daun
    • 2
  • Luis M. Villegas Silva
    • 2
    Email author
  1. 1.Posgrado en Ciencias de la ComplejidadUniversidad Autónoma de la Ciudad de MéxicoBenito JuarezMexico
  2. 2.Departamento de MatemáticasUniversidad Autónoma Metropolitana-IztapalapaIztapalapaMexico

Personalised recommendations